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Forces and motion - HackMD
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<div id="doc" class="markdown-body container-fluid"><h1 id="forces-and-motion"><a class="anchor hidden-xs" href="#forces-and-motion" title="forces-and-motion"><span class="octicon octicon-link"></span></a>Forces and motion</h1><h2 id="constant-acceleration-equations"><a class="anchor hidden-xs" href="#constant-acceleration-equations" title="constant-acceleration-equations"><span class="octicon octicon-link"></span></a>Constant acceleration equations</h2><p>The constant acceleration equations use the following variables</p><ul>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-1-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2"><span class="mi" id="MathJax-Span-3" style="font-family: STIXGeneral; font-style: italic;">v</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math></span></span><script type="math/tex" id="MathJax-Element-1">v</script></span> Final velocity</li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-2-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-4" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-5"><span class="mi" id="MathJax-Span-6" style="font-family: STIXGeneral; font-style: italic;">u</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math></span></span><script type="math/tex" id="MathJax-Element-2">u</script></span> Initial velocity</li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-3-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-7" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-8"><span class="mi" id="MathJax-Span-9" style="font-family: STIXGeneral; font-style: italic;">a</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math></span></span><script type="math/tex" id="MathJax-Element-3">a</script></span> Acceleration</li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-4-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-10" style="width: 0.31em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.259em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.847em 1000.26em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-11"><span class="mi" id="MathJax-Span-12" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.816em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></span></span><script type="math/tex" id="MathJax-Element-4">t</script></span> Time for the change in velocity</li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-5-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-13" style="width: 0.515em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-14"><span class="mi" id="MathJax-Span-15" style="font-family: STIXGeneral; font-style: italic;">s</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math></span></span><script type="math/tex" id="MathJax-Element-5">s</script></span> Displacement during the velocity change</li>
</ul><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-6-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mi>v</mi></mtd><mtd><mi></mi><mo>=</mo><mi>u</mi><mo>+</mo><mi>a</mi><mi>t</mi></mtd></mtr><mtr><mtd><msup><mi>v</mi><mn>2</mn></msup></mtd><mtd><mi></mi><mo>=</mo><msup><mi>u</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>s</mi></mtd></mtr><mtr><mtd><mi>s</mi></mtd><mtd><mi></mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>u</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mi>t</mi></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mi>u</mi><mi>t</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mi>v</mi><mi>t</mi><mo>&#x2212;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-16" style="width: 8.097em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.611em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-2.969em 1006.46em 6.867em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-17"><span class="mtable" id="MathJax-Span-18" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 6.304em; height: 0px;"><span style="position: absolute; clip: rect(4.716em 1000.87em 13.527em -999.997em); top: -9.526em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -8.245em; right: 0em;"><span class="mtd" id="MathJax-Span-19"><span class="mrow" id="MathJax-Span-20"><span class="mi" id="MathJax-Span-21" style="font-family: STIXGeneral; font-style: italic;">v</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.974em 1000.87em 4.152em -999.997em); top: -6.811em; right: 0em;"><span class="mtd" id="MathJax-Span-30"><span class="mrow" id="MathJax-Span-31"><span class="msubsup" id="MathJax-Span-32"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-33" style="font-family: STIXGeneral; font-style: italic;">v</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.464em;"><span class="mn" id="MathJax-Span-34" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -4.915em; right: 0em;"><span class="mtd" id="MathJax-Span-46"><span class="mrow" id="MathJax-Span-47"><span class="mi" id="MathJax-Span-48" style="font-family: STIXGeneral; font-style: italic;">s</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -2.508em; right: 0em;"><span class="mtd" id="MathJax-Span-62"><span class="mrow" id="MathJax-Span-63"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -0.151em; right: 0em;"><span class="mtd" id="MathJax-Span-78"><span class="mrow" id="MathJax-Span-79"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 9.531em;"></span></span><span style="position: absolute; clip: rect(5.382em 1005.43em 14.962em -999.997em); top: -10.295em; left: 0.873em;"><span style="display: inline-block; position: relative; width: 5.433em; height: 0px;"><span style="position: absolute; clip: rect(3.281em 1003.79em 4.203em -999.997em); top: -8.245em; left: 0em;"><span class="mtd" id="MathJax-Span-22"><span class="mrow" id="MathJax-Span-23"><span class="mi" id="MathJax-Span-24"></span><span class="mo" id="MathJax-Span-25" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-26" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">u</span><span class="mo" id="MathJax-Span-27" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-28" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">a</span><span class="mi" id="MathJax-Span-29" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.974em 1004.82em 4.203em -999.997em); top: -6.811em; left: 0em;"><span class="mtd" id="MathJax-Span-35"><span class="mrow" id="MathJax-Span-36"><span class="mi" id="MathJax-Span-37"></span><span class="mo" id="MathJax-Span-38" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-39" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-40" style="font-family: STIXGeneral; font-style: italic;">u</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.515em;"><span class="mn" id="MathJax-Span-41" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-42" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mn" id="MathJax-Span-43" style="font-family: STIXGeneral; padding-left: 0.259em;">2</span><span class="mi" id="MathJax-Span-44" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-45" style="font-family: STIXGeneral; font-style: italic;">s</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.513em 1005.28em 4.818em -999.997em); top: -4.915em; left: 0em;"><span class="mtd" id="MathJax-Span-49"><span class="mrow" id="MathJax-Span-50"><span class="mi" id="MathJax-Span-51"></span><span class="mo" id="MathJax-Span-52" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-53" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-54" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-55" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-56" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-57" style="font-family: STIXGeneral; font-style: italic;">u</span><span class="mo" id="MathJax-Span-58" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-59" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">v</span><span class="mo" id="MathJax-Span-60" style="font-family: STIXGeneral;">)</span><span class="mi" id="MathJax-Span-61" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.513em 1005.43em 4.818em -999.997em); top: -2.508em; left: 0em;"><span class="mtd" id="MathJax-Span-64"><span class="mrow" id="MathJax-Span-65"><span class="mi" id="MathJax-Span-66"></span><span class="mo" id="MathJax-Span-67" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-68" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">u</span><span class="mi" id="MathJax-Span-69" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-70" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mfrac" id="MathJax-Span-71" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-72" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-73" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-74" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="msubsup" id="MathJax-Span-75"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.281em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-76" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="mn" id="MathJax-Span-77" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.513em 1005.38em 4.818em -999.997em); top: -0.151em; left: 0em;"><span class="mtd" id="MathJax-Span-80"><span class="mrow" id="MathJax-Span-81"><span class="mi" id="MathJax-Span-82"></span><span class="mo" id="MathJax-Span-83" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-84" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">v</span><span class="mi" id="MathJax-Span-85" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-86" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mfrac" id="MathJax-Span-87" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-88" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-89" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-90" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="msubsup" id="MathJax-Span-91"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.281em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-92" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.361em;"><span class="mn" id="MathJax-Span-93" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 10.3em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -5.559em; border-left: 0px solid; width: 0px; height: 11.816em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mi>v</mi></mtd><mtd><mi></mi><mo>=</mo><mi>u</mi><mo>+</mo><mi>a</mi><mi>t</mi></mtd></mtr><mtr><mtd><msup><mi>v</mi><mn>2</mn></msup></mtd><mtd><mi></mi><mo>=</mo><msup><mi>u</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>s</mi></mtd></mtr><mtr><mtd><mi>s</mi></mtd><mtd><mi></mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>u</mi><mo>+</mo><mi>v</mi><mo stretchy="false">)</mo><mi>t</mi></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mi>u</mi><mi>t</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mi>v</mi><mi>t</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-6">\begin{align} v &= u + at \\
v^2&= u^2 + 2as \\ s &= \frac{1}{2}(u + v)t \\&= ut + \frac{1}{2}at^2 \\&= vt - \frac{1}{2}at^2 \end{align}</script></span></p><p>The final equation is not required for this course.</p><h2 id="free-fall-and-projectile-motion"><a class="anchor hidden-xs" href="#free-fall-and-projectile-motion" title="free-fall-and-projectile-motion"><span class="octicon octicon-link"></span></a>Free fall and projectile motion</h2><p><strong>Free fall</strong> The acceleration a body feels when falling in the presence of a gravitational field</p><p>Objects of different masses fall at the same rate under the influence of gravity, this is because the acceleration of a body is directly proportional to the net force, but inversely proportional to its mass.</p><p><strong>Trajectory</strong> The path of any moving object</p><p><strong>Projectile</strong> An object that is given an initial force, and then allowed to move freely through space</p><p><strong>Parabola</strong> The name given to the shape of a curve which a projectile follows when gravity is the only force acting upon it</p><p>In the real world air resistance also acts on projectiles, however this is ignored in order to allow the problems to be solved using the constant acceleration equations.</p><p>To solve a problem, the components of motion are split into two parts, horizontal and vertical.</p><h2 id="measurement-of-g"><a class="anchor hidden-xs" href="#measurement-of-g" title="measurement-of-g"><span class="octicon octicon-link"></span></a>Measurement of g</h2><p><strong>Light gate</strong> A device which measures the time between a beam of light being broken and being restored.</p><p><strong>Motion sensor</strong> Record displacement at regular time intervals</p><p>Both of these devices can be used to determine an experimental value for <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-7-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-94" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-95"><span class="mi" id="MathJax-Span-96" style="font-family: STIXGeneral; font-style: italic;">g</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></span></span><script type="math/tex" id="MathJax-Element-7">g</script></span>.</p><p>Measurements can either be direct or indirect</p><ul>
<li>Direct measurement is measurement which does not require manipulation, such as time a falling object</li>
<li>Indirect measurement requires manipulation, such as measuring the period of pendulum and rearranging to find <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-8-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-97" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-98"><span class="mi" id="MathJax-Span-99" style="font-family: STIXGeneral; font-style: italic;">g</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></span></span><script type="math/tex" id="MathJax-Element-8">g</script></span></li>
</ul><p>When explaining an investigation, state the following</p><ul>
<li>The measurements to be taken</li>
<li>The equipment with which to take the measurements</li>
<li>The variables required to determine <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-100" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-101"><span class="mi" id="MathJax-Span-102" style="font-family: STIXGeneral; font-style: italic;">g</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></span></span><script type="math/tex" id="MathJax-Element-9">g</script></span></li>
<li>The sources of uncertainty in the measurements</li>
<li>The possible methods to minimise uncertainty</li>
</ul><h3 id="investigation-use-of-a-trap-door-and-an-electromagnet-to-determine-g"><a class="anchor hidden-xs" href="#investigation-use-of-a-trap-door-and-an-electromagnet-to-determine-g" title="investigation-use-of-a-trap-door-and-an-electromagnet-to-determine-g"><span class="octicon octicon-link"></span></a>Investigation- Use of a trap door and an electromagnet to determine <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-286-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2286" style="width: 0.617em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.494em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.928em 1000.45em 2.83em -999.998em); top: -2.498em; left: 0em;"><span class="mrow" id="MathJax-Span-2287"><span class="mi" id="MathJax-Span-2288" style="font-family: STIXGeneral; font-style: italic;">g</span></span><span style="display: inline-block; width: 0px; height: 2.502em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.297em; border-left: 0px solid; width: 0px; height: 0.903em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></span></span><script type="math/tex" id="MathJax-Element-286">g</script></span></h3><p><strong>Equipment</strong><br>
An electromagnet supports a steel ball.<br>
The electromagnet is on the same circuit as a timer.</p><p><strong>Procedure</strong></p><p>When a switch is toggled the ball begins to fall a known distance, and the timer on the same circuit begins.</p><p>The ball falls onto a trap door, stopping the timer.</p><p><strong>Calculation</strong></p><p>The variables are the distance fallen, s, and the time taken, t.</p><p>Using the constant acceleration acceleration <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-11-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><msup><mi>t</mi><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-106" style="width: 4.46em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.64em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.64em 3.076em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-107"><span class="mi" id="MathJax-Span-108" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mo" id="MathJax-Span-109" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-110" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-111" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-112" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-113" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="msubsup" id="MathJax-Span-114"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.281em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-115" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.361em;"><span class="mn" id="MathJax-Span-116" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><msup><mi>t</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-11">s = \frac{1}{2} at^2</script></span></p><p>For any values of s and t we can then find a value for acceleration due to gravity.</p><p>Alternatively we could use the inferior and inaccurate method of drawing a graph of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-12-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-117" style="width: 0.515em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-118"><span class="mi" id="MathJax-Span-119" style="font-family: STIXGeneral; font-style: italic;">s</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math></span></span><script type="math/tex" id="MathJax-Element-12">s</script></span> against <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-13-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>t</mi><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-120" style="width: 0.976em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em 1000.77em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-121"><span class="msubsup" id="MathJax-Span-122"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.281em 1000.31em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-123" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.361em;"><span class="mn" id="MathJax-Span-124" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>t</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-13">t^2</script></span> which would have a gradient of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-14-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mn>2</mn></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-125" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.335em 1000.67em 2.718em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-126"><span class="mfrac" id="MathJax-Span-127"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.537em 1000.36em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-128" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-129" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-14">\frac{a}{2}</script></span>.</p><p><strong>Methods to increase accuracy and reduce error</strong></p><p>In order to increase the accuracy of the experiment we could take repeat readings to find a mean value for <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-15-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-130" style="width: 0.31em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.259em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.847em 1000.26em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-131"><span class="mi" id="MathJax-Span-132" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.816em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></span></span><script type="math/tex" id="MathJax-Element-15">t</script></span>, and measure t across different heights.</p><h2 id="car-stopping-distances"><a class="anchor hidden-xs" href="#car-stopping-distances" title="car-stopping-distances"><span class="octicon octicon-link"></span></a>Car stopping distances</h2><p><strong>Stopping distance</strong> The total distance that a vehicle travels before it stops</p><p><strong>Thinking distance</strong> The distance a vehicle travels between the operator seeing the hazard and applying the brakes</p><p><strong>Braking distance</strong> The distance that a vehicle travels between its brakes being applied and it coming to rest</p><table>
<thead>
<tr>
<th>Factors affecting thinking distance</th>
<th>Factors affecting braking distance</th>
</tr>
</thead>
<tbody>
<tr>
<td>Reflexes</td>
<td>Poor road conditions</td>
</tr>
<tr>
<td>Drugs</td>
<td>Road surface</td>
</tr>
<tr>
<td>Tiredness</td>
<td>Problems with the vehicle</td>
</tr>
<tr>
<td>Distractions</td>
<td>Mass of the vehicle</td>
</tr>
<tr>
<td>Speed of the vehicle</td>
<td></td>
</tr>
</tbody>
</table><h2 id="force-and-the-newton"><a class="anchor hidden-xs" href="#force-and-the-newton" title="force-and-the-newton"><span class="octicon octicon-link"></span></a>Force and the Newton</h2><p><strong>Resultant force</strong> A single force which has the same effect as the sum of all forces acting on a body<br>
<strong>Thrust</strong> A term used for the driving force provided by a jet engine<br>
<strong>Drag</strong> A force which acts in the opposite direction to motion, impeding it. Also called the dissipative force</p><p>Types of forces</p><ul>
<li>Contact- Two bodies touch when their repulsive molecular forces equal the force that is trying to bring them together</li>
<li>Friction- When two bodies are in contact their attractive molecular forces try to prevent their common surfaces moving relative to each other</li>
<li>Tension- The force exerted by a body when it is stretched. It is due to attractive molecular forces</li>
<li>Compression- The force exerted by a body when it is compressed. It is due to repulsive molecular forces</li>
<li>Fluid upthrust- The force exerted by a fluid on a body because of the weight of the fluid that has been displaced by the body. Archimedes’ principle states that the upthrust force is equal to the weight of fluid displaced.</li>
<li>Electrostatic- Attractive and repulsive forces due to bodies being charged</li>
<li>Magnetic- Attractive and repulsive forces due to moving electric charges</li>
<li>Electromagnetic- Attractive and repulsive forces due to bodies being charged. Contact, friction, tension, compression, fluid upthrust, electrostatic, and magnetic forces are all forms of electromagnetic foce</li>
<li>Weak nuclear- The force responsible for nuclear decay</li>
<li>Electro-weak- It is thought that both the electromagnetic and weak nuclear forces are both forms of this fundamental force</li>
<li>Strong nuclear- This fundamental force holds protons and neutrons together in the nucles</li>
<li>Gravitational- The fundamental force exerted on a body due to its mass</li>
</ul><h2 id="dynamics"><a class="anchor hidden-xs" href="#dynamics" title="dynamics"><span class="octicon octicon-link"></span></a>Dynamics</h2><p><strong>Dynamics</strong> A branch of mechanics concerned with the motion of bodies under the action of forces</p><p>While dynamics is the study of the motion of bodies and the forces that cause the movement, kinematics is the study of movement without any reference to external forces.</p><p>The normal contact force, or reaction force, acts perpendicular to the point of contact against a surface.</p><p>When two surfaces move next to each other, each surface is slightly deformed, meaning that the bonds between the atoms are stretched and compressed.</p><p>Friction occurs within all fluids.</p><h3 id="free-body-diagrams"><a class="anchor hidden-xs" href="#free-body-diagrams" title="free-body-diagrams"><span class="octicon octicon-link"></span></a>Free body diagrams</h3><p>Free body diagrams help to determine the net force on a body</p><p>They should always contain</p><ul>
<li>The body or point upon which the forces are acting</li>
<li>The direction in which each force acts</li>
<li>The type of each force</li>
<li>The size of each force</li>
</ul><p>The diagram below shows the free body diagram for an object on a rough inclined plane</p><p><img src="https://i.stack.imgur.com/lMCD8.png" alt=""></p><h2 id="drag-and-terminal-velocity"><a class="anchor hidden-xs" href="#drag-and-terminal-velocity" title="drag-and-terminal-velocity"><span class="octicon octicon-link"></span></a>Drag and terminal velocity</h2><p><strong>Fluid</strong> A substance which has no fixed shape, and yields easily to external pressure</p><p>Any object which moves through a fluid experiences a dissipative drag force.<br>
The drag force is called aerodynamic drag when the fluid is a gas, and hydrodynamic drag when the fluid is a liquid.</p><p>The frictional force on a body moving through a fluid is given by</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-16-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>d</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>&#x03C1;</mi><msub><mi>C</mi><mi>d</mi></msub><mi>A</mi><msup><mi>V</mi><mn>2</mn></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-133" style="width: 8.097em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.611em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1006.61em 3.384em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-134"><span class="msubsup" id="MathJax-Span-135"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-136" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-137" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-138" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-139" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-140" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-141" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-142" style="font-family: STIXGeneral; font-style: italic;">ρ</span><span class="msubsup" id="MathJax-Span-143"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-144" style="font-family: STIXGeneral; font-style: italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.669em;"><span class="mi" id="MathJax-Span-145" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-146" style="font-family: STIXGeneral; font-style: italic;">A</span><span class="msubsup" id="MathJax-Span-147"><span style="display: inline-block; position: relative; width: 1.181em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-148" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.771em;"><span class="mn" id="MathJax-Span-149" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.872em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>F</mi><mi>d</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>ρ</mi><msub><mi>C</mi><mi>d</mi></msub><mi>A</mi><msup><mi>V</mi><mn>2</mn></msup></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-16">F_d = \frac{1}{2} \rho C_d AV^2</script></span></p><p>Where<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-17-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>d</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-150" style="width: 1.335em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.08em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-151"><span class="msubsup" id="MathJax-Span-152"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-153" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-154" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>d</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-17">F_d</script></span> is the drag force <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-18-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>N</mtext><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-155" style="width: 1.693em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.386em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.34em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-156"><span class="mo" id="MathJax-Span-157" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-158" style="font-family: STIXGeneral;">N</span><span class="mo" id="MathJax-Span-159" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>N</mtext><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-18">(\text{N})</script></span><br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-19-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C1;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-160" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-161"><span class="mi" id="MathJax-Span-162" style="font-family: STIXGeneral; font-style: italic;">ρ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi></math></span></span><script type="math/tex" id="MathJax-Element-19">\rho</script></span> is the density of the fluid <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-20-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>kg m</mtext><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>3</mn></mrow></msup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-163" style="width: 4.408em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.589em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.54em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-164"><span class="mo" id="MathJax-Span-165" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-166"><span style="display: inline-block; position: relative; width: 2.923em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1002.05em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mtext" id="MathJax-Span-167" style="font-family: STIXGeneral;">kg m</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 2.052em;"><span class="texatom" id="MathJax-Span-168"><span class="mrow" id="MathJax-Span-169"><span class="mo" id="MathJax-Span-170" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-171" style="font-size: 70.7%; font-family: STIXGeneral;">3</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-172" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>kg m</mtext><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>3</mn></mrow></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-20">(\text{kg m}^{-3})</script></span><br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-21-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>&#x2212;</mo><mi>d</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-173" style="width: 2.974em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.41em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.41em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-174"><span class="mi" id="MathJax-Span-175" style="font-family: STIXGeneral; font-style: italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-176" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-177" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>−</mo><mi>d</mi></math></span></span><script type="math/tex" id="MathJax-Element-21">C-d</script></span> is the coefficient of drag<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-22-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-178" style="width: 0.771em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.57em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-179"><span class="mi" id="MathJax-Span-180" style="font-family: STIXGeneral; font-style: italic;">A</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-22">A</script></span> is the cross sectional area of the object <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-23-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>m</mtext><mn>2</mn></msup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-181" style="width: 2.308em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1001.85em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-182"><span class="mo" id="MathJax-Span-183" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-184"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mtext" id="MathJax-Span-185" style="font-family: STIXGeneral;">m</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.771em;"><span class="mn" id="MathJax-Span-186" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-187" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>m</mtext><mn>2</mn></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-23">(\text{m}^2)</script></span><br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-24-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-188" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-189"><span class="mi" id="MathJax-Span-190" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math></span></span><script type="math/tex" id="MathJax-Element-24">V</script></span> is the velocity of the object <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-25-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>m s</mtext><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-191" style="width: 3.64em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.974em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1002.92em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-192"><span class="mo" id="MathJax-Span-193" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-194"><span style="display: inline-block; position: relative; width: 2.308em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1001.39em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mtext" id="MathJax-Span-195" style="font-family: STIXGeneral;">m s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 1.437em;"><span class="texatom" id="MathJax-Span-196"><span class="mrow" id="MathJax-Span-197"><span class="mo" id="MathJax-Span-198" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-199" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-200" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>m s</mtext><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-25">(\text{m s}^{-1})</script></span></p><p>For an object of constant size, moving through a fluid of uniform density the drag force, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-26-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>d</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-201" style="width: 1.335em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.08em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-202"><span class="msubsup" id="MathJax-Span-203"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-204" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-205" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>d</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-26">F_d</script></span>, is proportional the the square of the velocity, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-27-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>V</mi><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-206" style="width: 1.437em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.181em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em 1001.18em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-207"><span class="msubsup" id="MathJax-Span-208"><span style="display: inline-block; position: relative; width: 1.181em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-209" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.771em;"><span class="mn" id="MathJax-Span-210" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>V</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-27">V^2</script></span>.</p><h3 id="terminal-velocity"><a class="anchor hidden-xs" href="#terminal-velocity" title="terminal-velocity"><span class="octicon octicon-link"></span></a>Terminal velocity</h3><p>When an object falls from a height, it initially accelerates because the force of gravity is greater than the resistive force.<br>
As it accelerates, its velocity increases, and as such the drag force increases proportional to the square of the velocity (<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-28-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>d</mi></msub><mo>&#x221D;</mo><msup><mi>V</mi><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-211" style="width: 4.306em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.537em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1003.54em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-212"><span class="msubsup" id="MathJax-Span-213"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-214" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-215" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-216" style="font-family: STIXGeneral; padding-left: 0.31em;">∝</span><span class="msubsup" id="MathJax-Span-217" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.181em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-218" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.771em;"><span class="mn" id="MathJax-Span-219" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>F</mi><mi>d</mi></msub><mo>∝</mo><msup><mi>V</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-28">F_d \propto V^2</script></span>). The increase in drag force results in a decrease in acceleration.</p><p>When the drag force is equal to the force of gravity, the object is at terminal velocity.</p><h2 id="equilibrium"><a class="anchor hidden-xs" href="#equilibrium" title="equilibrium"><span class="octicon octicon-link"></span></a>Equilibrium</h2><p><strong>Equilibrium</strong> If the resultant force acting on an object is 0, the object is said to be in equilibrium</p><p>The resultant force can be found by splitting each of the forces acting on a body into their respective horizontal and vertical components, before finding the resultant in each axis.</p><h2 id="turning-forces"><a class="anchor hidden-xs" href="#turning-forces" title="turning-forces"><span class="octicon octicon-link"></span></a>Turning forces</h2><p>The turning effect of a force is called a moment</p><p><strong>Couple</strong> Two equal antiparallel forces acting to produce rotation with no linear motion<br>
<strong>Torque</strong> The torque of a couple is defined as the moment or turning effect produced by a couple</p><p>The moment is the force multiplied by the perpendicular distance from the pivot around which it is acting</p><h3 id="principle-of-moments"><a class="anchor hidden-xs" href="#principle-of-moments" title="principle-of-moments"><span class="octicon octicon-link"></span></a>Principle of moments</h3><p>The principle of moments states that when in equilibrium the sum of the clockwise moments is equal to the sum of the anticlockwise moments.</p><p>The resultant force must also be 0 to ensure that there is no translational acceleration.</p><h2 id="centre-of-mass"><a class="anchor hidden-xs" href="#centre-of-mass" title="centre-of-mass"><span class="octicon octicon-link"></span></a>Centre of mass</h2><p><strong>Centre of mass</strong> The centre of mass of an object is the single point at which all of an objects mass can be assumed to be situated. For a symmetrical body of uniform density it is always situated at the centre of the object.</p><h3 id="method-for-determining-the-mass-of-irregular-shapes"><a class="anchor hidden-xs" href="#method-for-determining-the-mass-of-irregular-shapes" title="method-for-determining-the-mass-of-irregular-shapes"><span class="octicon octicon-link"></span></a>Method for determining the mass of irregular shapes</h3><ul>
<li>Make two holes in the objects</li>
<li>Hang the object from each of the holes and draw a line from the hole parallel to the ground</li>
<li>The centre of mass is the point at which the lines cross</li>
</ul><p><strong>Centre of gravity</strong> The centre of gravity is the single point through which the entire weight of the object can be thought to act</p><p>The stability of objects are determined by the position of their centres of mass and their base area.<br>
Objects with a low centre of mass and wide base area are more stable.</p><h2 id="density"><a class="anchor hidden-xs" href="#density" title="density"><span class="octicon octicon-link"></span></a>Density</h2><p><strong>Density</strong> The mass per unit volume of a substance</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-29-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>&#x03C1;</mi><mo>=</mo><mfrac><mi>m</mi><mi>v</mi></mfrac><mspace width="1em" /><mo stretchy="false">(</mo><msup><mtext>kg m</mtext><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>3</mn></mrow></msup><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-220" style="width: 9.224em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.533em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.283em 1007.48em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-221"><span class="mi" id="MathJax-Span-222" style="font-family: STIXGeneral; font-style: italic;">ρ</span><span class="mo" id="MathJax-Span-223" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-224" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.72em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.356em;"><span class="mi" id="MathJax-Span-225" style="font-family: STIXGeneral; font-style: italic;">m</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-226" style="font-family: STIXGeneral; font-style: italic;">v</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.82em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.822em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mspace" id="MathJax-Span-227" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-228" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-229"><span style="display: inline-block; position: relative; width: 2.923em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1002.05em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mtext" id="MathJax-Span-230" style="font-family: STIXGeneral;">kg m</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 2.052em;"><span class="texatom" id="MathJax-Span-231"><span class="mrow" id="MathJax-Span-232"><span class="mo" id="MathJax-Span-233" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-234" style="font-size: 70.7%; font-family: STIXGeneral;">3</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-235" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.378em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>ρ</mi><mo>=</mo><mfrac><mi>m</mi><mi>v</mi></mfrac><mspace width="1em"></mspace><mo stretchy="false">(</mo><msup><mtext>kg m</mtext><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>3</mn></mrow></msup><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-29">\rho = \frac{m}{v} \quad (\text{kg m}^{-3})</script></span></p><h2 id="pressure"><a class="anchor hidden-xs" href="#pressure" title="pressure"><span class="octicon octicon-link"></span></a>Pressure</h2><p><strong>Pressure</strong> The force per unit area</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-30-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo>=</mo><mfrac><mi>F</mi><mi>A</mi></mfrac><mspace width="1em" /><mrow><mo>(</mo><mfrac><mi>N</mi><msup><mi>m</mi><mn>2</mn></msup></mfrac><mo>)</mo></mrow><mtext>(Pa</mtext><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-236" style="width: 10.658em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.712em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.308em 1008.66em 4.716em -999.997em); top: -3.788em; left: 0em;"><span class="mrow" id="MathJax-Span-237"><span class="mi" id="MathJax-Span-238" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-239" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-240" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.305em;"><span class="mi" id="MathJax-Span-241" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.305em;"><span class="mi" id="MathJax-Span-242" style="font-family: STIXGeneral; font-style: italic;">A</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.77em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.771em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mspace" id="MathJax-Span-243" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mrow" id="MathJax-Span-244"><span class="mo" id="MathJax-Span-245" style="vertical-align: -0.459em;"><span><span style="font-size: 111%; font-family: STIXSizeTwoSym;">(</span></span></span><span class="mfrac" id="MathJax-Span-246"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.72em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.356em;"><span class="mi" id="MathJax-Span-247" style="font-family: STIXGeneral; font-style: italic;">N<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.076em 1001.13em 4.152em -999.997em); top: -3.276em; left: 50%; margin-left: -0.561em;"><span class="msubsup" id="MathJax-Span-248"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.72em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-249" style="font-family: STIXGeneral; font-style: italic;">m</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.301em; left: 0.72em;"><span class="mn" id="MathJax-Span-250" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.28em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.283em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-251" style="vertical-align: -0.459em;"><span><span style="font-size: 111%; font-family: STIXSizeTwoSym;">)</span></span></span></span><span class="mtext" id="MathJax-Span-252" style="font-family: STIXGeneral; padding-left: 0.207em;">(Pa</span><span class="mo" id="MathJax-Span-253" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.794em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.997em; border-left: 0px solid; width: 0px; height: 2.691em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo>=</mo><mfrac><mi>F</mi><mi>A</mi></mfrac><mspace width="1em"></mspace><mrow><mo>(</mo><mfrac><mi>N</mi><msup><mi>m</mi><mn>2</mn></msup></mfrac><mo>)</mo></mrow><mtext>(Pa</mtext><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-30">P = \frac{F}{A} \quad \left(\frac{N}{m^2}\right) \text{(Pa}) </script></span></p><p>For hydrostatic pressure</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-31-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo>=</mo><mi>h</mi><mi>&#x03C1;</mi><mi>g</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-254" style="width: 4.203em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.435em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.38em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-255"><span class="mi" id="MathJax-Span-256" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-257" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-258" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">h</span><span class="mi" id="MathJax-Span-259" style="font-family: STIXGeneral; font-style: italic;">ρ</span><span class="mi" id="MathJax-Span-260" style="font-family: STIXGeneral; font-style: italic;">g</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo>=</mo><mi>h</mi><mi>ρ</mi><mi>g</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-31">P = h\rho g</script></span><br>
Where<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-32-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-261" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-262"><span class="mi" id="MathJax-Span-263" style="font-family: STIXGeneral; font-style: italic;">h</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></span></span><script type="math/tex" id="MathJax-Element-32">h</script></span> is the depth of the fluid <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-33-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>m</mtext><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-264" style="width: 1.744em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.437em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.39em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-265"><span class="mo" id="MathJax-Span-266" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-267" style="font-family: STIXGeneral;">m</span><span class="mo" id="MathJax-Span-268" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>m</mtext><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-33">(\text{m})</script></span><br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-34-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C1;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-269" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-270"><span class="mi" id="MathJax-Span-271" style="font-family: STIXGeneral; font-style: italic;">ρ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi></math></span></span><script type="math/tex" id="MathJax-Element-34">\rho</script></span> is the density of the fluid <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-35-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>kg m</mtext><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>3</mn></mrow></msup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-272" style="width: 4.408em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.589em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.54em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-273"><span class="mo" id="MathJax-Span-274" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-275"><span style="display: inline-block; position: relative; width: 2.923em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1002.05em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mtext" id="MathJax-Span-276" style="font-family: STIXGeneral;">kg m</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 2.052em;"><span class="texatom" id="MathJax-Span-277"><span class="mrow" id="MathJax-Span-278"><span class="mo" id="MathJax-Span-279" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-280" style="font-size: 70.7%; font-family: STIXGeneral;">3</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-281" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>kg m</mtext><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>3</mn></mrow></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-35">(\text{kg m}^{-3})</script></span><br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-36-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-282" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-283"><span class="mi" id="MathJax-Span-284" style="font-family: STIXGeneral; font-style: italic;">g</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></span></span><script type="math/tex" id="MathJax-Element-36">g</script></span> is acceleration due to gravity <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-37-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>m s</mtext><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-285" style="width: 3.64em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.974em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1002.92em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-286"><span class="mo" id="MathJax-Span-287" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-288"><span style="display: inline-block; position: relative; width: 2.308em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1001.39em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mtext" id="MathJax-Span-289" style="font-family: STIXGeneral;">m s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 1.437em;"><span class="texatom" id="MathJax-Span-290"><span class="mrow" id="MathJax-Span-291"><span class="mo" id="MathJax-Span-292" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-293" style="font-size: 70.7%; font-family: STIXGeneral;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-294" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>m s</mtext><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-37">(\text{m s}^{-2})</script></span></p><h2 id="work-and-the-joule"><a class="anchor hidden-xs" href="#work-and-the-joule" title="work-and-the-joule"><span class="octicon octicon-link"></span></a>Work and the joule</h2><p><strong>Work</strong> The work done is the product of the force and the distance moved by the force in the direction of movement</p><p>One joule is defined as the energy exerted when a force of one newton is applied over a displacement of one metre.</p><p>The area under the curve of a force-distance graph is equal to the work done</p><h2 id="the-conservation-of-energy"><a class="anchor hidden-xs" href="#the-conservation-of-energy" title="the-conservation-of-energy"><span class="octicon octicon-link"></span></a>The conservation of energy</h2><p>Forms of energy</p><ul>
<li>Kinetic energy- Energy possessed by a moving body</li>
<li>Nuclear energy- Energy transferred during a nuclear reaction</li>
<li>Gravitational potential energy- Energy possessed by anything under the influence of a gravitational force</li>
<li>Elastic potential energy- Energy possessed by an object which has been stretched</li>
<li>Chemical energy- Energy stored in the bonds of a molecule</li>
<li>Electrical energy- Energy due to the flow of electrons</li>
<li>Light energy- Energy possessed by electromagnetic radiation</li>
<li>Sound energy- Energy possessed by sound waves</li>
<li>Radiation- Energy transferred by heated object in the infrared spectrum</li>
</ul><p>1st law of thermodynamics<br>
Energy cannot be created or destroyed in an isolated system</p><p>2nd law of thermodynamics<br>
The entropy of any isolated system always increases</p><p>3rd law of thermodynamics<br>
The entropy of a system approaches a constant value as the temperature of the system approaches absolute zero</p><h2 id="potential-and-kinetic-energy"><a class="anchor hidden-xs" href="#potential-and-kinetic-energy" title="potential-and-kinetic-energy"><span class="octicon octicon-link"></span></a>Potential and kinetic energy</h2><p>The kinetic energy, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-38-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>k</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-295" style="width: 1.283em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.03em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-296"><span class="msubsup" id="MathJax-Span-297"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-298" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-299" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>k</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-38">E_k</script></span> possessed by an object in motion is given by</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-39-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>E</mi><mi>k</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-300" style="width: 5.843em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.767em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1004.77em 3.384em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-301"><span class="msubsup" id="MathJax-Span-302"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-303" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-304" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-305" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-306" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-307" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-308" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-309" style="font-family: STIXGeneral; font-style: italic;">m</span><span class="msubsup" id="MathJax-Span-310"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-311" style="font-family: STIXGeneral; font-style: italic;">v</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.464em;"><span class="mn" id="MathJax-Span-312" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.872em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>E</mi><mi>k</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-39">E_k = \frac{1}{2}mv^2 </script></span></p><p>The gravitational potential energy, GPE, stored in an object by virtue of its position in a gravitational field is given by</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-40-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>G</mi><mi>P</mi><mi>E</mi><mo>=</mo><mi>m</mi><mi>g</mi><mi mathvariant="normal">&#x0394;</mi><mi>h</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-313" style="width: 6.97em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.689em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1005.69em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-314"><span class="mi" id="MathJax-Span-315" style="font-family: STIXGeneral; font-style: italic;">G</span><span class="mi" id="MathJax-Span-316" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mi" id="MathJax-Span-317" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-318" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-319" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">m</span><span class="mi" id="MathJax-Span-320" style="font-family: STIXGeneral; font-style: italic;">g</span><span class="mi" id="MathJax-Span-321" style="font-family: STIXGeneral;">Δ</span><span class="mi" id="MathJax-Span-322" style="font-family: STIXGeneral; font-style: italic;">h</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>G</mi><mi>P</mi><mi>E</mi><mo>=</mo><mi>m</mi><mi>g</mi><mi mathvariant="normal">Δ</mi><mi>h</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-40">GPE = mg\Delta h</script></span></p><p>When an object falls under the influence of a gravitational force, its gravitational potential energy is converted into kinetic energy.</p><p>When an object is launched upwards with an initial velocity, and therefore an initial kinetic energy, its kinetic energy is converted to gravitational potential energy until it has 0 velocity, and begins to fall.</p><h2 id="power-and-the-watt"><a class="anchor hidden-xs" href="#power-and-the-watt" title="power-and-the-watt"><span class="octicon octicon-link"></span></a>Power and the watt</h2><p><strong>Power</strong> The rate of work done <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-41-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>W</mtext><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-323" style="width: 1.949em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.591em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.54em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-324"><span class="mo" id="MathJax-Span-325" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-326" style="font-family: STIXGeneral;">W</span><span class="mo" id="MathJax-Span-327" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>W</mtext><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-41">(\text{W})</script></span> <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-42-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>J s</mtext><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-328" style="width: 3.23em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.615em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1002.56em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-329"><span class="mo" id="MathJax-Span-330" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-331"><span style="display: inline-block; position: relative; width: 1.949em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mtext" id="MathJax-Span-332" style="font-family: STIXGeneral;">J s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-333"><span class="mrow" id="MathJax-Span-334"><span class="mo" id="MathJax-Span-335" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-336" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-337" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>J s</mtext><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-42">(\text{J s}^{-1})</script></span></p><p>The power produced by an object moving at speed v under a force F produces a power given by</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-43-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo>=</mo><mi>F</mi><mi>v</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-338" style="width: 3.691em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.025em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.02em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-339"><span class="mi" id="MathJax-Span-340" style="font-family: STIXGeneral; font-style: italic;">P</span><span class="mo" id="MathJax-Span-341" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-342" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-343" style="font-family: STIXGeneral; font-style: italic;">v</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>P</mi><mo>=</mo><mi>F</mi><mi>v</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-43">P = Fv</script></span></p><h2 id="efficiency"><a class="anchor hidden-xs" href="#efficiency" title="efficiency"><span class="octicon octicon-link"></span></a>Efficiency</h2><p><strong>Efficiency</strong> The efficiency of a process is the ratio of useful work performed to the total energy expended</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-44-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>Efficiency</mtext><mo>=</mo><mfrac><mtext>Useful output energy</mtext><mtext>Total input energy</mtext></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-344" style="width: 17.267em; display: inline-block;"><span style="display: inline-block; position: relative; width: 14.142em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1014.14em 3.64em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-345"><span class="mtext" id="MathJax-Span-346" style="font-family: STIXGeneral;">Efficiency</span><span class="mo" id="MathJax-Span-347" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-348" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 8.558em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1008.4em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -4.198em;"><span class="mtext" id="MathJax-Span-349" style="font-family: STIXGeneral;">Useful output energy</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1007.38em 4.357em -999.997em); top: -3.327em; left: 50%; margin-left: -3.686em;"><span class="mtext" id="MathJax-Span-350" style="font-family: STIXGeneral;">Total input energy</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1008.56em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 8.558em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.184em; border-left: 0px solid; width: 0px; height: 2.878em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>Efficiency</mtext><mo>=</mo><mfrac><mtext>Useful output energy</mtext><mtext>Total input energy</mtext></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-44">\text{Efficiency} = \frac{\text{Useful output energy}}{\text{Total input energy}}</script></span></p><h2 id="deformation-of-materials"><a class="anchor hidden-xs" href="#deformation-of-materials" title="deformation-of-materials"><span class="octicon octicon-link"></span></a>Deformation of materials</h2><p><strong>Deformation</strong> The change in shape or size of an object</p><p><strong>Elasticity</strong> The tendency of a body to resume it’s original shape or size once a deforming force or stress has been removed</p><p><strong>Elastic deformation</strong> When a body returns to it’s original shape once a stress, strain, or force is removed</p><p><strong>Plastic deformation</strong> When a body does not return to it’s original shape once a stress, strain, or force is removed</p><h3 id="tensile-forces"><a class="anchor hidden-xs" href="#tensile-forces" title="tensile-forces"><span class="octicon octicon-link"></span></a>Tensile forces</h3><p>Tensile forces cause tension in an object.<br>
If there is tension in a wire, then there must be an equal and opposite force at both ends.<br>
These forces will cause the wire to increase in length, a positive extension.</p><p>Compressive forces act towards each other and have the opposite effect.</p><h3 id="measuring-the-stretch-of-a-wire"><a class="anchor hidden-xs" href="#measuring-the-stretch-of-a-wire" title="measuring-the-stretch-of-a-wire"><span class="octicon octicon-link"></span></a>Measuring the stretch of a wire</h3><ul>
<li>Attach a wire to a clamp</li>
<li>Run the wire horizontally, over the edge of a surface through a pulley</li>
<li>Attach mass to the wire so that it is taught</li>
<li>Place a mark on the wire, and a ruler underneath it inline with the mark</li>
<li>Progressively add more mass, measuring how far the mark moves</li>
</ul><h3 id="elastic-limit"><a class="anchor hidden-xs" href="#elastic-limit" title="elastic-limit"><span class="octicon octicon-link"></span></a>Elastic limit</h3><p><img src="http://www.cyberphysics.co.uk/graphics/graphs/stress_strain2.gif" alt=""></p><p>The important points are</p><ul>
<li>The limit of proportionality</li>
<li>The elastic limit</li>
<li>The fracture point</li>
<li>The plastic deformation region</li>
</ul><h2 id="hooke’s-law"><a class="anchor hidden-xs" href="#hooke’s-law" title="hooke’s-law"><span class="octicon octicon-link"></span></a>Hooke’s law</h2><p><strong>Hooke’s law</strong> The extension of an object is proportional to the force that causes it, provided that the elastic limit is not exceeded</p><p>Hooke’s law states</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-45-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>F</mi><mo>=</mo><mi>k</mi><mi>x</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-351" style="width: 3.537em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.871em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.87em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-352"><span class="mi" id="MathJax-Span-353" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-354" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-355" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-356" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>F</mi><mo>=</mo><mi>k</mi><mi>x</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-45">F = kx</script></span></p><p>The force (N) is equal to the spring constant <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-46-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>N m</mtext><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-357" style="width: 4.101em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.332em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.28em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-358"><span class="mo" id="MathJax-Span-359" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-360"><span style="display: inline-block; position: relative; width: 2.666em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1001.74em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mtext" id="MathJax-Span-361" style="font-family: STIXGeneral;">N m</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.744em;"><span class="texatom" id="MathJax-Span-362"><span class="mrow" id="MathJax-Span-363"><span class="mo" id="MathJax-Span-364" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-365" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-366" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>N m</mtext><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-46">(\text{N m}^{-1})</script></span> multiplied by the extension <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-47-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>m</mtext><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-367" style="width: 1.744em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.437em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.39em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-368"><span class="mo" id="MathJax-Span-369" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-370" style="font-family: STIXGeneral;">m</span><span class="mo" id="MathJax-Span-371" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>m</mtext><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-47">(\text{m})</script></span>.</p><h3 id="springs-in-series-and-combination"><a class="anchor hidden-xs" href="#springs-in-series-and-combination" title="springs-in-series-and-combination"><span class="octicon octicon-link"></span></a>Springs in series and combination</h3><p>When springs are suspended in series, each spring experiences the same pull from the weight of the mass it supports. Therefore each spring extends the same amount as an individual spring would do.</p><p>When springs support a weight by combination, they share the load and therefore are not stretched as much as if they were supporting the weight themselves.</p><h3 id="force-extension-graphs-and-work-done"><a class="anchor hidden-xs" href="#force-extension-graphs-and-work-done" title="force-extension-graphs-and-work-done"><span class="octicon octicon-link"></span></a>Force extension graphs and work done</h3><p>As stated previously, work done = force x distance.<br>
Therefore, the area under a force-extension graph is equal to the work done in applying the extension.<br>
This work done is equal to the elastic potential energy stored within the extended object.</p><p>Prior to the limit of proportionality, the area under the curve forms a triangle</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-48-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>F</mi><mi>x</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-372" style="width: 4.818em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.947em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.95em 3.076em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-373"><span class="mi" id="MathJax-Span-374" style="font-family: STIXGeneral; font-style: italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-375" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-376" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-377" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-378" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-379" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-380" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>F</mi><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-48">W = \frac{1}{2}Fx</script></span></p><p>as <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-49-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mi>k</mi><mi>x</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-381" style="width: 3.537em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.871em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.87em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-382"><span class="mi" id="MathJax-Span-383" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-384" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-385" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-386" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><mo>=</mo><mi>k</mi><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-49">F = kx</script></span>, we also have</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-50-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-387" style="width: 5.228em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.255em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1004.25em 3.076em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-388"><span class="mi" id="MathJax-Span-389" style="font-family: STIXGeneral; font-style: italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-390" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-391" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-392" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-393" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-394" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-395"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-396" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.515em;"><span class="mn" id="MathJax-Span-397" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>k</mi><msup><mi>x</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-50">W = \frac{1}{2}kx^2</script></span></p><p>Usually we plot the independent variable on the x-axis, and the dependent on the y-axis.<br>
Force extension graphs break this convention.</p><p>If we plot extension on the x, the gradient is then the spring constant</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-51-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>k</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">&#x0394;</mi><mi>F</mi></mrow><mrow><mi mathvariant="normal">&#x0394;</mi><mi>x</mi></mrow></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-398" style="width: 4.203em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.435em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1003.43em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-399"><span class="mi" id="MathJax-Span-400" style="font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-401" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-402" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.488em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1001.39em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.663em;"><span class="mrow" id="MathJax-Span-403"><span class="mi" id="MathJax-Span-404" style="font-family: STIXGeneral;">Δ</span><span class="mi" id="MathJax-Span-405" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1001.18em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.561em;"><span class="mrow" id="MathJax-Span-406"><span class="mi" id="MathJax-Span-407" style="font-family: STIXGeneral;">Δ</span><span class="mi" id="MathJax-Span-408" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.49em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.488em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>k</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>F</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>x</mi></mrow></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-51">k = \frac{\Delta F}{\Delta x}</script></span></p><h2 id="the-young-modulus"><a class="anchor hidden-xs" href="#the-young-modulus" title="the-young-modulus"><span class="octicon octicon-link"></span></a>The Young modulus</h2><p><strong>Stress</strong> Force per unit area <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-52-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>N m</mtext><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo><mtext>&#xA0;</mtext><mo stretchy="false">(</mo><mtext>Pa</mtext><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-409" style="width: 6.406em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.228em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1005.18em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-410"><span class="mo" id="MathJax-Span-411" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-412"><span style="display: inline-block; position: relative; width: 2.666em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1001.74em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mtext" id="MathJax-Span-413" style="font-family: STIXGeneral;">N m</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.744em;"><span class="texatom" id="MathJax-Span-414"><span class="mrow" id="MathJax-Span-415"><span class="mo" id="MathJax-Span-416" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-417" style="font-size: 70.7%; font-family: STIXGeneral;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-418" style="font-family: STIXGeneral;">)</span><span class="mtext" id="MathJax-Span-419" style="font-family: STIXGeneral;"> </span><span class="mo" id="MathJax-Span-420" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-421" style="font-family: STIXGeneral;">Pa</span><span class="mo" id="MathJax-Span-422" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>N m</mtext><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo><mtext> </mtext><mo stretchy="false">(</mo><mtext>Pa</mtext><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-52">(\text{N m}^{-2})\ (\text{Pa})</script></span> Given the symbol <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-53-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C3;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-423" style="width: 0.72em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.001em 1000.57em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-424"><span class="mi" id="MathJax-Span-425" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi></math></span></span><script type="math/tex" id="MathJax-Element-53">\sigma</script></span></p><p><strong>Strain</strong> Extension per unit length (Unitless) Given the symbol <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-54-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B5;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-426" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-427"><span class="mi" id="MathJax-Span-428" style="font-family: STIXGeneral; font-style: italic;">ε</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ε</mi></math></span></span><script type="math/tex" id="MathJax-Element-54">\varepsilon</script></span></p><p>The young modulus is the ratio of stress to strain</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-55-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>Young modulus</mtext><mo>=</mo><mfrac><mtext>tensile stress</mtext><mtext>tensile strain</mtext></mfrac><mspace linebreak="newline" /><mi>E</mi><mo>=</mo><mfrac><mi>&#x03C3;</mi><mi>&#x03B5;</mi></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-429" style="width: 100%; display: inline-block; min-width: 16.089em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(3.794em 1013.17em 8.455em -999.997em); top: -5.325em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-430"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.462em 1013.17em 4.869em -999.997em); top: -3.993em; left: 50%; margin-left: -6.555em;"><span class="mtext" id="MathJax-Span-431" style="font-family: STIXGeneral;">Young modulus</span><span class="mo" id="MathJax-Span-432" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-433" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 5.228em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1005.02em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -2.559em;"><span class="mtext" id="MathJax-Span-434" style="font-family: STIXGeneral;">tensile stress</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1005.07em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -2.559em;"><span class="mtext" id="MathJax-Span-435" style="font-family: STIXGeneral;">tensile strain</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1005.23em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 5.228em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.718em 1002.82em 4.869em -999.997em); top: -1.688em; left: 50%; margin-left: -1.432em;"><span class="mspace" id="MathJax-Span-436" style="height: 0em; vertical-align: 0em; width: 0em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-437" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-438" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-439" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.435em 1000.51em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-440" style="font-family: STIXGeneral; font-style: italic;">σ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-441" style="font-family: STIXGeneral; font-style: italic;">ε</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.67em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.669em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 5.33em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -3.684em; border-left: 0px solid; width: 0px; height: 5.441em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>Young modulus</mtext><mo>=</mo><mfrac><mtext>tensile stress</mtext><mtext>tensile strain</mtext></mfrac><mspace linebreak="newline"></mspace><mi>E</mi><mo>=</mo><mfrac><mi>σ</mi><mi>ε</mi></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-55">\text{Young modulus} = \frac{\text{tensile stress}}{\text{tensile strain}} \\ E = \frac{\sigma}{\varepsilon}</script></span></p><p>A stiff material needs more force to deform in comparison to a soft material. The Young’s modulus is a measure of the stiffness of a solid.</p><p><strong>Strength</strong> The amount of force that a material can withstand and still recover its original shape</p><p><strong>Hardness</strong> The relative resistance that the surface of a material imposes against the penetration of a harder body</p><p><strong>Toughness</strong> The amount of energy that a material can absorb before fracturing</p><p>As we know that</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-56-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>stress</mtext><mo>=</mo><mfrac><mtext>force</mtext><mtext>cross-sectional area</mtext></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-442" style="width: 14.347em; display: inline-block;"><span style="display: inline-block; position: relative; width: 11.734em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1011.73em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-443"><span class="mtext" id="MathJax-Span-444" style="font-family: STIXGeneral;">stress</span><span class="mo" id="MathJax-Span-445" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-446" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 7.994em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1002.05em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -1.022em;"><span class="mtext" id="MathJax-Span-447" style="font-family: STIXGeneral;">force</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1007.84em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -3.942em;"><span class="mtext" id="MathJax-Span-448" style="font-family: STIXGeneral;">cross-sectional area</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1007.99em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 7.994em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>stress</mtext><mo>=</mo><mfrac><mtext>force</mtext><mtext>cross-sectional area</mtext></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-56">\text{stress} = \frac{\text{force}}{\text{cross-sectional area}}</script></span></p><p>and</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-57-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>strain</mtext><mo>=</mo><mfrac><mtext>extension</mtext><mtext>original length</mtext></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-449" style="width: 11.888em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.736em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1009.74em 3.64em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-450"><span class="mtext" id="MathJax-Span-451" style="font-family: STIXGeneral;">strain</span><span class="mo" id="MathJax-Span-452" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-453" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 5.996em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1003.85em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -1.893em;"><span class="mtext" id="MathJax-Span-454" style="font-family: STIXGeneral;">extension</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1005.84em 4.357em -999.997em); top: -3.327em; left: 50%; margin-left: -2.918em;"><span class="mtext" id="MathJax-Span-455" style="font-family: STIXGeneral;">original length</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1006em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 5.996em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.184em; border-left: 0px solid; width: 0px; height: 2.878em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>strain</mtext><mo>=</mo><mfrac><mtext>extension</mtext><mtext>original length</mtext></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-57">\text{strain} = \frac{\text{extension}}{\text{original length}}</script></span></p><p>we can write the Young’s modulus as</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-58-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mtext>Young modulus</mtext></mtd><mtd><mi></mi><mo>=</mo><mfrac><mfrac><mtext>force</mtext><mtext>cross-sectional area</mtext></mfrac><mfrac><mtext>extension</mtext><mtext>original length</mtext></mfrac></mfrac></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mfrac><mrow><mtext>force</mtext><mo>&#x00D7;</mo><mtext>original length</mtext></mrow><mrow><mtext>extension</mtext><mo>&#x00D7;</mo><mtext>cross-sectional area</mtext></mrow></mfrac></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mfrac><mrow><mi>F</mi><mi>l</mi></mrow><mrow><mi>e</mi><mi>A</mi></mrow></mfrac></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-456" style="width: 25.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 20.955em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-2.2em 1020.8em 6.099em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-457"><span class="mtable" id="MathJax-Span-458" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 20.648em; height: 0px;"><span style="position: absolute; clip: rect(3.896em 1006.36em 10.402em -999.997em); top: -7.221em; left: 0em;"><span style="display: inline-block; position: relative; width: 6.406em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1006.36em 4.357em -999.997em); top: -6.452em; right: 0em;"><span class="mtd" id="MathJax-Span-459"><span class="mrow" id="MathJax-Span-460"><span class="mtext" id="MathJax-Span-461" style="font-family: STIXGeneral;">Young modulus</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -3.379em; right: 0em;"><span class="mtd" id="MathJax-Span-473"><span class="mrow" id="MathJax-Span-474"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -0.971em; right: 0em;"><span class="mtd" id="MathJax-Span-488"><span class="mrow" id="MathJax-Span-489"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 7.226em;"></span></span><span style="position: absolute; clip: rect(4.562em 1014.19em 12.861em -999.997em); top: -8.963em; left: 6.406em;"><span style="display: inline-block; position: relative; width: 14.193em; height: 0px;"><span style="position: absolute; clip: rect(2.257em 1007.59em 5.689em -999.997em); top: -6.606em; left: 0em;"><span class="mtd" id="MathJax-Span-462"><span class="mrow" id="MathJax-Span-463"><span class="mi" id="MathJax-Span-464"></span><span class="mo" id="MathJax-Span-465" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-466" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 6.048em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(2.974em 1005.89em 4.562em -999.997em); top: -4.864em; left: 50%; margin-left: -2.969em;"><span class="mfrac" id="MathJax-Span-467"><span style="display: inline-block; position: relative; width: 5.689em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1001.44em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.715em;"><span class="mtext" id="MathJax-Span-468" style="font-size: 70.7%; font-family: STIXGeneral;">force</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1005.53em 4.152em -999.997em); top: -3.584em; left: 50%; margin-left: -2.764em;"><span class="mtext" id="MathJax-Span-469" style="font-size: 70.7%; font-family: STIXGeneral;">cross-sectional area</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1005.69em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 5.689em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.974em 1004.51em 4.716em -999.997em); top: -3.122em; left: 50%; margin-left: -2.252em;"><span class="mfrac" id="MathJax-Span-470"><span style="display: inline-block; position: relative; width: 4.255em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1002.72em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -1.329em;"><span class="mtext" id="MathJax-Span-471" style="font-size: 70.7%; font-family: STIXGeneral;">extension</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1004.15em 4.306em -999.997em); top: -3.584em; left: 50%; margin-left: -2.047em;"><span class="mtext" id="MathJax-Span-472" style="font-size: 70.7%; font-family: STIXGeneral;">original length</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1004.25em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.255em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1006.05em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 6.048em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.152em;"></span></span><span style="position: absolute; clip: rect(2.462em 1014.19em 4.869em -999.997em); top: -3.379em; left: 0em;"><span class="mtd" id="MathJax-Span-475"><span class="mrow" id="MathJax-Span-476"><span class="mi" id="MathJax-Span-477"></span><span class="mo" id="MathJax-Span-478" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-479"><span style="display: inline-block; position: relative; width: 12.964em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1009.07em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -4.557em;"><span class="mrow" id="MathJax-Span-480"><span class="mtext" id="MathJax-Span-481" style="font-family: STIXGeneral;">force</span><span class="mo" id="MathJax-Span-482" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mtext" id="MathJax-Span-483" style="font-family: STIXGeneral; padding-left: 0.259em;">original length</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1012.86em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -6.401em;"><span class="mrow" id="MathJax-Span-484"><span class="mtext" id="MathJax-Span-485" style="font-family: STIXGeneral;">extension</span><span class="mo" id="MathJax-Span-486" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mtext" id="MathJax-Span-487" style="font-family: STIXGeneral; padding-left: 0.259em;">cross-sectional area</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1012.96em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 12.964em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.462em 1002.72em 4.869em -999.997em); top: -0.971em; left: 0em;"><span class="mtd" id="MathJax-Span-490"><span class="mrow" id="MathJax-Span-491"><span class="mi" id="MathJax-Span-492"></span><span class="mo" id="MathJax-Span-493" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-494" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.181em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.92em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.459em;"><span class="mrow" id="MathJax-Span-495"><span class="mi" id="MathJax-Span-496" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-497" style="font-family: STIXGeneral; font-style: italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1001.03em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-498"><span class="mi" id="MathJax-Span-499" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="mi" id="MathJax-Span-500" style="font-family: STIXGeneral; font-style: italic;">A</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.18em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.181em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 8.968em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -4.622em; border-left: 0px solid; width: 0px; height: 9.878em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mtext>Young modulus</mtext></mtd><mtd><mi></mi><mo>=</mo><mfrac><mfrac><mtext>force</mtext><mtext>cross-sectional area</mtext></mfrac><mfrac><mtext>extension</mtext><mtext>original length</mtext></mfrac></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mtext>force</mtext><mo>×</mo><mtext>original length</mtext></mrow><mrow><mtext>extension</mtext><mo>×</mo><mtext>cross-sectional area</mtext></mrow></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><mrow><mi>F</mi><mi>l</mi></mrow><mrow><mi>e</mi><mi>A</mi></mrow></mfrac></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-58">\begin{align}
\text{Young modulus} &= \frac{ \frac{\text{force}}{\text{cross-sectional area}} }{ \frac{\text{extension}}{\text{original length}}} \\
&= \frac{\text{force} \times \text{original length}}{\text{extension} \times \text{cross-sectional area}} \\
&= \frac{Fl}{eA}
\end{align}</script></span></p><h3 id="investigating-young-modulus"><a class="anchor hidden-xs" href="#investigating-young-modulus" title="investigating-young-modulus"><span class="octicon octicon-link"></span></a>Investigating Young modulus</h3><p>Searles apparatus</p><ul>
<li>Measures the original length of a wire before any load</li>
<li>Measures the diameter of the wire with a micrometre</li>
<li>Force of tension is then increased by adding masses to the right side</li>
<li>Extension is measured after each new mass is added</li>
<li>A stress vs strain graph is plotted from the data</li>
</ul><p><img src="https://i.imgsafe.org/496b25e74f.png" alt=""></p><p>With a ruler and mass</p><ul>
<li>Attach a wire to a clamp</li>
<li>Run the wire horizontally, over the edge of a surface through a pulley</li>
<li>Attach mass to the wire so that it is taught</li>
<li>Place a mark on the wire, and a ruler underneath it inline with the mark</li>
<li>Progressively add more mass, measuring how far the mark moves</li>
</ul><p>Use the collected values to calculate the Young modulus of the wire</p><p><strong>Ultimate tensile strength</strong> The maximum stress a material can withstand while being pulled or stretched before it breaks</p><p><strong>Elastic hysteresis</strong> The process in which energy is dissipated by the loading and unloading of a material</p><h2 id="categorisation-of-materials"><a class="anchor hidden-xs" href="#categorisation-of-materials" title="categorisation-of-materials"><span class="octicon octicon-link"></span></a>Categorisation of materials</h2><p><strong>Ductile</strong> A ductile material can be drawn into wires and will show plastic deformation under tensile stress before it breaks</p><p><strong>Malleable</strong> A malleable material can be hammered and beaten into flat sheets and will show extensive plastic deformation when subjected to compressive forces</p><p><strong>Brittle</strong> A brittle material will break with very little plastic deformation</p><p><strong>Polymeric</strong> A polymeric material has long chains of molecules that allow for elastic deformation</p><p>When a ductile material is stretched, it will increase in length and decrease in cross sectional area to compensate for this. This process is known as necking.</p><p><img src="http://spaceflight.esa.int/impress/text/education/Images/MechanicalProperties/Image006.png" alt=""></p><h2 id="three-laws-of-motion"><a class="anchor hidden-xs" href="#three-laws-of-motion" title="three-laws-of-motion"><span class="octicon octicon-link"></span></a>Three laws of motion</h2><h3 id="newton’s-first-law"><a class="anchor hidden-xs" href="#newton’s-first-law" title="newton’s-first-law"><span class="octicon octicon-link"></span></a>Newton’s first law</h3><p>A body will remain at rest or continue to move at constant velocity until an external force acts upon it.</p><h3 id="newton’s-second-law"><a class="anchor hidden-xs" href="#newton’s-second-law" title="newton’s-second-law"><span class="octicon octicon-link"></span></a>Newton’s second law</h3><p>The resultant force on an object is proportional to the rate of change of momentum, when the change in momentum takes place in the direction of the force.</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-59-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mover><mi>F</mi><mo>&#x2192;</mo></mover><mo>=</mo><mi>m</mi><mover><mi>a</mi><mo>&#x2192;</mo></mover></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-501" style="width: 4.818em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.947em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.232em 1003.9em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-502"><span class="munderover" id="MathJax-Span-503"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0.156em;"><span class="mi" id="MathJax-Span-504" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.87em 4.101em -999.997em); top: -4.711em; left: 0.054em;"><span class="mo" id="MathJax-Span-505" style=""><span style="font-family: STIXGeneral;">→</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-506" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-507" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">m</span><span class="munderover" id="MathJax-Span-508"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0.207em;"><span class="mi" id="MathJax-Span-509" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.87em 4.101em -999.997em); top: -4.506em; left: 0em;"><span class="mo" id="MathJax-Span-510" style=""><span style="font-family: STIXGeneral;">→</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.566em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mover><mi>F</mi><mo>→</mo></mover><mo>=</mo><mi>m</mi><mover><mi>a</mi><mo>→</mo></mover></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-59">\overrightarrow{F} = m \overrightarrow{a}</script></span></p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-60-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>F</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">&#x0394;</mi><mo stretchy="false">(</mo><mi>m</mi><mi>v</mi><mo stretchy="false">)</mo></mrow><mi>t</mi></mfrac><mo>=</mo><mi>m</mi><mfrac><mrow><mi mathvariant="normal">&#x0394;</mi><mi>v</mi></mrow><mi>t</mi></mfrac><mo>=</mo><mi>m</mi><mi>a</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-511" style="width: 13.271em; display: inline-block;"><span style="display: inline-block; position: relative; width: 10.863em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1010.86em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-512"><span class="mi" id="MathJax-Span-513" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-514" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-515" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.666em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1002.51em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -1.278em;"><span class="mrow" id="MathJax-Span-516"><span class="mi" id="MathJax-Span-517" style="font-family: STIXGeneral;">Δ</span><span class="mo" id="MathJax-Span-518" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-519" style="font-family: STIXGeneral; font-style: italic;">m</span><span class="mi" id="MathJax-Span-520" style="font-family: STIXGeneral; font-style: italic;">v</span><span class="mo" id="MathJax-Span-521" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.281em 1000.31em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-522" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.67em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.666em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-523" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-524" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">m</span><span class="mfrac" id="MathJax-Span-525"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1001.13em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.561em;"><span class="mrow" id="MathJax-Span-526"><span class="mi" id="MathJax-Span-527" style="font-family: STIXGeneral;">Δ</span><span class="mi" id="MathJax-Span-528" style="font-family: STIXGeneral; font-style: italic;">v</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.281em 1000.31em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-529" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.28em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.283em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-530" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-531" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">m</span><span class="mi" id="MathJax-Span-532" style="font-family: STIXGeneral; font-style: italic;">a</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>F</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><mi>m</mi><mi>v</mi><mo stretchy="false">)</mo></mrow><mi>t</mi></mfrac><mo>=</mo><mi>m</mi><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>v</mi></mrow><mi>t</mi></mfrac><mo>=</mo><mi>m</mi><mi>a</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-60">F = \frac{\Delta (mv)}{t} = m \frac{\Delta v}{t} = ma</script></span></p><h3 id="newton’s-third-law"><a class="anchor hidden-xs" href="#newton’s-third-law" title="newton’s-third-law"><span class="octicon octicon-link"></span></a>Newton’s third law</h3><p>For every action there is an equal and opposite reaction</p><h2 id="momentum"><a class="anchor hidden-xs" href="#momentum" title="momentum"><span class="octicon octicon-link"></span></a>Momentum</h2><p><strong>Linear momentum</strong> The mass of an object multiplied by its velocity</p><p><strong>Conservation of momentum</strong> The total momentum before a collision is always equal to the total momentum after a collision, provided that no external forces are involved</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-61-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>p</mi><mo>=</mo><mi>m</mi><mi>v</mi><mspace width="1em" /><mo stretchy="false">(</mo><msup><mtext>kg m s</mtext><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-533" style="width: 10.197em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.353em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1008.3em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-534"><span class="mi" id="MathJax-Span-535" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-536" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-537" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">m</span><span class="mi" id="MathJax-Span-538" style="font-family: STIXGeneral; font-style: italic;">v</span><span class="mspace" id="MathJax-Span-539" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-540" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-541"><span style="display: inline-block; position: relative; width: 3.589em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1002.62em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mtext" id="MathJax-Span-542" style="font-family: STIXGeneral;">kg m s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 2.666em;"><span class="texatom" id="MathJax-Span-543"><span class="mrow" id="MathJax-Span-544"><span class="mo" id="MathJax-Span-545" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-546" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-547" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>p</mi><mo>=</mo><mi>m</mi><mi>v</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><msup><mtext>kg m s</mtext><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-61">p = mv\quad (\text{kg m s}^{-1})</script></span></p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-62-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>F</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">&#x0394;</mi><mi>p</mi></mrow><mrow><mi mathvariant="normal">&#x0394;</mi><mi>t</mi></mrow></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-548" style="width: 4.255em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.486em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1003.49em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-549"><span class="mi" id="MathJax-Span-550" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-551" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-552" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.335em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1001.18em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -0.612em;"><span class="mrow" id="MathJax-Span-553"><span class="mi" id="MathJax-Span-554" style="font-family: STIXGeneral;">Δ</span><span class="mi" id="MathJax-Span-555" style="font-family: STIXGeneral; font-style: italic;">p</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1001.03em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-556"><span class="mi" id="MathJax-Span-557" style="font-family: STIXGeneral;">Δ</span><span class="mi" id="MathJax-Span-558" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.34em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.335em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>F</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>p</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-62">F = \frac{\Delta p}{\Delta t}</script></span></p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-63-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>F</mi><mi>t</mi><mo>=</mo><mi>m</mi><mi>v</mi><mo>&#x2212;</mo><mi>m</mi><mi>u</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-559" style="width: 7.123em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.843em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1005.84em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-560"><span class="mi" id="MathJax-Span-561" style="font-family: STIXGeneral; font-style: italic;">F<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-562" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-563" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-564" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">m</span><span class="mi" id="MathJax-Span-565" style="font-family: STIXGeneral; font-style: italic;">v</span><span class="mo" id="MathJax-Span-566" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mi" id="MathJax-Span-567" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">m</span><span class="mi" id="MathJax-Span-568" style="font-family: STIXGeneral; font-style: italic;">u</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>F</mi><mi>t</mi><mo>=</mo><mi>m</mi><mi>v</mi><mo>−</mo><mi>m</mi><mi>u</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-63">Ft = mv - mu</script></span></p><h2 id="force-and-impulse"><a class="anchor hidden-xs" href="#force-and-impulse" title="force-and-impulse"><span class="octicon octicon-link"></span></a>Force and impulse</h2><h3 id="elastic-and-inelastic-collisions"><a class="anchor hidden-xs" href="#elastic-and-inelastic-collisions" title="elastic-and-inelastic-collisions"><span class="octicon octicon-link"></span></a>Elastic and inelastic collisions</h3><p><strong>Elastic collision</strong> A collision in which the momentum and the kinetic energy are conserved. No energy is transferred to any other forms</p><p><strong>Inelastic collisions</strong> A collision in which momentum is conserved but some of the kinetic energy is transferred to to other forms in the collision</p><h2 id="electrons-waves-and-photons"><a class="anchor hidden-xs" href="#electrons-waves-and-photons" title="electrons-waves-and-photons"><span class="octicon octicon-link"></span></a>Electrons, Waves, and Photons</h2><h2 id="electric-circuit-components"><a class="anchor hidden-xs" href="#electric-circuit-components" title="electric-circuit-components"><span class="octicon octicon-link"></span></a>Electric circuit components</h2><p><img src="https://upload.wikimedia.org/wikibooks/en/thumb/1/12/Circuit_Symbols_for_A-level-OCR-Physics_A.png/600px-Circuit_Symbols_for_A-level-OCR-Physics_A.png" alt=""></p><p>All electric circuits require the following</p><ul>
<li>An energy source- A source of electromotive force</li>
<li>A closed path or complete circuit- For electrons to flow in a complete loop from a negative electrode to a positive electrode</li>
<li>Electrical components</li>
</ul><p><strong>Conventional current</strong> Behaves as if the positive charge carries cause current flow from the positive terminal to the negative terminal</p><p><strong>Electron flow</strong> As electrons are negatively charged, they flow from the negative electrode to the positive electrode</p><h2 id="electric-current-and-charge"><a class="anchor hidden-xs" href="#electric-current-and-charge" title="electric-current-and-charge"><span class="octicon octicon-link"></span></a>Electric current and charge</h2><p>Electrons protons and neutrons have mass and charge</p><table>
<thead>
<tr>
<th>****</th>
<th>Mass</th>
<th>Charge</th>
</tr>
</thead>
<tbody>
<tr>
<td>p</td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-64-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1.673</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>27</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-569" style="width: 6.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.638em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1005.64em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-570"><span class="mn" id="MathJax-Span-571" style="font-family: STIXGeneral;">1.673</span><span class="mo" id="MathJax-Span-572" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-573" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-574" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-575"><span class="mrow" id="MathJax-Span-576"><span class="mo" id="MathJax-Span-577" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-578" style="font-size: 70.7%; font-family: STIXGeneral;">27</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1.673</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>27</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-64">1.673\times10^{-27}</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-65-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mn>1.6</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>19</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-579" style="width: 6.509em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.33em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1005.33em 2.769em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-580"><span class="mo" id="MathJax-Span-581" style="font-family: STIXGeneral;">+</span><span class="mn" id="MathJax-Span-582" style="font-family: STIXGeneral;">1.6</span><span class="mo" id="MathJax-Span-583" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-584" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-585" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-586"><span class="mrow" id="MathJax-Span-587"><span class="mo" id="MathJax-Span-588" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-589" style="font-size: 70.7%; font-family: STIXGeneral;">19</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.122em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mn>1.6</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>19</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-65">+1.6\times10^{-19}</script></span></td>
</tr>
<tr>
<td>n</td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-66-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1.675</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>27</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-590" style="width: 6.919em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.638em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1005.64em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-591"><span class="mn" id="MathJax-Span-592" style="font-family: STIXGeneral;">1.675</span><span class="mo" id="MathJax-Span-593" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-594" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-595" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-596"><span class="mrow" id="MathJax-Span-597"><span class="mo" id="MathJax-Span-598" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-599" style="font-size: 70.7%; font-family: STIXGeneral;">27</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1.675</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>27</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-66">1.675\times10^{-27}</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-67-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-600" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-601"><span class="mn" id="MathJax-Span-602" style="font-family: STIXGeneral;">0</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-67">0</script></span></td>
</tr>
<tr>
<td>e</td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-68-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9.1</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>31</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-603" style="width: 5.689em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.664em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1004.66em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-604"><span class="mn" id="MathJax-Span-605" style="font-family: STIXGeneral;">9.1</span><span class="mo" id="MathJax-Span-606" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-607" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-608" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-609"><span class="mrow" id="MathJax-Span-610"><span class="mo" id="MathJax-Span-611" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-612" style="font-size: 70.7%; font-family: STIXGeneral;">31</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9.1</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>31</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-68">9.1\times10^{-31}</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-69-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2212;</mo><mn>1.6</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>19</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-613" style="width: 6.509em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.33em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1005.33em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-614"><span class="mo" id="MathJax-Span-615" style="font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-616" style="font-family: STIXGeneral;">1.6</span><span class="mo" id="MathJax-Span-617" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-618" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-619" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-620"><span class="mrow" id="MathJax-Span-621"><span class="mo" id="MathJax-Span-622" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-623" style="font-size: 70.7%; font-family: STIXGeneral;">19</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>1.6</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>19</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-69">-1.6\times10^{-19}</script></span></td>
</tr>
</tbody>
</table><p>These are normalised to</p><table>
<thead>
<tr>
<th>****</th>
<th>Mass</th>
<th>Charge</th>
</tr>
</thead>
<tbody>
<tr>
<td>p</td>
<td>1</td>
<td>+1</td>
</tr>
<tr>
<td>n</td>
<td>1</td>
<td>0</td>
</tr>
<tr>
<td>e</td>
<td>0</td>
<td>-1</td>
</tr>
</tbody>
</table><p><strong>Electric current</strong> The rate of flow of charge</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-70-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>I</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">&#x0394;</mi><mi>Q</mi></mrow><mrow><mi mathvariant="normal">&#x0394;</mi><mi>t</mi></mrow></mfrac><mspace width="1em" /><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-624" style="width: 7.226em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.894em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1005.84em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-625"><span class="mi" id="MathJax-Span-626" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-627" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-628" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.591em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1001.44em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -0.715em;"><span class="mrow" id="MathJax-Span-629"><span class="mi" id="MathJax-Span-630" style="font-family: STIXGeneral;">Δ</span><span class="mi" id="MathJax-Span-631" style="font-family: STIXGeneral; font-style: italic;">Q</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1001.03em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="mrow" id="MathJax-Span-632"><span class="mi" id="MathJax-Span-633" style="font-family: STIXGeneral;">Δ</span><span class="mi" id="MathJax-Span-634" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.59em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.591em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mspace" id="MathJax-Span-635" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-636" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-637" style="font-family: STIXGeneral; font-style: italic;">A</span><span class="mo" id="MathJax-Span-638" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>I</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>Q</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-70">I = \frac{\Delta Q}{\Delta t} \quad (A)</script></span></p><p>This can also be written as</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-71-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>I</mi><mo>=</mo><mfrac><mrow><mi>n</mi><mi>e</mi></mrow><mi>t</mi></mfrac><mspace width="1em" /><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-639" style="width: 6.56em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.382em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.283em 1005.33em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-640"><span class="mi" id="MathJax-Span-641" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-642" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-643" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.92em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.459em;"><span class="mrow" id="MathJax-Span-644"><span class="mi" id="MathJax-Span-645" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mi" id="MathJax-Span-646" style="font-family: STIXGeneral; font-style: italic;">e</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.281em 1000.31em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-647" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mspace" id="MathJax-Span-648" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mo" id="MathJax-Span-649" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-650" style="font-family: STIXGeneral; font-style: italic;">A</span><span class="mo" id="MathJax-Span-651" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.316em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>I</mi><mo>=</mo><mfrac><mrow><mi>n</mi><mi>e</mi></mrow><mi>t</mi></mfrac><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-71">I = \frac{ne}{t} \quad (A)</script></span></p><p>where n is the number of electrons.</p><p>The unit of charge is the coulomb (<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-72-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-652" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.62em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-653"><span class="mtext" id="MathJax-Span-654" style="font-family: STIXGeneral;">C</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math></span></span><script type="math/tex" id="MathJax-Element-72">\text{C}</script></span>).<br>
One coulomb is the quantity of charge that passes a fixed point in one second when a current of one ampere is flowing</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-73-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>1</mn><mtext>&#xA0;C</mtext><mo>=</mo><mn>1</mn><mtext>&#xA0;A s</mtext></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-655" style="width: 5.894em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.818em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1004.77em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-656"><span class="mn" id="MathJax-Span-657" style="font-family: STIXGeneral;">1</span><span class="mtext" id="MathJax-Span-658" style="font-family: STIXGeneral;"> C</span><span class="mo" id="MathJax-Span-659" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-660" style="font-family: STIXGeneral; padding-left: 0.31em;">1</span><span class="mtext" id="MathJax-Span-661" style="font-family: STIXGeneral;"> A s</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>1</mn><mtext> C</mtext><mo>=</mo><mn>1</mn><mtext> A s</mtext></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-73">1\text{ C} = 1\text{ A s}</script></span></p><h3 id="kirchhoff’s-first-law0"><a class="anchor hidden-xs" href="#kirchhoff’s-first-law0" title="kirchhoff’s-first-law0"><span class="octicon octicon-link"></span></a>Kirchhoff’s first law</h3><p>At any junction in a circuit, the sum of the currents arriving at the junction is equal to the sum of the currents leaving the junction.</p><h2 id="electron-drift-velocity"><a class="anchor hidden-xs" href="#electron-drift-velocity" title="electron-drift-velocity"><span class="octicon octicon-link"></span></a>Electron drift velocity</h2><p>Electrons move in random directions.<br>
When a current is flowing, the electrons have a mean velocity in the direction of electron flow.</p><h3 id="calculating-electron-drift-velocity"><a class="anchor hidden-xs" href="#calculating-electron-drift-velocity" title="calculating-electron-drift-velocity"><span class="octicon octicon-link"></span></a>Calculating electron drift velocity</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-74-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>I</mi><mo>=</mo><mi>n</mi><mi>A</mi><mi>v</mi><mi>q</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-662" style="width: 4.562em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.742em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.74em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-663"><span class="mi" id="MathJax-Span-664" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-665" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-666" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">n</span><span class="mi" id="MathJax-Span-667" style="font-family: STIXGeneral; font-style: italic;">A</span><span class="mi" id="MathJax-Span-668" style="font-family: STIXGeneral; font-style: italic;">v</span><span class="mi" id="MathJax-Span-669" style="font-family: STIXGeneral; font-style: italic;">q</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>I</mi><mo>=</mo><mi>n</mi><mi>A</mi><mi>v</mi><mi>q</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-74">I = nAvq</script></span></p><p>where q is the charge on each charge carrier</p><h2 id="potential-difference-and-emf"><a class="anchor hidden-xs" href="#potential-difference-and-emf" title="potential-difference-and-emf"><span class="octicon octicon-link"></span></a>Potential difference and E.M.F</h2><h3 id="the-difference-between-potential-difference-and-electromotive-force"><a class="anchor hidden-xs" href="#the-difference-between-potential-difference-and-electromotive-force" title="the-difference-between-potential-difference-and-electromotive-force"><span class="octicon octicon-link"></span></a>The difference between potential difference and electromotive force</h3><p>p.d can be measured in several places in a circuit, across each of its components. It is the energy transferred per unit charge as they pass through the component.</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-75-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>p</mi><mo>.</mo><mi>d</mi><mo>=</mo><mfrac><mtext>work done</mtext><mtext>charge</mtext></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-670" style="width: 9.019em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.38em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1007.38em 3.64em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-671"><span class="mi" id="MathJax-Span-672" style="font-family: STIXGeneral; font-style: italic;">p</span><span class="mo" id="MathJax-Span-673" style="font-family: STIXGeneral;">.</span><span class="mi" id="MathJax-Span-674" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-675" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-676" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 4.357em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1004.25em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -2.098em;"><span class="mtext" id="MathJax-Span-677" style="font-family: STIXGeneral;">work done</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1002.67em 4.357em -999.997em); top: -3.327em; left: 50%; margin-left: -1.329em;"><span class="mtext" id="MathJax-Span-678" style="font-family: STIXGeneral;">charge</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1004.36em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 4.357em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.184em; border-left: 0px solid; width: 0px; height: 2.878em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>p</mi><mo>.</mo><mi>d</mi><mo>=</mo><mfrac><mtext>work done</mtext><mtext>charge</mtext></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-75">p.d = \frac{\text{work done}}{\text{charge}}</script></span></p><p>e.m.f is only associated with the electrical energy source. It is the energy gained per unit charge passing through the source.</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-76-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><mo>.</mo><mi>m</mi><mo>.</mo><mi>f</mi><mo>=</mo><mfrac><mtext>energy gained</mtext><mtext>charge</mtext></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-679" style="width: 11.888em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.736em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1009.74em 3.64em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-680"><span class="mi" id="MathJax-Span-681" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="mo" id="MathJax-Span-682" style="font-family: STIXGeneral;">.</span><span class="mi" id="MathJax-Span-683" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">m</span><span class="mo" id="MathJax-Span-684" style="font-family: STIXGeneral;">.</span><span class="mi" id="MathJax-Span-685" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-686" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-687" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 5.74em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1005.64em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -2.815em;"><span class="mtext" id="MathJax-Span-688" style="font-family: STIXGeneral;">energy gained</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1002.67em 4.357em -999.997em); top: -3.327em; left: 50%; margin-left: -1.329em;"><span class="mtext" id="MathJax-Span-689" style="font-family: STIXGeneral;">charge</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1005.74em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 5.74em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.184em; border-left: 0px solid; width: 0px; height: 2.878em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><mo>.</mo><mi>m</mi><mo>.</mo><mi>f</mi><mo>=</mo><mfrac><mtext>energy gained</mtext><mtext>charge</mtext></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-76">e.m.f = \frac{\text{energy gained}}{\text{charge}}</script></span></p><p>Mathematically they are the same, because they are both measured in volts.</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-77-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo>=</mo><mfrac><mi>W</mi><mi>Q</mi></mfrac><mo>=</mo><mfrac><mi>E</mi><mi>Q</mi></mfrac><mo>=</mo><mi>&#x03B5;</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-690" style="width: 8.968em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.328em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1007.33em 3.589em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-691"><span class="mi" id="MathJax-Span-692" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-693" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-694" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.92em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.459em;"><span class="mi" id="MathJax-Span-695" style="font-family: STIXGeneral; font-style: italic;">W<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.72em 4.357em -999.997em); top: -3.327em; left: 50%; margin-left: -0.356em;"><span class="mi" id="MathJax-Span-696" style="font-family: STIXGeneral; font-style: italic;">Q</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.03em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.027em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-697" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-698" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.305em;"><span class="mi" id="MathJax-Span-699" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.72em 4.357em -999.997em); top: -3.327em; left: 50%; margin-left: -0.356em;"><span class="mi" id="MathJax-Span-700" style="font-family: STIXGeneral; font-style: italic;">Q</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.82em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.822em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-701" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-702" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">ε</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.122em; border-left: 0px solid; width: 0px; height: 2.816em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo>=</mo><mfrac><mi>W</mi><mi>Q</mi></mfrac><mo>=</mo><mfrac><mi>E</mi><mi>Q</mi></mfrac><mo>=</mo><mi>ε</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-77">V = \frac{W}{Q} = \frac{E}{Q} = \varepsilon</script></span></p><h2 id="resistance-and-ohm’s-law"><a class="anchor hidden-xs" href="#resistance-and-ohm’s-law" title="resistance-and-ohm’s-law"><span class="octicon octicon-link"></span></a>Resistance and Ohm’s law</h2><h3 id="ohm’s-law"><a class="anchor hidden-xs" href="#ohm’s-law" title="ohm’s-law"><span class="octicon octicon-link"></span></a>Ohm’s law</h3><p>The current through a conductor is directly proportional to the potential difference across it, provided that physical conditions, such as temperature, remain constant.</p><h3 id="resistance"><a class="anchor hidden-xs" href="#resistance" title="resistance"><span class="octicon octicon-link"></span></a>Resistance</h3><p><strong>Resistance</strong> The ratio of voltage applied to a material to the current which flows through it (<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-78-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">&#x03A9;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-703" style="width: 0.976em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-704"><span class="mi" id="MathJax-Span-705" style="font-family: STIXGeneral;">Ω</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Ω</mi></math></span></span><script type="math/tex" id="MathJax-Element-78">\Omega</script></span>)</p><p>Resistance depends upon the type of material, and the volume of the material.<br>
Longer wires will have a smaller potential gradient. The longer the wire, the lower the ‘voltage per metre’. This will lower the drift velocity and hence result in a smaller current.<br>
This means that the resistance of the wire increases in direct proportion to the length of the wire.</p><p>Resistance is inversely proportional to the cross sectional area of the wire. There is no change in potential gradient, but there is a large volume of electrons that are available to flow at the same drift velocity.</p><p>Resistance also depends on the temperature of the wire.<br>
When the temperature increases, the fixed metal ions vibrate more creating an obstacle for the flow of electrons.</p><h3 id="resistivity"><a class="anchor hidden-xs" href="#resistivity" title="resistivity"><span class="octicon octicon-link"></span></a>Resistivity</h3><p><strong>Resistivity</strong> A measure of the resisting power of a material to the flow of electric current through it</p><p>Resistivity applies to a particular component.<br>
While resistance is measured in ohms <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-79-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi mathvariant="normal">&#x03A9;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-706" style="width: 1.693em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.386em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.34em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-707"><span class="mo" id="MathJax-Span-708" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-709" style="font-family: STIXGeneral;">Ω</span><span class="mo" id="MathJax-Span-710" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-79">(\Omega)</script></span>, resistivity is measured in ohm metres <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-80-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi mathvariant="normal">&#x03A9;</mi><mtext>m</mtext><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-711" style="width: 2.718em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.15em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-712"><span class="mo" id="MathJax-Span-713" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-714" style="font-family: STIXGeneral;">Ω</span><span class="mtext" id="MathJax-Span-715" style="font-family: STIXGeneral;">m</span><span class="mo" id="MathJax-Span-716" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mtext>m</mtext><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-80">(\Omega\text{m})</script></span></p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax MathJax_FullWidth" id="MathJax-Element-81-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>R</mi><mo>=</mo><mfrac><mrow><mi>&#x03C1;</mi><mi>l</mi></mrow><mi>A</mi></mfrac><mspace linebreak="newline" /><mi>&#x03C1;</mi><mo>=</mo><mfrac><mrow><mi>R</mi><mi>A</mi></mrow><mi>l</mi></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-717" style="width: 100%; display: inline-block; min-width: 4.152em;"><span style="display: inline-block; position: relative; width: 100%; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(4.05em 1003.38em 8.968em -999.997em); top: -5.581em; left: 0em; width: 100%;"><span class="mrow" id="MathJax-Span-718"><span style="display: inline-block; position: relative; width: 100%; height: 0px;"><span style="position: absolute; clip: rect(2.462em 1003.08em 4.818em -999.997em); top: -3.993em; left: 50%; margin-left: -1.534em;"><span class="mi" id="MathJax-Span-719" style="font-family: STIXGeneral; font-style: italic;">R</span><span class="mo" id="MathJax-Span-720" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-721" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.77em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -0.407em;"><span class="mrow" id="MathJax-Span-722"><span class="mi" id="MathJax-Span-723" style="font-family: STIXGeneral; font-style: italic;">ρ</span><span class="mi" id="MathJax-Span-724" style="font-family: STIXGeneral; font-style: italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.305em;"><span class="mi" id="MathJax-Span-725" style="font-family: STIXGeneral; font-style: italic;">A</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.92em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.925em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.513em 1003.38em 4.869em -999.997em); top: -1.483em; left: 50%; margin-left: -1.688em;"><span class="mspace" id="MathJax-Span-726" style="height: 0em; vertical-align: 0em; width: 0em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-727" style="font-family: STIXGeneral; font-style: italic;">ρ</span><span class="mo" id="MathJax-Span-728" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-729" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.335em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1001.18em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.612em;"><span class="mrow" id="MathJax-Span-730"><span class="mi" id="MathJax-Span-731" style="font-family: STIXGeneral; font-style: italic;">R</span><span class="mi" id="MathJax-Span-732" style="font-family: STIXGeneral; font-style: italic;">A</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.26em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-733" style="font-family: STIXGeneral; font-style: italic;">l<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.34em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.335em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 5.587em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -3.997em; border-left: 0px solid; width: 0px; height: 5.691em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>R</mi><mo>=</mo><mfrac><mrow><mi>ρ</mi><mi>l</mi></mrow><mi>A</mi></mfrac><mspace linebreak="newline"></mspace><mi>ρ</mi><mo>=</mo><mfrac><mrow><mi>R</mi><mi>A</mi></mrow><mi>l</mi></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-81">R = \frac{\rho l}{A} \\ \rho = \frac{RA}{l}</script></span></p><p>where<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-82-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C1;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-734" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-735"><span class="mi" id="MathJax-Span-736" style="font-family: STIXGeneral; font-style: italic;">ρ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ρ</mi></math></span></span><script type="math/tex" id="MathJax-Element-82">\rho</script></span> = resistivity <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-83-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi mathvariant="normal">&#x03A9;</mi><mtext>&#xA0;m</mtext><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-737" style="width: 3.025em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.462em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.41em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-738"><span class="mo" id="MathJax-Span-739" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-740" style="font-family: STIXGeneral;">Ω</span><span class="mtext" id="MathJax-Span-741" style="font-family: STIXGeneral;"> m</span><span class="mo" id="MathJax-Span-742" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mtext> m</mtext><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-83">(\Omega\text{ m})</script></span><br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-84-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-743" style="width: 0.771em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.62em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-744"><span class="mi" id="MathJax-Span-745" style="font-family: STIXGeneral; font-style: italic;">R</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math></span></span><script type="math/tex" id="MathJax-Element-84">R</script></span> = resistance <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-85-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi mathvariant="normal">&#x03A9;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-746" style="width: 1.693em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.386em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.34em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-747"><span class="mo" id="MathJax-Span-748" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-749" style="font-family: STIXGeneral;">Ω</span><span class="mo" id="MathJax-Span-750" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi mathvariant="normal">Ω</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-85">(\Omega)</script></span><br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-86-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-751" style="width: 0.771em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.57em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-752"><span class="mi" id="MathJax-Span-753" style="font-family: STIXGeneral; font-style: italic;">A</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></span></span><script type="math/tex" id="MathJax-Element-86">A</script></span> = area <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-87-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>m</mtext><mn>2</mn></msup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-754" style="width: 2.308em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1001.85em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-755"><span class="mo" id="MathJax-Span-756" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-757"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.77em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mtext" id="MathJax-Span-758" style="font-family: STIXGeneral;">m</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.771em;"><span class="mn" id="MathJax-Span-759" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-760" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>m</mtext><mn>2</mn></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-87">(\text{m}^2)</script></span><br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-88-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-761" style="width: 0.72em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.57em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-762"><span class="mi" id="MathJax-Span-763" style="font-family: STIXGeneral; font-style: italic;">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math></span></span><script type="math/tex" id="MathJax-Element-88">L</script></span> = length <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-89-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>m</mtext><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-764" style="width: 1.744em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.437em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.39em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-765"><span class="mo" id="MathJax-Span-766" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-767" style="font-family: STIXGeneral;">m</span><span class="mo" id="MathJax-Span-768" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mtext>m</mtext><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-89">(\text{m})</script></span></p><h3 id="the-effect-of-temperature-on-resistivity"><a class="anchor hidden-xs" href="#the-effect-of-temperature-on-resistivity" title="the-effect-of-temperature-on-resistivity"><span class="octicon octicon-link"></span></a>The effect of temperature on resistivity</h3><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-90-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>&#x03C1;</mi><mi>T</mi></msub><mo>=</mo><msub><mi>&#x03C1;</mi><mn>0</mn></msub><mo stretchy="false">[</mo><mn>1</mn><mo>+</mo><mi>&#x03B1;</mi><mo stretchy="false">(</mo><mi>T</mi><mo>&#x2212;</mo><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-769" style="width: 11.837em; display: inline-block;"><span style="display: inline-block; position: relative; width: 9.685em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1009.58em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-770"><span class="msubsup" id="MathJax-Span-771"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-772" style="font-family: STIXGeneral; font-style: italic;">ρ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-773" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-774" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-775" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-776" style="font-family: STIXGeneral; font-style: italic;">ρ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-777" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-778" style="font-family: STIXGeneral;">[</span><span class="mn" id="MathJax-Span-779" style="font-family: STIXGeneral;">1</span><span class="mo" id="MathJax-Span-780" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-781" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">α</span><span class="mo" id="MathJax-Span-782" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-783" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-784" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="msubsup" id="MathJax-Span-785" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-786" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.566em;"><span class="mn" id="MathJax-Span-787" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-788" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-789" style="font-family: STIXGeneral;">]</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>ρ</mi><mi>T</mi></msub><mo>=</mo><msub><mi>ρ</mi><mn>0</mn></msub><mo stretchy="false">[</mo><mn>1</mn><mo>+</mo><mi>α</mi><mo stretchy="false">(</mo><mi>T</mi><mo>−</mo><msub><mi>T</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-90">\rho _T = \rho _0 [1 + \alpha (T - T_0)]</script></span></p><p>where<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-91-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03C1;</mi><mi>T</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-790" style="width: 1.283em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1001.03em 2.564em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-791"><span class="msubsup" id="MathJax-Span-792"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-793" style="font-family: STIXGeneral; font-style: italic;">ρ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mi" id="MathJax-Span-794" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mi>T</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-91">\rho_T</script></span> = resistivity of the material at temperature <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-92-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-795" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-796"><span class="mi" id="MathJax-Span-797" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-92">T</script></span><br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-93-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03C1;</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-798" style="width: 1.13em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1000.92em 2.564em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-799"><span class="msubsup" id="MathJax-Span-800"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-801" style="font-family: STIXGeneral; font-style: italic;">ρ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-802" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ρ</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-93">\rho_0</script></span>= resistivity of the material at temperature <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-94-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-803" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-804"><span class="mi" id="MathJax-Span-805" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-94">T</script></span><br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-95-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B1;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-806" style="width: 0.72em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.57em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-807"><span class="mi" id="MathJax-Span-808" style="font-family: STIXGeneral; font-style: italic;">α</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math></span></span><script type="math/tex" id="MathJax-Element-95">\alpha</script></span> = the temperature coefficient<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-96-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-809" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-810"><span class="mi" id="MathJax-Span-811" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-96">T</script></span> = the temperature of the material<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-97-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-812" style="width: 1.181em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1000.98em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-813"><span class="msubsup" id="MathJax-Span-814"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-815" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.566em;"><span class="mn" id="MathJax-Span-816" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>T</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-97">T_0</script></span> The reference temperature at which the resistivity of the material is quoted (Usually room temperature)</p><h2 id="electrical-power"><a class="anchor hidden-xs" href="#electrical-power" title="electrical-power"><span class="octicon octicon-link"></span></a>Electrical power</h2><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-98-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mi>P</mi></mtd><mtd><mi></mi><mo>=</mo><mi>V</mi><mi>I</mi></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><msup><mi>I</mi><mn>2</mn></msup><mi>R</mi></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mfrac><msup><mi>V</mi><mn>2</mn></msup><mi>R</mi></mfrac></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-817" style="width: 4.664em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.794em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-0.663em 1003.64em 4.613em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-818"><span class="mtable" id="MathJax-Span-819" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 3.486em; height: 0px;"><span style="position: absolute; clip: rect(2.359em 1000.62em 6.816em -999.997em); top: -5.12em; left: 0em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -5.94em; right: 0em;"><span class="mtd" id="MathJax-Span-820"><span class="mrow" id="MathJax-Span-821"><span class="mi" id="MathJax-Span-822" style="font-family: STIXGeneral; font-style: italic;">P</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -4.506em; right: 0em;"><span class="mtd" id="MathJax-Span-829"><span class="mrow" id="MathJax-Span-830"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -2.456em; right: 0em;"><span class="mtd" id="MathJax-Span-839"><span class="mrow" id="MathJax-Span-840"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.126em;"></span></span><span style="position: absolute; clip: rect(3.076em 1002.87em 8.251em -999.997em); top: -5.838em; left: 0.617em;"><span style="display: inline-block; position: relative; width: 2.871em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1002.36em 4.152em -999.997em); top: -5.94em; left: 0em;"><span class="mtd" id="MathJax-Span-823"><span class="mrow" id="MathJax-Span-824"><span class="mi" id="MathJax-Span-825"></span><span class="mo" id="MathJax-Span-826" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-827" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-828" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.974em 1002.77em 4.152em -999.997em); top: -4.506em; left: 0em;"><span class="mtd" id="MathJax-Span-831"><span class="mrow" id="MathJax-Span-832"><span class="mi" id="MathJax-Span-833"></span><span class="mo" id="MathJax-Span-834" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-835" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-836" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.464em;"><span class="mn" id="MathJax-Span-837" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-838" style="font-family: STIXGeneral; font-style: italic;">R</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.308em 1002.87em 4.818em -999.997em); top: -2.456em; left: 0em;"><span class="mtd" id="MathJax-Span-841"><span class="mrow" id="MathJax-Span-842"><span class="mi" id="MathJax-Span-843"></span><span class="mo" id="MathJax-Span-844" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-845" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.335em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.025em 1001.18em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.612em;"><span class="msubsup" id="MathJax-Span-846"><span style="display: inline-block; position: relative; width: 1.181em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-847" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.771em;"><span class="mn" id="MathJax-Span-848" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.305em;"><span class="mi" id="MathJax-Span-849" style="font-family: STIXGeneral; font-style: italic;">R</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.34em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.335em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.843em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.809em; border-left: 0px solid; width: 0px; height: 6.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mi>P</mi></mtd><mtd><mi></mi><mo>=</mo><mi>V</mi><mi>I</mi></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><msup><mi>I</mi><mn>2</mn></msup><mi>R</mi></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><msup><mi>V</mi><mn>2</mn></msup><mi>R</mi></mfrac></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-98">\begin{align} P &= VI \\ &= I^2 R \\ &= \frac{V^2}{R} \end{align}</script></span></p><h3 id="the-cost-of-electrical-energy"><a class="anchor hidden-xs" href="#the-cost-of-electrical-energy" title="the-cost-of-electrical-energy"><span class="octicon octicon-link"></span></a>The cost of electrical energy</h3><p><strong>Kilowatt-hour</strong> The kilowatt hour (<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-99-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>k Wh</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-850" style="width: 2.718em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.21em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-851"><span class="mtext" id="MathJax-Span-852" style="font-family: STIXGeneral;">k Wh</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>k Wh</mtext></math></span></span><script type="math/tex" id="MathJax-Element-99">\text{k Wh}</script></span>) is a unit of energy used by electricity companies in the context of billing household customers</p><p>1 kilowatt hour is the energy expended by a task drawing 1 kilowatt of power for 1 hour.</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-100-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>1</mn><mtext>&#xA0;k Wh</mtext><mo>=</mo><mn>1000</mn><mtext>&#xA0;W</mtext><mo>&#x00D7;</mo><mn>3600</mn><mtext>s</mtext><mo>=</mo><mn>3</mn><mo>,</mo><mn>600</mn><mo>,</mo><mn>000</mn><mtext>J</mtext></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-853" style="width: 20.853em; display: inline-block;"><span style="display: inline-block; position: relative; width: 17.062em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1017.06em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-854"><span class="mn" id="MathJax-Span-855" style="font-family: STIXGeneral;">1</span><span class="mtext" id="MathJax-Span-856" style="font-family: STIXGeneral;"> k Wh</span><span class="mo" id="MathJax-Span-857" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-858" style="font-family: STIXGeneral; padding-left: 0.31em;">1000</span><span class="mtext" id="MathJax-Span-859" style="font-family: STIXGeneral;"> W</span><span class="mo" id="MathJax-Span-860" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mn" id="MathJax-Span-861" style="font-family: STIXGeneral; padding-left: 0.259em;">3600</span><span class="mtext" id="MathJax-Span-862" style="font-family: STIXGeneral;">s</span><span class="mo" id="MathJax-Span-863" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-864" style="font-family: STIXGeneral; padding-left: 0.31em;">3</span><span class="mo" id="MathJax-Span-865" style="font-family: STIXGeneral;">,</span><span class="mn" id="MathJax-Span-866" style="font-family: STIXGeneral; padding-left: 0.207em;">600</span><span class="mo" id="MathJax-Span-867" style="font-family: STIXGeneral;">,</span><span class="mn" id="MathJax-Span-868" style="font-family: STIXGeneral; padding-left: 0.207em;">000</span><span class="mtext" id="MathJax-Span-869" style="font-family: STIXGeneral;">J</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>1</mn><mtext> k Wh</mtext><mo>=</mo><mn>1000</mn><mtext> W</mtext><mo>×</mo><mn>3600</mn><mtext>s</mtext><mo>=</mo><mn>3</mn><mo>,</mo><mn>600</mn><mo>,</mo><mn>000</mn><mtext>J</mtext></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-100">1 \text{ k Wh} = 1000\text{ W} \times 3600\text{s} = 3,600,000 \text{J}</script></span></p><h2 id="kirchhoff’s-first-and-second-laws"><a class="anchor hidden-xs" href="#kirchhoff’s-first-and-second-laws" title="kirchhoff’s-first-and-second-laws"><span class="octicon octicon-link"></span></a>Kirchhoff’s first and second laws</h2><h3 id="kirchhoff’s-first-law"><a class="anchor hidden-xs" href="#kirchhoff’s-first-law" title="kirchhoff’s-first-law"><span class="octicon octicon-link"></span></a>Kirchhoff’s first law</h3><p>At any node in an electrical circuit, the sum of currents flowing into that node is equal to the sum of the currents flowing out of that node</p><h3 id="kirchhoff’s-second-law"><a class="anchor hidden-xs" href="#kirchhoff’s-second-law" title="kirchhoff’s-second-law"><span class="octicon octicon-link"></span></a>Kirchhoff’s second law</h3><p>The directed sum of the potential differences around any closed network is zero.</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-101-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>I</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-870" style="width: 4.869em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.998em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.822em 1004em 3.998em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-871"><span class="munderover" id="MathJax-Span-872"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(2.871em 1001.18em 4.613em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-873" style="font-family: STIXSizeOneSym; vertical-align: -0.51em;">∑</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.08em 4.255em -999.997em); top: -2.866em; left: 0.054em;"><span class="texatom" id="MathJax-Span-874"><span class="mrow" id="MathJax-Span-875"><span class="mi" id="MathJax-Span-876" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-877" style="font-size: 70.7%; font-family: STIXGeneral;">=</span><span class="mn" id="MathJax-Span-878" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.435em 1000.36em 4.152em -999.997em); top: -5.172em; left: 0.464em;"><span class="mi" id="MathJax-Span-879" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-880" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-881" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.361em;"><span class="mi" id="MathJax-Span-882" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-883" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-884" style="font-family: STIXGeneral; padding-left: 0.31em;">0</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.622em; border-left: 0px solid; width: 0px; height: 3.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>I</mi><mi>k</mi></msub><mo>=</mo><mn>0</mn></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-101">\sum _{k=1} ^n I_k = 0</script></span></p><p>More simply, the sum of the e.m.fs around a circuit is equal to the sum of the potential differences.</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-102-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>&#x2211;</mo><mi>p</mi><mo>.</mo><mi>d</mi><mo>=</mo><mo>&#x2211;</mo><mi>I</mi><mi>R</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-885" style="width: 8.353em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.816em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.847em 1006.82em 3.589em -999.997em); top: -2.969em; left: 0em;"><span class="mrow" id="MathJax-Span-886"><span class="mo" id="MathJax-Span-887" style="font-family: STIXSizeOneSym; vertical-align: -0.51em;">∑</span><span class="mi" id="MathJax-Span-888" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">p</span><span class="mo" id="MathJax-Span-889" style="font-family: STIXGeneral;">.</span><span class="mi" id="MathJax-Span-890" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-891" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-892" style="font-family: STIXSizeOneSym; vertical-align: -0.51em; padding-left: 0.31em;">∑</span><span class="mi" id="MathJax-Span-893" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-894" style="font-family: STIXGeneral; font-style: italic;">R</span></span><span style="display: inline-block; width: 0px; height: 2.974em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.622em; border-left: 0px solid; width: 0px; height: 1.878em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo>∑</mo><mi>p</mi><mo>.</mo><mi>d</mi><mo>=</mo><mo>∑</mo><mi>I</mi><mi>R</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-102">\sum p.d = \sum IR</script></span></p><h2 id="series-circuits"><a class="anchor hidden-xs" href="#series-circuits" title="series-circuits"><span class="octicon octicon-link"></span></a>Series circuits</h2><h3 id="resistance-in-series-circuits"><a class="anchor hidden-xs" href="#resistance-in-series-circuits" title="resistance-in-series-circuits"><span class="octicon octicon-link"></span></a>Resistance in series circuits</h3><p>The total resistance of a set of resistors in series is equal to the sum of the individual resistances of the resistors</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-103-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>R</mi><mi>T</mi></msub><mo>=</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>R</mi><mi>k</mi></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-895" style="width: 5.996em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.921em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.822em 1004.92em 3.998em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-896"><span class="msubsup" id="MathJax-Span-897"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-898" style="font-family: STIXGeneral; font-style: italic;">R</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-899" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-900" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="munderover" id="MathJax-Span-901" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(2.871em 1001.18em 4.613em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-902" style="font-family: STIXSizeOneSym; vertical-align: -0.51em;">∑</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.08em 4.255em -999.997em); top: -2.866em; left: 0.054em;"><span class="texatom" id="MathJax-Span-903"><span class="mrow" id="MathJax-Span-904"><span class="mi" id="MathJax-Span-905" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-906" style="font-size: 70.7%; font-family: STIXGeneral;">=</span><span class="mn" id="MathJax-Span-907" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.435em 1000.36em 4.152em -999.997em); top: -5.172em; left: 0.464em;"><span class="mi" id="MathJax-Span-908" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-909" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-910" style="font-family: STIXGeneral; font-style: italic;">R</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-911" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.622em; border-left: 0px solid; width: 0px; height: 3.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>R</mi><mi>T</mi></msub><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>R</mi><mi>k</mi></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-103">R_T = \sum _{k=1} ^n R_k</script></span></p><h3 id="current-in-series-circuits"><a class="anchor hidden-xs" href="#current-in-series-circuits" title="current-in-series-circuits"><span class="octicon octicon-link"></span></a>Current in series circuits</h3><p>The amperage at any point in a series circuit is the same</p><h3 id="voltage-in-series-circuits"><a class="anchor hidden-xs" href="#voltage-in-series-circuits" title="voltage-in-series-circuits"><span class="octicon octicon-link"></span></a>Voltage in series circuits</h3><p>The voltage applied in a series circuit is equal to the sum of the individual voltage drops</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-104-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>V</mi><mi>T</mi></msub><mo>=</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>V</mi><mi>k</mi></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-912" style="width: 5.996em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.921em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.822em 1004.92em 3.998em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-913"><span class="msubsup" id="MathJax-Span-914"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-915" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-916" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-917" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="munderover" id="MathJax-Span-918" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(2.871em 1001.18em 4.613em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-919" style="font-family: STIXSizeOneSym; vertical-align: -0.51em;">∑</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.08em 4.255em -999.997em); top: -2.866em; left: 0.054em;"><span class="texatom" id="MathJax-Span-920"><span class="mrow" id="MathJax-Span-921"><span class="mi" id="MathJax-Span-922" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-923" style="font-size: 70.7%; font-family: STIXGeneral;">=</span><span class="mn" id="MathJax-Span-924" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.435em 1000.36em 4.152em -999.997em); top: -5.172em; left: 0.464em;"><span class="mi" id="MathJax-Span-925" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-926" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-927" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-928" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.622em; border-left: 0px solid; width: 0px; height: 3.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>V</mi><mi>T</mi></msub><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>V</mi><mi>k</mi></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-104">V_T = \sum _{k=1} ^n V_k</script></span></p><h2 id="parallel-circuits"><a class="anchor hidden-xs" href="#parallel-circuits" title="parallel-circuits"><span class="octicon octicon-link"></span></a>Parallel circuits</h2><h3 id="resistance-in-parallel-circuits"><a class="anchor hidden-xs" href="#resistance-in-parallel-circuits" title="resistance-in-parallel-circuits"><span class="octicon octicon-link"></span></a>Resistance in parallel circuits</h3><p>The reciprocal of the total resistance in a parallel circuit is equal to the sum of the reciprocals of the individual resistances of the resistors</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-105-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mn>1</mn><msub><mi>R</mi><mi>T</mi></msub></mfrac><mo>=</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><mn>1</mn><msub><mi>R</mi><mi>k</mi></msub></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-929" style="width: 6.816em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.587em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.822em 1005.59em 3.998em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-930"><span class="mfrac" id="MathJax-Span-931"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-932" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1001.13em 4.306em -999.997em); top: -3.327em; left: 50%; margin-left: -0.561em;"><span class="msubsup" id="MathJax-Span-933"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-934" style="font-family: STIXGeneral; font-style: italic;">R</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-935" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.23em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.232em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-936" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="munderover" id="MathJax-Span-937" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(2.871em 1001.18em 4.613em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-938" style="font-family: STIXSizeOneSym; vertical-align: -0.51em;">∑</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.08em 4.255em -999.997em); top: -2.866em; left: 0.054em;"><span class="texatom" id="MathJax-Span-939"><span class="mrow" id="MathJax-Span-940"><span class="mi" id="MathJax-Span-941" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-942" style="font-size: 70.7%; font-family: STIXGeneral;">=</span><span class="mn" id="MathJax-Span-943" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.435em 1000.36em 4.152em -999.997em); top: -5.172em; left: 0.464em;"><span class="mi" id="MathJax-Span-944" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mfrac" id="MathJax-Span-945" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-946" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1001.03em 4.306em -999.997em); top: -3.327em; left: 50%; margin-left: -0.51em;"><span class="msubsup" id="MathJax-Span-947"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.57em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-948" style="font-family: STIXGeneral; font-style: italic;">R</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-949" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.13em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.13em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.622em; border-left: 0px solid; width: 0px; height: 3.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mn>1</mn><msub><mi>R</mi><mi>T</mi></msub></mfrac><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfrac><mn>1</mn><msub><mi>R</mi><mi>k</mi></msub></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-105">\frac{1}{R_T} = \sum _{k=1} ^n \frac{1}{R_k}</script></span></p><h3 id="current-in-parallel-circuits"><a class="anchor hidden-xs" href="#current-in-parallel-circuits" title="current-in-parallel-circuits"><span class="octicon octicon-link"></span></a>Current in parallel circuits</h3><p>The sum of the currents in the branches of a parallel circuit are equal to the total current of the circuit</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-106-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>I</mi><mi>T</mi></msub><mo>=</mo><munderover><mo>&#x2211;</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>I</mi><mi>k</mi></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-950" style="width: 5.33em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.357em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.822em 1004.36em 3.998em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-951"><span class="msubsup" id="MathJax-Span-952"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-953" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.361em;"><span class="mi" id="MathJax-Span-954" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-955" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="munderover" id="MathJax-Span-956" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px;"><span style="position: absolute; clip: rect(2.871em 1001.18em 4.613em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-957" style="font-family: STIXSizeOneSym; vertical-align: -0.51em;">∑</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.08em 4.255em -999.997em); top: -2.866em; left: 0.054em;"><span class="texatom" id="MathJax-Span-958"><span class="mrow" id="MathJax-Span-959"><span class="mi" id="MathJax-Span-960" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-961" style="font-size: 70.7%; font-family: STIXGeneral;">=</span><span class="mn" id="MathJax-Span-962" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.435em 1000.36em 4.152em -999.997em); top: -5.172em; left: 0.464em;"><span class="mi" id="MathJax-Span-963" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-964" style="padding-left: 0.207em;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-965" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.361em;"><span class="mi" id="MathJax-Span-966" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.622em; border-left: 0px solid; width: 0px; height: 3.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>I</mi><mi>T</mi></msub><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>I</mi><mi>k</mi></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-106">I_T = \sum _{k=1} ^n I_k</script></span></p><h3 id="voltage-in-parallel-circuits"><a class="anchor hidden-xs" href="#voltage-in-parallel-circuits" title="voltage-in-parallel-circuits"><span class="octicon octicon-link"></span></a>Voltage in parallel circuits</h3><p>The voltage across each component s the same as the voltage of the battery.</p><h2 id="ammeters-and-voltmeters-in-circuits"><a class="anchor hidden-xs" href="#ammeters-and-voltmeters-in-circuits" title="ammeters-and-voltmeters-in-circuits"><span class="octicon octicon-link"></span></a>Ammeters and voltmeters in circuits</h2><h3 id="ammeters"><a class="anchor hidden-xs" href="#ammeters" title="ammeters"><span class="octicon octicon-link"></span></a>Ammeters</h3><p>Ammeters are always connected in series with components to measure the size of the current flowing through them. Ammeters are designed to have negligible resistance.</p><h3 id="voltmeters"><a class="anchor hidden-xs" href="#voltmeters" title="voltmeters"><span class="octicon octicon-link"></span></a>Voltmeters</h3><p>Voltmeters are always arranged in parallel with components in a circuit. They are designed to have close to infinite resistance so that none of the current flows through them.</p><h2 id="the-potential-divider"><a class="anchor hidden-xs" href="#the-potential-divider" title="the-potential-divider"><span class="octicon octicon-link"></span></a>The potential divider</h2><p><strong>Potential divider</strong> A circuit which uses two resistors to split up or divide the voltage of the supply in a chosen ratio</p><p><img src="http://www.allaboutcircuits.com/uploads/thumbnails/voltage-divider-main-circuit.png" alt=""></p><p>The size of the output voltage is given by</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-107-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>V</mi><mrow class="MJX-TeXAtom-ORD"><mtext>out</mtext></mrow></msub><mo>=</mo><msub><mi>V</mi><mrow class="MJX-TeXAtom-ORD"><mtext>in</mtext></mrow></msub><mo>&#x00D7;</mo><mfrac><msub><mi>R</mi><mn>2</mn></msub><mrow><msub><mi>R</mi><mn>1</mn></msub><mo>+</mo><msub><mi>R</mi><mn>2</mn></msub></mrow></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-967" style="width: 10.812em; 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mode=display" id="MathJax-Element-107">V_{\text{out}} = V_{\text{in}} \times \frac{R_2}{R_1 + R_2}</script></span></p><p>The ratio of the potential differences across each of the resistors is equal to the ratio of the resistances of the resistors.</p><h3 id="using-a-variable-resistor"><a class="anchor hidden-xs" href="#using-a-variable-resistor" title="using-a-variable-resistor"><span class="octicon octicon-link"></span></a>Using a variable resistor</h3><p>The potential divider can be used with a variable resistor across the output voltage, allowing <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-108-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mrow class="MJX-TeXAtom-ORD"><mtext>out</mtext></mrow></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-993" style="width: 1.949em; 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display: none;"></span><span class="MathJax" id="MathJax-Element-109-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mrow class="MJX-TeXAtom-ORD"><mtext>out</mtext></mrow></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1000" style="width: 1.949em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.591em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.59em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1001"><span class="msubsup" id="MathJax-Span-1002"><span style="display: inline-block; position: relative; width: 1.591em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1003" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="texatom" id="MathJax-Span-1004"><span class="mrow" id="MathJax-Span-1005"><span class="mtext" id="MathJax-Span-1006" style="font-size: 70.7%; font-family: STIXGeneral;">out</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mrow class="MJX-TeXAtom-ORD"><mtext>out</mtext></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-109">V_{\text{out}}</script></span> the circuit can be used to control a heater on an on/off cycle.</p><h3 id="using-a-light-dependent-resistor-in-an-illumination-control-circuit"><a class="anchor hidden-xs" href="#using-a-light-dependent-resistor-in-an-illumination-control-circuit" title="using-a-light-dependent-resistor-in-an-illumination-control-circuit"><span class="octicon octicon-link"></span></a>Using a light-dependent resistor in an illumination control circuit</h3><p>An LDR has a high resistance when the incident light intensity is very low. As the intensity of the falling light increases the resistance falls.<br>
This means that the LDR can be used to switch lights on as the incident light decreases.</p><h2 id="internal-resistance"><a class="anchor hidden-xs" href="#internal-resistance" title="internal-resistance"><span class="octicon octicon-link"></span></a>Internal resistance</h2><p><strong>Internal resistance</strong> The internal resistance of a source of e.m.f is the resistance to electric current of the materials inside a cell</p><p><strong>Terminal p.d</strong> The potential difference recorded across the terminals of a cell.</p><p><strong>Lost volts</strong> The difference between e.m.f and the terminal p.d when charge flows in a cell</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-110-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="script">E</mi></mrow></mtd><mtd><mi></mi><mo>=</mo><mi>I</mi><mi>R</mi><mo>+</mo><mi>I</mi><mi>r</mi></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><mi>R</mi><mo>+</mo><mi>r</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><msub><mi>V</mi><mi>R</mi></msub><mo>+</mo><mi>I</mi><mi>r</mi></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1007" style="width: 6.919em; 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padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-1034" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1035" style="font-family: STIXGeneral;">)</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1004.41em 4.306em -999.997em); top: -2.61em; left: 0em;"><span class="mtd" id="MathJax-Span-1038"><span class="mrow" id="MathJax-Span-1039"><span class="mi" id="MathJax-Span-1040"></span><span class="mo" id="MathJax-Span-1041" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-1042" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1043" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-1044" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">R</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1045" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-1046" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1047" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.46em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.997em; border-left: 0px solid; width: 0px; height: 4.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="script">E</mi></mrow></mtd><mtd><mi></mi><mo>=</mo><mi>I</mi><mi>R</mi><mo>+</mo><mi>I</mi><mi>r</mi></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mi>I</mi><mo stretchy="false">(</mo><mi>R</mi><mo>+</mo><mi>r</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><msub><mi>V</mi><mi>R</mi></msub><mo>+</mo><mi>I</mi><mi>r</mi></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-110">\begin{align} \mathscr{E} &= IR + Ir \\ &= I(R + r) \\ &= V_R + Ir\end{align}</script></span></p><p>So when no current flows, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-111-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mn>0</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1048" style="width: 2.718em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.205em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.21em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1049"><span class="mi" id="MathJax-Span-1050" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1051" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-1052" style="font-family: STIXGeneral; padding-left: 0.31em;">0</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-111">I = 0</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-112-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B5;</mi><mo>=</mo><msub><mi>V</mi><mi>R</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1053" style="width: 3.537em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.871em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.87em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1054"><span class="mi" id="MathJax-Span-1055" style="font-family: STIXGeneral; font-style: italic;">ε</span><span class="mo" id="MathJax-Span-1056" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-1057" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1058" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-1059" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">R</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ε</mi><mo>=</mo><msub><mi>V</mi><mi>R</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-112">\varepsilon = V_R</script></span></p><p>The e.m.f and internal resistance of a cell are constant values. It is therefore possible to have a situation in which different currents will flow for different load resistances.<br>
The values can be solved using simultaneous equations.</p><h3 id="finding-the-internal-resistance-of-a-cell"><a class="anchor hidden-xs" href="#finding-the-internal-resistance-of-a-cell" title="finding-the-internal-resistance-of-a-cell"><span class="octicon octicon-link"></span></a>Finding the internal resistance of a cell</h3><p>I we rearrange <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-113-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B5;</mi><mo>=</mo><mi>I</mi><mi>R</mi><mo>+</mo><mi>I</mi><mi>r</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1060" style="width: 5.791em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.716em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1004.72em 2.769em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1061"><span class="mi" id="MathJax-Span-1062" style="font-family: STIXGeneral; font-style: italic;">ε</span><span class="mo" id="MathJax-Span-1063" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1064" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1065" style="font-family: STIXGeneral; font-style: italic;">R</span><span class="mo" id="MathJax-Span-1066" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-1067" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1068" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.122em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ε</mi><mo>=</mo><mi>I</mi><mi>R</mi><mo>+</mo><mi>I</mi><mi>r</mi></math></span></span><script type="math/tex" id="MathJax-Element-113">\varepsilon = IR + Ir</script></span> and substitute <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-114-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>R</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1069" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1070"><span class="msubsup" id="MathJax-Span-1071"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1072" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-1073" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">R</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>R</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-114">V_R</script></span> for <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-115-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mi>R</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1074" style="width: 1.181em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.98em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1075"><span class="mi" id="MathJax-Span-1076" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1077" style="font-family: STIXGeneral; font-style: italic;">R</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mi>R</mi></math></span></span><script type="math/tex" id="MathJax-Element-115">IR</script></span> we have <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-116-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>R</mi></msub><mo>=</mo><mo>&#x2212;</mo><mi>I</mi><mi>r</mi><mo>+</mo><mi>&#x03B5;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1078" style="width: 6.765em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.535em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1005.53em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1079"><span class="msubsup" id="MathJax-Span-1080"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1081" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-1082" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">R</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1083" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mo" id="MathJax-Span-1084" style="font-family: STIXGeneral; padding-left: 0.31em;">−</span><span class="mi" id="MathJax-Span-1085" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1086" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1087" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-1088" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">ε</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>R</mi></msub><mo>=</mo><mo>−</mo><mi>I</mi><mi>r</mi><mo>+</mo><mi>ε</mi></math></span></span><script type="math/tex" id="MathJax-Element-116">V_R = -Ir + \varepsilon</script></span> .</p><p>This equations has the form <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-117-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>c</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1089" style="width: 5.587em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.562em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.847em 1004.56em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1090"><span class="mi" id="MathJax-Span-1091" style="font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-1092" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1093" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">m</span><span class="mi" id="MathJax-Span-1094" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1095" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-1096" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">c</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>c</mi></math></span></span><script type="math/tex" id="MathJax-Element-117">y = mx + c</script></span>, so if we plot a graph of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-118-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1097" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1098"><span class="mi" id="MathJax-Span-1099" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math></span></span><script type="math/tex" id="MathJax-Element-118">V</script></span> against <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-119-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1100" style="width: 0.515em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1101"><span class="mi" id="MathJax-Span-1102" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math></span></span><script type="math/tex" id="MathJax-Element-119">I</script></span> we will have a straight line with negative gradient and positive intercepts.<br>
The <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-120-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1103" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1104"><span class="mi" id="MathJax-Span-1105" style="font-family: STIXGeneral; font-style: italic;">y</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></span></span><script type="math/tex" id="MathJax-Element-120">y</script></span> intercept will be equal to the e.m.f of the cell.<br>
The gradient of the graph equals the internal resistance of the cell.</p><h2 id="circuit-analysis"><a class="anchor hidden-xs" href="#circuit-analysis" title="circuit-analysis"><span class="octicon octicon-link"></span></a>Circuit analysis</h2><h3 id="analysis-of-series-and-parallel-hybrid-circuits"><a class="anchor hidden-xs" href="#analysis-of-series-and-parallel-hybrid-circuits" title="analysis-of-series-and-parallel-hybrid-circuits"><span class="octicon octicon-link"></span></a>Analysis of series and parallel hybrid circuits</h3><p>Many circuits have a parallel branch that is connected in series with other components.<br>
It is helpful to calculate the resistance of the parallel network first, before adding it to the resistance of the other components that are connected in series with it.</p><h3 id="series-circuits-involving-two-sources-of-emf"><a class="anchor hidden-xs" href="#series-circuits-involving-two-sources-of-emf" title="series-circuits-involving-two-sources-of-emf"><span class="octicon octicon-link"></span></a>Series circuits involving two sources of e.m.f</h3><p>If two sources of e.m.f have opposite polarities the overall e.m.f can be found by taking one source to be the negative direction and summing them.</p><h3 id="parallel-circuits-involving-two-sources-of-emf"><a class="anchor hidden-xs" href="#parallel-circuits-involving-two-sources-of-emf" title="parallel-circuits-involving-two-sources-of-emf"><span class="octicon octicon-link"></span></a>Parallel circuits involving two sources of e.m.f</h3><p>The overall e.m.f available to the circuit can be found in the same manner as for a series circuit.</p><p>To calculate the combined effective resistance of the network, arrange as would normally be done for a parallel circuit.</p><h2 id="wave-motion"><a class="anchor hidden-xs" href="#wave-motion" title="wave-motion"><span class="octicon octicon-link"></span></a>Wave motion</h2><p>Waves transfer energy from one place to another without any net transfer of matter.</p><p>Waves can be longitudinal or transverse.<br>
In longitudinal waves the vibrations are parallel to the direction of energy transfer, resulting in the formation of compressions and rarefactions.<br>
In transverse waves the vibrations are perpendicular to the direction of energy transfer.</p><p><strong>Progressive waves</strong> Waves which transfer energy away from a source</p><h2 id="wave-terminology"><a class="anchor hidden-xs" href="#wave-terminology" title="wave-terminology"><span class="octicon octicon-link"></span></a>Wave terminology</h2><p><strong>Wavelength</strong> The wavelength, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-121-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi><mo stretchy="false">(</mo><mtext>&#xA0;m</mtext><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1106" style="width: 2.666em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.1em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1107"><span class="mi" id="MathJax-Span-1108" style="font-family: STIXGeneral; font-style: italic;">λ</span><span class="mo" id="MathJax-Span-1109" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-1110" style="font-family: STIXGeneral;"> m</span><span class="mo" id="MathJax-Span-1111" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo stretchy="false">(</mo><mtext> m</mtext><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-121">\lambda (\text{ m} )</script></span> of a wave is the distance between two successive identical points that have the same pattern of oscillation.</p><p><strong>Period</strong> The period, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-122-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mtext>&#xA0;</mtext><mo stretchy="false">(</mo><mtext>s</mtext><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1112" style="width: 2.41em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.949em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.9em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1113"><span class="mi" id="MathJax-Span-1114" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mtext" id="MathJax-Span-1115" style="font-family: STIXGeneral;"> </span><span class="mo" id="MathJax-Span-1116" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-1117" style="font-family: STIXGeneral;">s</span><span class="mo" id="MathJax-Span-1118" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mtext> </mtext><mo stretchy="false">(</mo><mtext>s</mtext><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-122">T\ (\text{s})</script></span>, is the time taken for the wave to complete one pattern of oscillation</p><p><strong>Frequency</strong> The frequency, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-123-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mtext>&#xA0;</mtext><mo stretchy="false">(</mo><mtext>Hz</mtext><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1119" style="width: 3.076em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.513em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1120"><span class="mi" id="MathJax-Span-1121" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mtext" id="MathJax-Span-1122" style="font-family: STIXGeneral;"> </span><span class="mo" id="MathJax-Span-1123" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-1124" style="font-family: STIXGeneral;">Hz</span><span class="mo" id="MathJax-Span-1125" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mtext> </mtext><mo stretchy="false">(</mo><mtext>Hz</mtext><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-123">f\ (\text{Hz})</script></span>, is the number of oscillations per unit time at any point. It is the inverse of the period of the wave</p><p><strong>Displacement</strong> The displacement, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-124-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mtext>&#xA0;</mtext><mo stretchy="false">(</mo><mtext>m</mtext><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1126" style="width: 2.666em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.1em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1127"><span class="mi" id="MathJax-Span-1128" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mtext" id="MathJax-Span-1129" style="font-family: STIXGeneral;"> </span><span class="mo" id="MathJax-Span-1130" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-1131" style="font-family: STIXGeneral;">m</span><span class="mo" id="MathJax-Span-1132" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mtext> </mtext><mo stretchy="false">(</mo><mtext>m</mtext><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-124">x\ (\text{m})</script></span>, is the distance of any part of the wave from its mean position</p><p><strong>Amplitude</strong> The amplitude, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-125-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>0</mn></msub><mtext>&#xA0;</mtext><mo stretchy="false">(</mo><mtext>m</mtext><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1133" style="width: 3.128em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.564em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.51em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1134"><span class="msubsup" id="MathJax-Span-1135"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1136" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-1137" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mtext" id="MathJax-Span-1138" style="font-family: STIXGeneral;"> </span><span class="mo" id="MathJax-Span-1139" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-1140" style="font-family: STIXGeneral;">m</span><span class="mo" id="MathJax-Span-1141" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>0</mn></msub><mtext> </mtext><mo stretchy="false">(</mo><mtext>m</mtext><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-125">x_0\ (\text{m})</script></span>, is the maximum displacement that the wave reaches from its mean position</p><p><strong>Phase difference</strong> The phase difference, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-126-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03C6;</mi><mtext>&#xA0;</mtext><mo stretchy="false">(</mo><mtext>rad</mtext><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1142" style="width: 3.435em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.82em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.77em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1143"><span class="mi" id="MathJax-Span-1144" style="font-family: STIXGeneral; font-style: italic;">φ</span><span class="mtext" id="MathJax-Span-1145" style="font-family: STIXGeneral;"> </span><span class="mo" id="MathJax-Span-1146" style="font-family: STIXGeneral;">(</span><span class="mtext" id="MathJax-Span-1147" style="font-family: STIXGeneral;">rad</span><span class="mo" id="MathJax-Span-1148" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>φ</mi><mtext> </mtext><mo stretchy="false">(</mo><mtext>rad</mtext><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-126">\varphi\ (\text{rad})</script></span>, the difference in the pattern of vibration between two points in the wave. Two points that have exactly the same pattern of oscillation are said to be in phase, there is zero phase difference between them</p><h2 id="using-an-oscilloscope"><a class="anchor hidden-xs" href="#using-an-oscilloscope" title="using-an-oscilloscope"><span class="octicon octicon-link"></span></a>Using an oscilloscope</h2><p>An oscilloscope displays a voltage-time signal and can be used as a voltmeter to display and measure the output from a microphone or signal generator.</p><p>The frequency of a wave displayed on an oscilloscope can be determined if you know the settings of the time base on the oscilloscope.<br>
This is the time taken for the luminous dot produced by the CRT to move a horizontal distance of 1cm across the oscilloscope’s screen.</p><h3 id="oscilloscope-controls"><a class="anchor hidden-xs" href="#oscilloscope-controls" title="oscilloscope-controls"><span class="octicon octicon-link"></span></a>Oscilloscope controls</h3><p>Each horizontal division on the oscilloscope represents a unit of time. The time base control varies the seconds or milliseconds per division.<br>
The uncertainty in the frequency measurement can be reduced by altering the time base such that one full wave has the widest possible range in the x direction.</p><p>If the time base is turned off, the spot no longer moves across the screen. This is useful for observing the intensity of the wave.</p><p>Each vertical division on the oscilloscope screen represents a unit of voltage. The sensitivity control varies the volts per division.</p><h2 id="wave-speed-and-the-wave-equation"><a class="anchor hidden-xs" href="#wave-speed-and-the-wave-equation" title="wave-speed-and-the-wave-equation"><span class="octicon octicon-link"></span></a>Wave speed and the wave equation</h2><h3 id="the-wave-equation"><a class="anchor hidden-xs" href="#the-wave-equation" title="the-wave-equation"><span class="octicon octicon-link"></span></a>The wave equation</h3><p>$\text{speed} = \frac{\text{distance}}{\text{time}} $ can also be applied to movement</p><p>In a time equal to one period, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-127-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1149" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1150"><span class="mi" id="MathJax-Span-1151" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-127">T</script></span>, the wave travels one wavelength, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-128-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1152" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1153"><span class="mi" id="MathJax-Span-1154" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-128">\lambda</script></span>. Therefore the speed, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-129-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1155" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1156"><span class="mi" id="MathJax-Span-1157" style="font-family: STIXGeneral; font-style: italic;">v</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math></span></span><script type="math/tex" id="MathJax-Element-129">v</script></span>, of the wave is</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-130-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>v</mi><mo>=</mo><mfrac><mi>&#x03BB;</mi><mi>T</mi></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1158" style="width: 3.332em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.718em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1002.72em 3.384em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1159"><span class="mi" id="MathJax-Span-1160" style="font-family: STIXGeneral; font-style: italic;">v</span><span class="mo" id="MathJax-Span-1161" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1162" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1163" style="font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.305em;"><span class="mi" id="MathJax-Span-1164" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.77em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.771em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.872em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>v</mi><mo>=</mo><mfrac><mi>λ</mi><mi>T</mi></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-130">v = \frac{\lambda}{T}</script></span></p><p>We know that the frequency is the reciprocal of the period, so this can also be written</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-131-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>v</mi><mo>=</mo><mi>f</mi><mo>&#x00D7;</mo><mi>&#x03BB;</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1165" style="width: 4.664em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.794em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.74em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1166"><span class="mi" id="MathJax-Span-1167" style="font-family: STIXGeneral; font-style: italic;">v</span><span class="mo" id="MathJax-Span-1168" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1169" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-1170" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mi" id="MathJax-Span-1171" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>v</mi><mo>=</mo><mi>f</mi><mo>×</mo><mi>λ</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-131">v = f \times \lambda</script></span></p><h3 id="energy-transfer-by-waves"><a class="anchor hidden-xs" href="#energy-transfer-by-waves" title="energy-transfer-by-waves"><span class="octicon octicon-link"></span></a>Energy transfer by waves</h3><p>A progressive wave transfers energy from one place to another. From a point source producing waves, this energy spreads out in all directions.</p><p><strong>Intensity</strong> Power per unit area <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-132-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>W m</mtext><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1172" style="width: 4.306em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.537em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.49em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1173"><span class="mo" id="MathJax-Span-1174" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-1175"><span style="display: inline-block; position: relative; width: 2.871em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1001.95em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mtext" id="MathJax-Span-1176" style="font-family: STIXGeneral;">W m</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.949em;"><span class="texatom" id="MathJax-Span-1177"><span class="mrow" id="MathJax-Span-1178"><span class="mo" id="MathJax-Span-1179" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1180" style="font-size: 70.7%; font-family: STIXGeneral;">2</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1181" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>W m</mtext><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-132">(\text{W m}^{-2})</script></span></p><p>The intensity is given by</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-133-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mtext>intensity</mtext><mo>=</mo><mfrac><mfrac><mtext>energy</mtext><mtext>time</mtext></mfrac><mtext>area</mtext></mfrac></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mtext>power</mtext><mtext>area</mtext></mfrac></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1182" style="width: 9.378em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.687em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-0.51em 1007.53em 4.408em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1183"><span class="mtable" id="MathJax-Span-1184" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 7.38em; height: 0px;"><span style="position: absolute; clip: rect(2.923em 1007.38em 7.841em -999.997em); top: -5.633em; left: 0em;"><span style="display: inline-block; position: relative; width: 7.38em; height: 0px;"><span style="position: absolute; clip: rect(2.103em 1007.38em 4.869em -999.997em); top: -4.813em; right: 0em;"><span class="mtd" id="MathJax-Span-1185"><span class="mrow" id="MathJax-Span-1186"><span class="mtext" id="MathJax-Span-1187" style="font-family: STIXGeneral;">intensity</span><span class="mo" id="MathJax-Span-1188" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1189" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.41em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(2.974em 1002.31em 4.562em -999.997em); top: -4.864em; left: 50%; margin-left: -1.124em;"><span class="mfrac" id="MathJax-Span-1190"><span style="display: inline-block; position: relative; width: 2.052em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.537em 1001.9em 4.306em -999.997em); top: -4.557em; left: 50%; margin-left: -0.971em;"><span class="mtext" id="MathJax-Span-1191" style="font-size: 70.7%; font-family: STIXGeneral;">energy</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.23em 4.152em -999.997em); top: -3.584em; left: 50%; margin-left: -0.612em;"><span class="mtext" id="MathJax-Span-1192" style="font-size: 70.7%; font-family: STIXGeneral;">time</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.05em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.052em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.64em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.817em;"><span class="mtext" id="MathJax-Span-1193" style="font-family: STIXGeneral;">area</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.41em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.41em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.718em 1003.85em 4.869em -999.997em); top: -2.61em; right: 0em;"><span class="mtd" id="MathJax-Span-1194"><span class="mrow" id="MathJax-Span-1195"><span class="mo" id="MathJax-Span-1196" style="font-family: STIXGeneral;">=</span><span class="mfrac" id="MathJax-Span-1197" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.615em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1002.51em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -1.227em;"><span class="mtext" id="MathJax-Span-1198" style="font-family: STIXGeneral;">power</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.64em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.817em;"><span class="mtext" id="MathJax-Span-1199" style="font-family: STIXGeneral;">area</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.615em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.638em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.559em; border-left: 0px solid; width: 0px; height: 5.753em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mtext>intensity</mtext><mo>=</mo><mfrac><mfrac><mtext>energy</mtext><mtext>time</mtext></mfrac><mtext>area</mtext></mfrac></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mtext>power</mtext><mtext>area</mtext></mfrac></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-133">\begin{align} \text{intensity} = \frac{\frac{\text{energy}}{\text{time}}}{\text{area}} \\ = \frac{\text{power}}{\text{area}}\end{align}</script></span></p><p>If the power spreads out equally in all directions this can be written</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-134-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>intensity</mtext><mo>=</mo><mfrac><mtext>power</mtext><mrow><mn>4</mn><mi>&#x03C0;</mi><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1200" style="width: 9.275em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.585em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.283em 1007.59em 3.486em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1201"><span class="mtext" id="MathJax-Span-1202" style="font-family: STIXGeneral;">intensity</span><span class="mo" id="MathJax-Span-1203" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1204" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 2.615em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1002.51em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -1.227em;"><span class="mtext" id="MathJax-Span-1205" style="font-family: STIXGeneral;">power</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.076em 1001.95em 4.152em -999.997em); top: -3.276em; left: 50%; margin-left: -0.971em;"><span class="mrow" id="MathJax-Span-1206"><span class="mn" id="MathJax-Span-1207" style="font-family: STIXGeneral;">4</span><span class="mi" id="MathJax-Span-1208" style="font-family: STIXGeneral; font-style: italic;">π<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="msubsup" id="MathJax-Span-1209"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1210" style="font-family: STIXGeneral; font-style: italic;">r<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.301em; left: 0.464em;"><span class="mn" id="MathJax-Span-1211" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1002.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 2.615em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.997em; border-left: 0px solid; width: 0px; height: 2.441em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>intensity</mtext><mo>=</mo><mfrac><mtext>power</mtext><mrow><mn>4</mn><mi>π</mi><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-134">\text{intensity} = \frac{\text{power}}{4\pi r^2}</script></span></p><h3 id="intensity-and-amplitude"><a class="anchor hidden-xs" href="#intensity-and-amplitude" title="intensity-and-amplitude"><span class="octicon octicon-link"></span></a>Intensity and amplitude</h3><p>The amplitude of a wave decreases as the wave spreads out from a source.<br>
The energy of a wave is proportional to the square of its amplitude. Hence the intensity of a progressive wave is also proportional to the square of its amplitude. $I \propto A^2 $</p><h2 id="common-properties-of-waves"><a class="anchor hidden-xs" href="#common-properties-of-waves" title="common-properties-of-waves"><span class="octicon octicon-link"></span></a>Common properties of waves</h2><h3 id="reflection"><a class="anchor hidden-xs" href="#reflection" title="reflection"><span class="octicon octicon-link"></span></a>Reflection</h3><p>All waves can be reflected.</p><p><strong>Ray</strong> A single line to illustrate the direction of a waves travel. Rays are always drawn at right angles to the wavefronts.</p><h3 id="refraction"><a class="anchor hidden-xs" href="#refraction" title="refraction"><span class="octicon octicon-link"></span></a>Refraction</h3><p>Refraction occurs when a wave moves from one material into another.<br>
In terms of refraction of light, we say that the two materials have different optical densities.</p><p>Two observations an be made when refraction occurs</p><ul>
<li>The wave will change its speed</li>
<li>There may be a change in direction</li>
</ul><h3 id="investigating-refraction-and-reflection-in-a-ripple-tank"><a class="anchor hidden-xs" href="#investigating-refraction-and-reflection-in-a-ripple-tank" title="investigating-refraction-and-reflection-in-a-ripple-tank"><span class="octicon octicon-link"></span></a>Investigating refraction and reflection in a ripple tank</h3><p>In a ripple tank, a motorised straight edge bar produces plane waves, while a small dipper produces circular waves.<br>
When light is shone from above through the waves produced, the bright bands or curves of light seen on on the screen below the tank show the wave crests. This makes it possible to measure the wavelength of the water waves, and investigate the angles of reflection and refraction.</p><p>Reflection at plane and curved surfaces can be investigated, and the angles of incidence and reflection measured with respect to the normal.</p><p>A glass sheet is used to decrease the water depth and so produce a region with a different wave speed. The water level can also be adjusted.<br>
If the separation of the wavefronts decreases this shows they are travelling more slowly. If the wavefronts are at non-zero angle when they cross the barrier, the waves also changes direction.</p><h3 id="diffraction"><a class="anchor hidden-xs" href="#diffraction" title="diffraction"><span class="octicon octicon-link"></span></a>Diffraction</h3><p><strong>Diffraction</strong> The spreading out of a wave after passing around an obstacle or through a gap</p><p>Diffraction is most pronounced when the wavelength of the wave being diffracted is the same size as the gap that they are travelling through.</p><h3 id="interference"><a class="anchor hidden-xs" href="#interference" title="interference"><span class="octicon octicon-link"></span></a>Interference</h3><p><strong>Interference</strong> The addition of two more more waves (superposition) that results in a new wave pattern</p><h2 id="electromagnetic-waves"><a class="anchor hidden-xs" href="#electromagnetic-waves" title="electromagnetic-waves"><span class="octicon octicon-link"></span></a>Electromagnetic waves</h2><h3 id="common-properties-of-electromagnetic-waves"><a class="anchor hidden-xs" href="#common-properties-of-electromagnetic-waves" title="common-properties-of-electromagnetic-waves"><span class="octicon octicon-link"></span></a>Common properties of electromagnetic waves</h3><p>The electromagnetic spectrum has a range of values for wavelength of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-135-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>16</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1212" style="width: 2.769em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1002.26em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1213"><span class="msubsup" id="MathJax-Span-1214"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1215" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1216"><span class="mrow" id="MathJax-Span-1217"><span class="mo" id="MathJax-Span-1218" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1219" style="font-size: 70.7%; font-family: STIXGeneral;">16</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>16</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-135">10^{-16}</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-136-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mn>4</mn></msup><mtext>&#xA0;m</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1220" style="width: 3.025em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.462em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1002.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1221"><span class="msubsup" id="MathJax-Span-1222"><span style="display: inline-block; position: relative; width: 1.437em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1223" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="mn" id="MathJax-Span-1224" style="font-size: 70.7%; font-family: STIXGeneral;">4</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mtext" id="MathJax-Span-1225" style="font-family: STIXGeneral;"> m</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mn>4</mn></msup><mtext> m</mtext></math></span></span><script type="math/tex" id="MathJax-Element-136">10^4 \text{ m}</script></span>.</p><p>Visible light is a small part of the electromagnetic spectrum ranging from approximately <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-137-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>370</mn><mtext>&#xA0;nm</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1226" style="width: 3.691em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.025em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.02em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1227"><span class="mn" id="MathJax-Span-1228" style="font-family: STIXGeneral;">370</span><span class="mtext" id="MathJax-Span-1229" style="font-family: STIXGeneral;"> nm</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>370</mn><mtext> nm</mtext></math></span></span><script type="math/tex" id="MathJax-Element-137">370 \text { nm}</script></span> for violet to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-138-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>740</mn><mtext>&#xA0;nm</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1230" style="width: 3.691em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.025em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.02em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1231"><span class="mn" id="MathJax-Span-1232" style="font-family: STIXGeneral;">740</span><span class="mtext" id="MathJax-Span-1233" style="font-family: STIXGeneral;"> nm</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>740</mn><mtext> nm</mtext></math></span></span><script type="math/tex" id="MathJax-Element-138">740 \text{ nm}</script></span> for red.</p><p>All electromagnetic waves share the following properties</p><ul>
<li>They can travel through a vacuum</li>
<li>They possess both a magnetic wave and an electrical wave interlocked perpendicular to each other</li>
<li>They travel at the speed of light in free space, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-139-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>2.98</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mn>8</mn></msup><msup><mtext>&#xA0;m s</mtext><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1234" style="width: 10.556em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.66em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1008.66em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1235"><span class="mi" id="MathJax-Span-1236" style="font-family: STIXGeneral; font-style: italic;">c</span><span class="mo" id="MathJax-Span-1237" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-1238" style="font-family: STIXGeneral; padding-left: 0.31em;">2.98</span><span class="mo" id="MathJax-Span-1239" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1240" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.437em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1241" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="mn" id="MathJax-Span-1242" style="font-size: 70.7%; font-family: STIXGeneral;">8</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="msubsup" id="MathJax-Span-1243"><span style="display: inline-block; position: relative; width: 2.564em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1001.64em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mtext" id="MathJax-Span-1244" style="font-family: STIXGeneral;"> m s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 1.693em;"><span class="texatom" id="MathJax-Span-1245"><span class="mrow" id="MathJax-Span-1246"><span class="mo" id="MathJax-Span-1247" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1248" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>2.98</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup><msup><mtext> m s</mtext><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-139">c = 2.98\times10^8\text{ m s}^{-1}</script></span></li>
<li>They are all transverse waves</li>
<li>They can all be reflected, refracted, and diffracted</li>
<li>They can all demonstrate interference</li>
<li>They can all be polarised</li>
</ul><table>
<thead>
<tr>
<th>Wavelength (m)</th>
<th>Frequency (Hz)</th>
<th>Method of production</th>
<th>Method of detection</th>
<th>Uses</th>
</tr>
</thead>
<tbody>
<tr>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-140-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1249" style="width: 2.308em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1001.9em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1250"><span class="msubsup" id="MathJax-Span-1251"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1252" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1253"><span class="mrow" id="MathJax-Span-1254"><span class="mo" id="MathJax-Span-1255" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1256" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-140">10^{-1}</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-141-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mn>4</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1257" style="width: 1.744em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.437em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1001.44em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1258"><span class="msubsup" id="MathJax-Span-1259"><span style="display: inline-block; position: relative; width: 1.437em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1260" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="mn" id="MathJax-Span-1261" style="font-size: 70.7%; font-family: STIXGeneral;">4</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mn>4</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-141">10^4</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-142-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mn>9</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1262" style="width: 3.794em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.076em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.08em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1263"><span class="mn" id="MathJax-Span-1264" style="font-family: STIXGeneral;">3</span><span class="mo" id="MathJax-Span-1265" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1266" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.437em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1267" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="mn" id="MathJax-Span-1268" style="font-size: 70.7%; font-family: STIXGeneral;">9</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><msup><mn>10</mn><mn>9</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-142">3\times10^9</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-143-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mn>4</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1269" style="width: 3.794em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.076em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.08em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1270"><span class="mn" id="MathJax-Span-1271" style="font-family: STIXGeneral;">3</span><span class="mo" id="MathJax-Span-1272" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1273" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.437em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1274" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="mn" id="MathJax-Span-1275" style="font-size: 70.7%; font-family: STIXGeneral;">4</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-143">3\times10^4</script></span></td>
<td>Electrons oscillated by electric fields in aerials</td>
<td>Resonance in electronic circuits</td>
<td>TV, Radio, and telecommunications</td>
</tr>
<tr>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-144-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>4</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1276" style="width: 2.308em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1001.9em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1277"><span class="msubsup" id="MathJax-Span-1278"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1279" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1280"><span class="mrow" id="MathJax-Span-1281"><span class="mo" id="MathJax-Span-1282" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1283" style="font-size: 70.7%; font-family: STIXGeneral;">4</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>4</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-144">10^{-4}</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-145-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1284" style="width: 2.308em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1001.9em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1285"><span class="msubsup" id="MathJax-Span-1286"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1287" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1288"><span class="mrow" id="MathJax-Span-1289"><span class="mo" id="MathJax-Span-1290" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1291" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-145">10^{-1}</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-146-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>12</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1292" style="width: 4.203em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.435em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.43em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1293"><span class="mn" id="MathJax-Span-1294" style="font-family: STIXGeneral;">3</span><span class="mo" id="MathJax-Span-1295" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1296" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1297" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1298"><span class="mrow" id="MathJax-Span-1299"><span class="mn" id="MathJax-Span-1300" style="font-size: 70.7%; font-family: STIXGeneral;">12</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>12</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-146">3\times10^{12}</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-147-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>9</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1301" style="width: 3.794em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.076em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.08em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1302"><span class="mn" id="MathJax-Span-1303" style="font-family: STIXGeneral;">3</span><span class="mo" id="MathJax-Span-1304" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1305" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.437em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1306" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1307"><span class="mrow" id="MathJax-Span-1308"><span class="mn" id="MathJax-Span-1309" style="font-size: 70.7%; font-family: STIXGeneral;">9</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>9</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-147">3\times10^{9}</script></span></td>
<td>Magnetron, klystron oscillators, using electrons to set up oscillations in a cavity</td>
<td>Heating effect, electronic circuits</td>
<td>Radar, mobile phones, microwave ovens, satellite navigation</td>
</tr>
<tr>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-148-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7.4</mn><msup><mo>&#x00D7;</mo><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>7</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1310" style="width: 3.384em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.769em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1002.77em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1311"><span class="mn" id="MathJax-Span-1312" style="font-family: STIXGeneral;">7.4</span><span class="msubsup" id="MathJax-Span-1313"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px;"><span style="position: absolute; clip: rect(3.332em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-1314" style="font-family: STIXGeneral;">×</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.617em;"><span class="texatom" id="MathJax-Span-1315"><span class="mrow" id="MathJax-Span-1316"><span class="mo" id="MathJax-Span-1317" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1318" style="font-size: 70.7%; font-family: STIXGeneral;">7</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7.4</mn><msup><mo>×</mo><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>7</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-148">7.4\times^{-7}</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-149-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>3</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1319" style="width: 2.308em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1001.9em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1320"><span class="msubsup" id="MathJax-Span-1321"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1322" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1323"><span class="mrow" id="MathJax-Span-1324"><span class="mo" id="MathJax-Span-1325" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1326" style="font-size: 70.7%; font-family: STIXGeneral;">3</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>3</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-149">10^{-3}</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-150-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>14</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1327" style="width: 4.203em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.435em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.43em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1328"><span class="mn" id="MathJax-Span-1329" style="font-family: STIXGeneral;">4</span><span class="mo" id="MathJax-Span-1330" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1331" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1332" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1333"><span class="mrow" id="MathJax-Span-1334"><span class="mn" id="MathJax-Span-1335" style="font-size: 70.7%; font-family: STIXGeneral;">14</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>14</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-150">4\times10^{14}</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-151-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>11</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1336" style="width: 4.203em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.435em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.43em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1337"><span class="mn" id="MathJax-Span-1338" style="font-family: STIXGeneral;">3</span><span class="mo" id="MathJax-Span-1339" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1340" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1341" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1342"><span class="mrow" id="MathJax-Span-1343"><span class="mn" id="MathJax-Span-1344" style="font-size: 70.7%; font-family: STIXGeneral;">11</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>11</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-151">3\times10^{11}</script></span></td>
<td>Oscillation of molecules, from all objects at any temperature</td>
<td>Photographic film, thermopile, heating of skin</td>
<td>Heaters, night vision, remote controls</td>
</tr>
<tr>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-152-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3.7</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>7</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1345" style="width: 4.716em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.845em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.85em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1346"><span class="mn" id="MathJax-Span-1347" style="font-family: STIXGeneral;">3.7</span><span class="mo" id="MathJax-Span-1348" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1349" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.437em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1350" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1351"><span class="mrow" id="MathJax-Span-1352"><span class="mn" id="MathJax-Span-1353" style="font-size: 70.7%; font-family: STIXGeneral;">7</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3.7</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>7</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-152">3.7\times10^{7}</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-153-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7.4</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>7</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1354" style="width: 5.279em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.306em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1004.31em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1355"><span class="mn" id="MathJax-Span-1356" style="font-family: STIXGeneral;">7.4</span><span class="mo" id="MathJax-Span-1357" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1358" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1359" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1360"><span class="mrow" id="MathJax-Span-1361"><span class="mo" id="MathJax-Span-1362" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1363" style="font-size: 70.7%; font-family: STIXGeneral;">7</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7.4</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>7</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-153">7.4\times10^{-7}</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-154-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>14</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1364" style="width: 4.203em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.435em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.43em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1365"><span class="mn" id="MathJax-Span-1366" style="font-family: STIXGeneral;">8</span><span class="mo" id="MathJax-Span-1367" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1368" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1369" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1370"><span class="mrow" id="MathJax-Span-1371"><span class="mn" id="MathJax-Span-1372" style="font-size: 70.7%; font-family: STIXGeneral;">14</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>14</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-154">8\times10^{14}</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-155-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>14</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1373" style="width: 4.203em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.435em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.43em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1374"><span class="mn" id="MathJax-Span-1375" style="font-family: STIXGeneral;">4</span><span class="mo" id="MathJax-Span-1376" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1377" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1378" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1379"><span class="mrow" id="MathJax-Span-1380"><span class="mn" id="MathJax-Span-1381" style="font-size: 70.7%; font-family: STIXGeneral;">14</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>14</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-155">4\times10^{14}</script></span></td>
<td>From high temperature solids, gases, and lasers</td>
<td>Photographic film, the retina</td>
<td>Sight, communication</td>
</tr>
<tr>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-156-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>9</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1382" style="width: 2.308em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1001.9em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1383"><span class="msubsup" id="MathJax-Span-1384"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1385" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1386"><span class="mrow" id="MathJax-Span-1387"><span class="mo" id="MathJax-Span-1388" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1389" style="font-size: 70.7%; font-family: STIXGeneral;">9</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>9</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-156">10^{-9}</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-157-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3.7</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>7</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1390" style="width: 5.279em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.306em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1004.31em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1391"><span class="mn" id="MathJax-Span-1392" style="font-family: STIXGeneral;">3.7</span><span class="mo" id="MathJax-Span-1393" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1394" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1395" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1396"><span class="mrow" id="MathJax-Span-1397"><span class="mo" id="MathJax-Span-1398" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1399" style="font-size: 70.7%; font-family: STIXGeneral;">7</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3.7</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>7</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-157">3.7\times10^{-7}</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-158-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>17</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1400" style="width: 4.203em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.435em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.43em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1401"><span class="mn" id="MathJax-Span-1402" style="font-family: STIXGeneral;">3</span><span class="mo" id="MathJax-Span-1403" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1404" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1405" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1406"><span class="mrow" id="MathJax-Span-1407"><span class="mn" id="MathJax-Span-1408" style="font-size: 70.7%; font-family: STIXGeneral;">17</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>17</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-158">3\times10^{17}</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-159-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>14</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1409" style="width: 4.203em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.435em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.43em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1410"><span class="mn" id="MathJax-Span-1411" style="font-family: STIXGeneral;">8</span><span class="mo" id="MathJax-Span-1412" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1413" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1414" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1415"><span class="mrow" id="MathJax-Span-1416"><span class="mn" id="MathJax-Span-1417" style="font-size: 70.7%; font-family: STIXGeneral;">14</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>14</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-159">8\times10^{14}</script></span></td>
<td>From high temperature solids, gases, and lasers</td>
<td>Photographic film, phosphors, sunburn</td>
<td>Disco lights, tanning studios, counterfiet detection, detergents</td>
</tr>
<tr>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-160-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>12</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1418" style="width: 2.769em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1002.26em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1419"><span class="msubsup" id="MathJax-Span-1420"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1421" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1422"><span class="mrow" id="MathJax-Span-1423"><span class="mo" id="MathJax-Span-1424" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1425" style="font-size: 70.7%; font-family: STIXGeneral;">12</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>12</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-160">10^{-12}</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-161-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>7</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1426" style="width: 2.308em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1001.9em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1427"><span class="msubsup" id="MathJax-Span-1428"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1429" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1430"><span class="mrow" id="MathJax-Span-1431"><span class="mo" id="MathJax-Span-1432" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1433" style="font-size: 70.7%; font-family: STIXGeneral;">7</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>7</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-161">10^{-7}</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-162-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>20</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1434" style="width: 4.203em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.435em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.43em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1435"><span class="mn" id="MathJax-Span-1436" style="font-family: STIXGeneral;">3</span><span class="mo" id="MathJax-Span-1437" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1438" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1439" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1440"><span class="mrow" id="MathJax-Span-1441"><span class="mn" id="MathJax-Span-1442" style="font-size: 70.7%; font-family: STIXGeneral;">20</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>20</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-162">3\times10^{20}</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-163-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>15</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1443" style="width: 4.203em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.435em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.43em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1444"><span class="mn" id="MathJax-Span-1445" style="font-family: STIXGeneral;">3</span><span class="mo" id="MathJax-Span-1446" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1447" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1448" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1449"><span class="mrow" id="MathJax-Span-1450"><span class="mn" id="MathJax-Span-1451" style="font-size: 70.7%; font-family: STIXGeneral;">15</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>15</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-163">3\times10^{15}</script></span></td>
<td>Bombarding metals with high-energy electrons</td>
<td>Photographic film, fluorescence</td>
<td>Computer-aided tomography scans, X-ray photography, crystal structure analysis</td>
</tr>
<tr>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-164-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>16</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1452" style="width: 2.769em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1002.26em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1453"><span class="msubsup" id="MathJax-Span-1454"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1455" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1456"><span class="mrow" id="MathJax-Span-1457"><span class="mo" id="MathJax-Span-1458" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1459" style="font-size: 70.7%; font-family: STIXGeneral;">16</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>16</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-164">10^{-16}</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-165-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>9</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1460" style="width: 2.308em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.181em 1001.9em 2.359em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1461"><span class="msubsup" id="MathJax-Span-1462"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1463" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1464"><span class="mrow" id="MathJax-Span-1465"><span class="mo" id="MathJax-Span-1466" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1467" style="font-size: 70.7%; font-family: STIXGeneral;">9</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>9</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-165">10^{-9}</script></span></td>
<td><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-166-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>24</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1468" style="width: 4.203em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.435em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.43em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1469"><span class="mn" id="MathJax-Span-1470" style="font-family: STIXGeneral;">3</span><span class="mo" id="MathJax-Span-1471" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1472" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1473" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1474"><span class="mrow" id="MathJax-Span-1475"><span class="mn" id="MathJax-Span-1476" style="font-size: 70.7%; font-family: STIXGeneral;">24</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>24</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-166">3\times10^{24}</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-167-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>17</mn></mrow></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1477" style="width: 4.203em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.435em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.43em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1478"><span class="mn" id="MathJax-Span-1479" style="font-family: STIXGeneral;">3</span><span class="mo" id="MathJax-Span-1480" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1481" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 1.796em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1482" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1483"><span class="mrow" id="MathJax-Span-1484"><span class="mn" id="MathJax-Span-1485" style="font-size: 70.7%; font-family: STIXGeneral;">17</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mn>17</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-167">3\times10^{17}</script></span></td>
<td>Nuclear decay or in a nuclear accelerator</td>
<td>Photographic film, Geiger tube</td>
<td>Diagnosis and cancer treatment</td>
</tr>
</tbody>
</table><h3 id="non-ionising-electromagnetic-radiation"><a class="anchor hidden-xs" href="#non-ionising-electromagnetic-radiation" title="non-ionising-electromagnetic-radiation"><span class="octicon octicon-link"></span></a>Non-ionising electromagnetic radiation</h3><p>Radio waves, microwaves, and visible light do not have enough photon energy to remove electrons from the shells of atoms. For this reason they are deemed to be relatively safe.</p><h3 id="ionising-electromagnetic-radiation"><a class="anchor hidden-xs" href="#ionising-electromagnetic-radiation" title="ionising-electromagnetic-radiation"><span class="octicon octicon-link"></span></a>Ionising electromagnetic radiation</h3><p>Ultraviolet rays, X-rays, and gamma radiation all have high photon energies. This means that they can cause ionisation.<br>
This can lead to human cell mutation and is therefore dangerous.</p><p>Ultraviolet radiation emitted by the Sun is often divided into three regions</p><ul>
<li><strong>UV-A</strong> 315-400nm causes tanning when skin is exposed. Accounts for 99% of UV light</li>
<li><strong>UV-B</strong> 280-315nm causes damage such as sunburn and skin cancer</li>
<li><strong>UV-C</strong> 100-280nm filtered out by the atmosphere and doesn’t reach Earth</li>
</ul><h3 id="production-of-x-rays"><a class="anchor hidden-xs" href="#production-of-x-rays" title="production-of-x-rays"><span class="octicon octicon-link"></span></a>Production of X-Rays</h3><p><img src="http://www.schoolphysics.co.uk/age16-19/Medical%20physics/text/X_rays/images/1.png" alt=""></p><h2 id="polarisation"><a class="anchor hidden-xs" href="#polarisation" title="polarisation"><span class="octicon octicon-link"></span></a>Polarisation</h2><p><strong>Plane polarised wave</strong> A wave in which the oscillations of the field and the direction of travel are confined to a single plane</p><p><strong>Polarising filter</strong> A filter which produces plane-polarised waves by selective absorption of one component of the incident oscillations.</p><h2 id="rotating-the-plane-of-polarisation"><a class="anchor hidden-xs" href="#rotating-the-plane-of-polarisation" title="rotating-the-plane-of-polarisation"><span class="octicon octicon-link"></span></a>Rotating the plane of polarisation</h2><p>A second polariser called the analyser an be used to rotate the plane of the waves polarisation to a direction parallel to its long edge.</p><p>If the amplitude of the light wave approaching the analyser sheet is A, then after it has had its plane of polarisation rotated by angle <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-168-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1486" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1487"><span class="mi" id="MathJax-Span-1488" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-168">\theta</script></span> the amplitude will be <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-169-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1489" style="width: 3.794em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.076em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.02em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1490"><span class="mi" id="MathJax-Span-1491" style="font-family: STIXGeneral; font-style: italic;">A</span><span class="mi" id="MathJax-Span-1492" style="font-family: STIXGeneral; font-style: italic;">c</span><span class="mi" id="MathJax-Span-1493" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-1494" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mo" id="MathJax-Span-1495" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1496" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1497" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-169">Acos(\theta)</script></span> .<br>
Because the intensity of a wave is proportional to its amplitude squared, the intensity after the analyser is proportional to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-170-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1498" style="width: 3.589em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.923em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1002.92em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1499"><span class="mi" id="MathJax-Span-1500" style="font-family: STIXGeneral; font-style: italic;">c</span><span class="mi" id="MathJax-Span-1501" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-1502" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mo" id="MathJax-Span-1503" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1504" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-1505"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-1506" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.361em;"><span class="mn" id="MathJax-Span-1507" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>θ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-170">cos(\theta)^2</script></span>.</p><p>This is know as Malus’ law.<br>
It states that when a perfect polariser is put in a beam of polarised light, the intensity, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-171-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1508" style="width: 0.515em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1509"><span class="mi" id="MathJax-Span-1510" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math></span></span><script type="math/tex" id="MathJax-Element-171">I</script></span>, of the light that passes through it is given by <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-172-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mtext>max</mtext></mrow></msub><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1511" style="width: 7.636em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.253em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1006.25em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1512"><span class="mi" id="MathJax-Span-1513" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mo" id="MathJax-Span-1514" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-1515" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.642em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1516" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.361em;"><span class="texatom" id="MathJax-Span-1517"><span class="mrow" id="MathJax-Span-1518"><span class="mtext" id="MathJax-Span-1519" style="font-size: 70.7%; font-family: STIXGeneral;">max</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mi" id="MathJax-Span-1520" style="font-family: STIXGeneral; font-style: italic;">c</span><span class="mi" id="MathJax-Span-1521" style="font-family: STIXGeneral; font-style: italic;">o</span><span class="mi" id="MathJax-Span-1522" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mo" id="MathJax-Span-1523" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1524" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-1525"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.31em 4.306em -999.997em); top: -3.993em; left: 0em;"><span class="mo" id="MathJax-Span-1526" style="font-family: STIXGeneral;">)</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.361em;"><span class="mn" id="MathJax-Span-1527" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.378em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mtext>max</mtext></mrow></msub><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>θ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-172">I = I_{\text{max}} cos(\theta)^2</script></span> , where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-173-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1528" style="width: 0.515em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1529"><span class="mi" id="MathJax-Span-1530" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math></span></span><script type="math/tex" id="MathJax-Element-173">I</script></span> is the intensity transmitted at angle <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-174-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1531" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1532"><span class="mi" id="MathJax-Span-1533" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-174">\theta</script></span> .<br>
Also <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-175-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mtext>max</mtext></mrow></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1534" style="width: 2.001em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.642em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.64em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1535"><span class="msubsup" id="MathJax-Span-1536"><span style="display: inline-block; position: relative; width: 1.642em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.36em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1537" style="font-family: STIXGeneral; font-style: italic;">I<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.361em;"><span class="texatom" id="MathJax-Span-1538"><span class="mrow" id="MathJax-Span-1539"><span class="mtext" id="MathJax-Span-1540" style="font-size: 70.7%; font-family: STIXGeneral;">max</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mrow class="MJX-TeXAtom-ORD"><mtext>max</mtext></mrow></msub></math></span></span><script type="math/tex" id="MathJax-Element-175">I_{\text{max}}</script></span> is the maximum intensity transmitted (at <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-176-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi><mo>=</mo><mn>0</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1541" style="width: 2.82em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.308em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.31em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1542"><span class="mi" id="MathJax-Span-1543" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1544" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-1545" style="font-family: STIXGeneral; padding-left: 0.31em;">0</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-176">\theta = 0</script></span>).<br>
This law also shows that if the analyser is at right angles to the polariser no light will pass through. This situation is known as ‘crossed polaroids’.</p><h3 id="uses-of-polarisation"><a class="anchor hidden-xs" href="#uses-of-polarisation" title="uses-of-polarisation"><span class="octicon octicon-link"></span></a>Uses of polarisation</h3><p>Polarisation can be used for strain analysis. Certain plastics, such as those used for making rulers, protractors and Sellotape, contain long chains of molecules which become aligned during manufacture.<br>
These materials are able to rotate the plane of polarisation so that the transmitted light is polarised.<br>
When these plastics are placed between two crossed polaroids, coloured images are produced that change as the plastics are stretched or squashed. This is because the rotation of the plane of polarisation is different for different wavelengths.</p><h2 id="refraction-of-light"><a class="anchor hidden-xs" href="#refraction-of-light" title="refraction-of-light"><span class="octicon octicon-link"></span></a>Refraction of light</h2><p>For all types of electromagnetic radiation, the speed of the wave will decrease as it moves from a material of lower refractive index into a material of higher refractive index.<br>
The speed of an electromagnetic wave in a material is related to the refractive index of the material by</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-177-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mtext>refractive index n</mtext></mtd><mtd><mi></mi><mo>=</mo><mfrac><mtext>speed of light in a vacuum</mtext><mtext>speed of light in the material</mtext></mfrac></mtd></mtr><mtr><mtd /><mtd><mi></mi><mo>=</mo><mfrac><mi>c</mi><mi>v</mi></mfrac></mtd></mtr></mtable></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1546" style="width: 25.003em; display: inline-block;"><span style="display: inline-block; position: relative; width: 20.494em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(-0.407em 1020.34em 4.306em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1547"><span class="mtable" id="MathJax-Span-1548" style="padding-right: 0.156em; padding-left: 0.156em;"><span style="display: inline-block; position: relative; width: 20.187em; height: 0px;"><span style="position: absolute; clip: rect(2.103em 1007.02em 5.484em -999.997em); top: -4.045em; left: 0em;"><span style="display: inline-block; position: relative; width: 7.072em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1007.02em 4.152em -999.997em); top: -5.069em; right: 0em;"><span class="mtd" id="MathJax-Span-1549"><span class="mrow" id="MathJax-Span-1550"><span class="mtext" id="MathJax-Span-1551" style="font-family: STIXGeneral;">refractive index n</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.845em 1000em 4.152em -999.997em); top: -2.713em; right: 0em;"><span class="mtd" id="MathJax-Span-1559"><span class="mrow" id="MathJax-Span-1560"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 4.05em;"></span></span><span style="position: absolute; clip: rect(2.82em 1013.12em 7.533em -999.997em); top: -5.428em; left: 7.072em;"><span style="display: inline-block; position: relative; width: 13.117em; height: 0px;"><span style="position: absolute; clip: rect(2.462em 1013.12em 5.074em -999.997em); top: -5.069em; left: 0em;"><span class="mtd" id="MathJax-Span-1552"><span class="mrow" id="MathJax-Span-1553"><span class="mi" id="MathJax-Span-1554"></span><span class="mo" id="MathJax-Span-1555" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1556" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 11.58em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1010.61em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -5.274em;"><span class="mtext" id="MathJax-Span-1557" style="font-family: STIXGeneral;">speed of light in a vacuum</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1011.48em 4.357em -999.997em); top: -3.327em; left: 50%; margin-left: -5.735em;"><span class="mtext" id="MathJax-Span-1558" style="font-family: STIXGeneral;">speed of light in the material</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1011.58em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 11.58em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.718em 1002.1em 4.869em -999.997em); top: -2.713em; left: 0em;"><span class="mtd" id="MathJax-Span-1561"><span class="mrow" id="MathJax-Span-1562"><span class="mi" id="MathJax-Span-1563"></span><span class="mo" id="MathJax-Span-1564" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1565" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1566" style="font-family: STIXGeneral; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1567" style="font-family: STIXGeneral; font-style: italic;">v</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.57em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.566em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 5.433em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -2.434em; border-left: 0px solid; width: 0px; height: 5.566em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mtr><mtd><mtext>refractive index n</mtext></mtd><mtd><mi></mi><mo>=</mo><mfrac><mtext>speed of light in a vacuum</mtext><mtext>speed of light in the material</mtext></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mfrac><mi>c</mi><mi>v</mi></mfrac></mtd></mtr></mtable></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-177">\begin{align} \text{refractive index n} &= \frac{\text{speed of light in a vacuum}}{\text{speed of light in the material}} \\ &= \frac{c}{v} \end{align}</script></span></p><p>A vacuum has a refractive index of 1, and air has a refractive index of 1.00028 ≈ 1.</p><h3 id="snell’s-law"><a class="anchor hidden-xs" href="#snell’s-law" title="snell’s-law"><span class="octicon octicon-link"></span></a>Snell’s law</h3><p>There is a link between the refractive indices of two materials and the directions at which the incident and refracted rays will travel with respect to the normal.<br>
An electromagnetic wave travelling at an angle, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-178-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03B8;</mi><mn>1</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1568" style="width: 1.13em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1000.92em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1569"><span class="msubsup" id="MathJax-Span-1570"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1571" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-1572" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-178">\theta_1</script></span>, to the normal in a material, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-179-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mn>1</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1573" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1574"><span class="msubsup" id="MathJax-Span-1575"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.72em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1576" style="font-family: STIXGeneral; font-style: italic;">m</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-1577" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-179">m_1</script></span>, will travel at an angle, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-180-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&#x03B8;</mi><mn>2</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1578" style="width: 1.13em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1000.92em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1579"><span class="msubsup" id="MathJax-Span-1580"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1581" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-1582" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>θ</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-180">\theta_2</script></span>, to the normal in a second material, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-181-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mn>2</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1583" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1001.13em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1584"><span class="msubsup" id="MathJax-Span-1585"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.72em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1586" style="font-family: STIXGeneral; font-style: italic;">m</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.72em;"><span class="mn" id="MathJax-Span-1587" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-181">m_2</script></span>.</p><p>This is explained by Snell’s law, which is given by</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-182-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>n</mi><mn>1</mn></msub><mo>&#x00D7;</mo><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><msub><mi>&#x03B8;</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>n</mi><mn>2</mn></msub><mo>&#x00D7;</mo><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><msub><mi>&#x03B8;</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1588" style="width: 13.373em; display: inline-block;"><span style="display: inline-block; position: relative; width: 10.966em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1010.91em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1589"><span class="msubsup" id="MathJax-Span-1590"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1591" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-1592" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1593" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mi" id="MathJax-Span-1594" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">s</span><span class="mi" id="MathJax-Span-1595" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-1596" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1597" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-1598"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1599" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-1600" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1601" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1602" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="msubsup" id="MathJax-Span-1603" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1604" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-1605" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1606" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mi" id="MathJax-Span-1607" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">s</span><span class="mi" id="MathJax-Span-1608" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-1609" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1610" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-1611"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1612" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-1613" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1614" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>n</mi><mn>1</mn></msub><mo>×</mo><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><msub><mi>θ</mi><mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>n</mi><mn>2</mn></msub><mo>×</mo><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><msub><mi>θ</mi><mn>2</mn></msub><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-182">n_1 \times sin(\theta _1) = n_2 \times sin(\theta _2)</script></span></p><p>or that</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-183-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>n</mi><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>constant</mtext></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1615" style="width: 9.07em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.431em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1007.43em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1616"><span class="mi" id="MathJax-Span-1617" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mi" id="MathJax-Span-1618" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mi" id="MathJax-Span-1619" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-1620" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1621" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1622" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1623" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1624" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mtext" id="MathJax-Span-1625" style="font-family: STIXGeneral; padding-left: 0.31em;">constant</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>n</mi><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mtext>constant</mtext></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-183">n sin(\theta) = \text{constant}</script></span></p><p>The angle of incidence and the angle of refraction are always measured to the normal line, not to the boundary.</p><h2 id="total-internal-reflection"><a class="anchor hidden-xs" href="#total-internal-reflection" title="total-internal-reflection"><span class="octicon octicon-link"></span></a>Total internal reflection</h2><p><strong>Total internal reflection</strong> The complete reflection of a light ray reaching an interface with a less dense medium when the angle of incidence exceeds the critical value.</p><h3 id="the-critical-angle"><a class="anchor hidden-xs" href="#the-critical-angle" title="the-critical-angle"><span class="octicon octicon-link"></span></a>The critical angle</h3><p>In the specific case of total internal reflection between two materials, we know that <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-184-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>1</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1626" style="width: 1.13em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1000.92em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1627"><span class="msubsup" id="MathJax-Span-1628"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1629" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-1630" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-184">n_1</script></span> is the refractive index of the material of higher refractive index.<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-185-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><msub><mi>&#x03B8;</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1631" style="width: 3.384em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.769em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.72em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1632"><span class="mi" id="MathJax-Span-1633" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mi" id="MathJax-Span-1634" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-1635" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1636" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-1637"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.51em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1638" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.464em;"><span class="mn" id="MathJax-Span-1639" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1640" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><msub><mi>θ</mi><mn>1</mn></msub><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-185">sin(\theta_1)</script></span> is the sine of the critical angle.</p><p>The critical angle is given by</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-186-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><msub><mi>n</mi><mn>2</mn></msub><msub><mi>n</mi><mn>1</mn></msub></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1641" style="width: 6.15em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.023em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.283em 1005.02em 3.537em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1642"><span class="mi" id="MathJax-Span-1643" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mi" id="MathJax-Span-1644" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-1645" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1646" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1647" style="font-family: STIXGeneral; font-style: italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1648" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1649" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1650" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.92em 4.306em -999.997em); top: -4.659em; left: 50%; margin-left: -0.459em;"><span class="msubsup" id="MathJax-Span-1651"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1652" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-1653" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.92em 4.306em -999.997em); top: -3.327em; left: 50%; margin-left: -0.459em;"><span class="msubsup" id="MathJax-Span-1654"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1655" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-1656" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.03em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.027em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.059em; border-left: 0px solid; width: 0px; height: 2.503em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><msub><mi>n</mi><mn>2</mn></msub><msub><mi>n</mi><mn>1</mn></msub></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-186">sin(C) = \frac{n_2}{n_1}</script></span></p><p>The ratio of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-187-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>2</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1657" style="width: 1.13em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1000.92em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1658"><span class="msubsup" id="MathJax-Span-1659"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1660" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-1661" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-187">n_2</script></span> to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-188-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>1</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1662" style="width: 1.13em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1000.92em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1663"><span class="msubsup" id="MathJax-Span-1664"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1665" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-1666" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-188">n_1</script></span> must be less than 1, so <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-189-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>2</mn></msub><mo>&lt;</mo><msub><mi>n</mi><mn>1</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1667" style="width: 3.845em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.128em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.898em 1003.13em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1668"><span class="msubsup" id="MathJax-Span-1669"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1670" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-1671" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1672" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="msubsup" id="MathJax-Span-1673" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1674" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-1675" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>2</mn></msub><mo><</mo><msub><mi>n</mi><mn>1</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-189">n_2 < n_1</script></span> . The smaller refractive index must be <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-190-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>2</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1676" style="width: 1.13em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1000.92em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1677"><span class="msubsup" id="MathJax-Span-1678"><span style="display: inline-block; position: relative; width: 0.925em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-1679" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.515em;"><span class="mn" id="MathJax-Span-1680" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mn>2</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-190">n_2</script></span>.</p><h3 id="determining-the-critical-angle-for-a-transport-material"><a class="anchor hidden-xs" href="#determining-the-critical-angle-for-a-transport-material" title="determining-the-critical-angle-for-a-transport-material"><span class="octicon octicon-link"></span></a>Determining the critical angle for a transport material</h3><p>To determine the critical angle of a material, use a semicircular shaped block of the material.<br>
Draw around the shape on a piece of paper, making a mark at the centre of the straight edge and draw a normal at that point.<br>
Shine a ray from a ray box towards the curved edge of the block in an arc until the ray emerges along the surface of the boundary between the material and air.<br>
The angle the ray makes with the normal is the critical angle.</p><p>The critical angle can be used to find the refractive index of the material in question using</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-191-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1681" style="width: 5.638em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.613em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1004.61em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1682"><span class="mi" id="MathJax-Span-1683" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mi" id="MathJax-Span-1684" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-1685" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1686" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1687" style="font-family: STIXGeneral; font-style: italic;">C<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1688" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1689" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1690" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-1691" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-1692" style="font-family: STIXGeneral; font-style: italic;">n</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>n</mi></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-191">sin(C) = \frac{1}{n}</script></span></p><h2 id="interference1"><a class="anchor hidden-xs" href="#interference1" title="interference1"><span class="octicon octicon-link"></span></a>Interference</h2><h3 id="the-principle-of-superposition"><a class="anchor hidden-xs" href="#the-principle-of-superposition" title="the-principle-of-superposition"><span class="octicon octicon-link"></span></a>The principle of superposition</h3><p>The principle of superposition states that when two or more waves of the same type meet, the resultant wave can be found by adding the displacements of the individual waves.</p><h3 id="interference2"><a class="anchor hidden-xs" href="#interference2" title="interference2"><span class="octicon octicon-link"></span></a>Interference</h3><p>If two waves, A and B, with the same amplitude exist at the same point and are travelling in phase, the amplitude of the resultant wave will be twice that of the individual waves. This is constructive interference.</p><p>If the waves are in antiphase they will cancel each other out. This is destructive interference.</p><h3 id="coherence"><a class="anchor hidden-xs" href="#coherence" title="coherence"><span class="octicon octicon-link"></span></a>Coherence</h3><p>Generally, waves will not be in perfect phase or antiphase.<br>
So long as the two waves have a constant phase difference, they are coherent.</p><p><strong>Path difference</strong> The path difference is the difference in metres between the lengths of two paths. It is often written in terms of the wavelength of two the wave</p><p><strong>Phase difference</strong> The phase difference is the difference in radians of the phases of two waves of the same frequency</p><table>
<thead>
<tr>
<th>****</th>
<th>Path difference</th>
<th>Phase difference</th>
</tr>
</thead>
<tbody>
<tr>
<td>Constructive</td>
<td>Integer number of wavelengths e.g. 0, λ, 2λ, …</td>
<td>0, 360, 720, … or 0, 2π, 4π, …</td>
</tr>
<tr>
<td>Destructive</td>
<td>Odd number of half wavelengths e.g. λ/2, 3λ/2, …</td>
<td>180, 540, 900, … or π, 3π, 5π…</td>
</tr>
</tbody>
</table><h3 id="interference-using-sound-waves"><a class="anchor hidden-xs" href="#interference-using-sound-waves" title="interference-using-sound-waves"><span class="octicon octicon-link"></span></a>Interference using sound waves</h3><p>Interference using sound waves can be demonstrated using two loudspeakers connected to the same signal generator. As you walk in front of the loudspeakers you will hear a loud sound where the sound waves reinforce one another, and a quiet sound where the waves partially cancel one another out.<br>
This variation is clear if you cover one ear.<br>
The distance between the loud and quiet regions is longer for lower frequencies.</p><h2 id="the-young-double-split-experiment"><a class="anchor hidden-xs" href="#the-young-double-split-experiment" title="the-young-double-split-experiment"><span class="octicon octicon-link"></span></a>The Young double-split experiment</h2><p>The coloured patterns patterns seen on an oil spill on water are a result of the interference of light waves.<br>
In order to make measurements of the wavelength of light, two conditions must be satisfied</p><ul>
<li>The light must be monochromatic</li>
<li>There must be an accurate method of producing very small path differences, and of measuring these differences</li>
</ul><h3 id="the-experiment"><a class="anchor hidden-xs" href="#the-experiment" title="the-experiment"><span class="octicon octicon-link"></span></a>The experiment</h3><p>Light is produced by a monochromatic source and diffracts outwards, before reaching a second set of two narrow parallel slits.<br>
The light from these two slits is coherent because it comes from a single slit.</p><p>When the waves reach an obstacle, alternate bright and dark vertical bands or ‘fringes’ will be seen.</p><p><img src="http://philschatz.com/physics-book/resources/Figure_28_03_06a.jpg" alt=""></p><p>For optimal results, the experiment should be carried out in a darkened room.<br>
To reduce percentage error in the fringe separation it is best to measure across all the fringes and then divide by the number of fringes.<br>
Increasing the distance from the slit to the screen will increase the fringe separation, but reduces light intensity at the screen.</p><h3 id="finding-the-wavelength-of-the-light-source"><a class="anchor hidden-xs" href="#finding-the-wavelength-of-the-light-source" title="finding-the-wavelength-of-the-light-source"><span class="octicon octicon-link"></span></a>Finding the wavelength of the light source</h3><p>For constructive interference to take place, the path different must be a whole number of wavelengths.<br>
For destructive interference to take place, the waves must arrive out of phase with path difference of half a wavelength.<br>
<a href="https://postimg.org/image/fop9tkaxl/" target="_blank"><img src="https://s28.postimg.org/kncs83eql/double_slit.png" alt=""></a></p><p>In the diagram above, the two triangles are a similar shape, so their angles are equal.<br>
If θ is the angle away from the central fringe</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-192-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>&#x03BB;</mi><mi>a</mi></mfrac><mspace width="1em" /><mi>a</mi><mi>n</mi><mi>d</mi><mtext>&#xA0;</mtext><mi>t</mi><mi>a</mi><mi>n</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>x</mi><mi>D</mi></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1693" style="width: 14.808em; display: inline-block;"><span style="display: inline-block; position: relative; width: 12.144em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1012.14em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1694"><span class="mi" id="MathJax-Span-1695" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mi" id="MathJax-Span-1696" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-1697" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1698" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1699" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1700" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1701" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1702" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1703" style="font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-1704" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mspace" id="MathJax-Span-1705" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mi" id="MathJax-Span-1706" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1707" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mi" id="MathJax-Span-1708" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mtext" id="MathJax-Span-1709" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-1710" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-1711" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1712" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1713" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1714" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1715" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1716" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1717" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1718" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.72em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.356em;"><span class="mi" id="MathJax-Span-1719" style="font-family: STIXGeneral; font-style: italic;">D</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.82em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.822em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>λ</mi><mi>a</mi></mfrac><mspace width="1em"></mspace><mi>a</mi><mi>n</mi><mi>d</mi><mtext> </mtext><mi>t</mi><mi>a</mi><mi>n</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>x</mi><mi>D</mi></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-192">sin(\theta) = \frac{\lambda}{a}\quad and\ tan(\theta) = \frac{x}{D}</script></span></p><p>If the angle <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-193-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1720" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1721"><span class="mi" id="MathJax-Span-1722" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-193">\theta</script></span> is small enough <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-194-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo>&#x2061;</mo><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo>&#x2248;</mo><mi>t</mi><mi>a</mi><mi>n</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1723" style="width: 7.38em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.048em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1006em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1724"><span class="mi" id="MathJax-Span-1725" style="font-family: STIXGeneral;">sin</span><span class="mo" id="MathJax-Span-1726"></span><span class="mo" id="MathJax-Span-1727" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1728" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1729" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1730" style="font-family: STIXGeneral; padding-left: 0.31em;">≈</span><span class="mi" id="MathJax-Span-1731" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-1732" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1733" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1734" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1735" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1736" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo></mo><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>≈</mo><mi>t</mi><mi>a</mi><mi>n</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-194">\sin(\theta) \approx tan(\theta)</script></span> so</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-195-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>&#x03BB;</mi><mi>a</mi></mfrac><mo>=</mo><mfrac><mi>x</mi><mi>D</mi></mfrac><mspace width="1em" /><mtext>giving</mtext><mtext>&#xA0;</mtext><mi>&#x03BB;</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mi>x</mi></mrow><mi>D</mi></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1737" style="width: 12.4em; display: inline-block;"><span style="display: inline-block; position: relative; width: 10.146em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1010.15em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1738"><span class="mfrac" id="MathJax-Span-1739"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1740" style="font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-1741" style="font-family: STIXGeneral; font-style: italic;">a</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-1742" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1743" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.46em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1744" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.72em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.356em;"><span class="mi" id="MathJax-Span-1745" style="font-family: STIXGeneral; font-style: italic;">D</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.82em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.822em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mspace" id="MathJax-Span-1746" style="height: 0em; vertical-align: 0em; width: 1.13em; display: inline-block; overflow: hidden;"></span><span class="mtext" id="MathJax-Span-1747" style="font-family: STIXGeneral;">giving</span><span class="mtext" id="MathJax-Span-1748" style="font-family: STIXGeneral;"> </span><span class="mi" id="MathJax-Span-1749" style="font-family: STIXGeneral; font-style: italic;">λ</span><span class="mo" id="MathJax-Span-1750" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1751" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.98em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.459em;"><span class="mrow" id="MathJax-Span-1752"><span class="mi" id="MathJax-Span-1753" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1754" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.72em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.356em;"><span class="mi" id="MathJax-Span-1755" style="font-family: STIXGeneral; font-style: italic;">D</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>λ</mi><mi>a</mi></mfrac><mo>=</mo><mfrac><mi>x</mi><mi>D</mi></mfrac><mspace width="1em"></mspace><mtext>giving</mtext><mtext> </mtext><mi>λ</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mi>x</mi></mrow><mi>D</mi></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-195">\frac{\lambda}{a} = \frac{x}{D} \quad \text{giving}\ \lambda = \frac{ax}{D}</script></span></p><p>The equation only applies if <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-196-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>&#x226A;</mo><mi>D</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1756" style="width: 3.384em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.769em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.77em 2.769em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1757"><span class="mi" id="MathJax-Span-1758" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mo" id="MathJax-Span-1759" style="font-family: STIXGeneral; padding-left: 0.31em;">≪</span><span class="mi" id="MathJax-Span-1760" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">D</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.122em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≪</mo><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-196">a \ll D</script></span> , or the angle away from the fringe is less than 10°.</p><h2 id="the-diffraction-grating"><a class="anchor hidden-xs" href="#the-diffraction-grating" title="the-diffraction-grating"><span class="octicon octicon-link"></span></a>The diffraction grating</h2><p>A diffraction grating is a piece of optical equipment made from glass, onto which many thousands of thin, parallel, and equally spaced grooves have been engraved using a diamond.<br>
Light that passes through the grating will be diffracted at different angles based on the wavelength of the incident light and the separation of the grooves.</p><h3 id="multiple-slits"><a class="anchor hidden-xs" href="#multiple-slits" title="multiple-slits"><span class="octicon octicon-link"></span></a>Multiple slits</h3><p>It is difficult to accurately determine wavelengths using the Young double-slit experiment. The diffraction fringes are quite blurred, which makes measuring the fringe width difficult.<br>
This can be overcome by using multiple slits.<br>
Increasing the number of slits that the light has to pass through improves the brightness and sharpness of the maxima and makes it easier to measure an accurate value for the wavelength of light.<br>
The maxima are also further apart, so the angle can be measured with a lower percentage uncertainty.</p><p>Because the length of n wavelengths, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-197-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>&#x03BB;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1761" style="width: 1.181em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.92em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1762"><span class="mi" id="MathJax-Span-1763" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mi" id="MathJax-Span-1764" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-197">n\lambda</script></span>, must be equal to , we get the diffraction equation</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-198-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>n</mi><mi>&#x03BB;</mi><mo>=</mo><mi>d</mi><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1765" style="width: 6.253em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.126em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1005.07em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1766"><span class="mi" id="MathJax-Span-1767" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mi" id="MathJax-Span-1768" style="font-family: STIXGeneral; font-style: italic;">λ</span><span class="mo" id="MathJax-Span-1769" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1770" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1771" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mi" id="MathJax-Span-1772" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-1773" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1774" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1775" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1776" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>n</mi><mi>λ</mi><mo>=</mo><mi>d</mi><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-198">n\lambda = d sin(\theta)</script></span></p><p>where n is called the order of the maximum, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-199-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1777" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1778"><span class="mi" id="MathJax-Span-1779" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-199">\lambda</script></span> is the wavelength of the incident monochromatic light, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-200-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1780" style="width: 0.72em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.57em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1781"><span class="mi" id="MathJax-Span-1782" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span><script type="math/tex" id="MathJax-Element-200">d</script></span> is the separation of the slits in the grating, and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-201-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1783" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1784"><span class="mi" id="MathJax-Span-1785" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-201">\theta</script></span> is the angle that the beam makes with the grating.</p><h3 id="measuring-the-wavelength-of-light-using-a-diffraction-grating"><a class="anchor hidden-xs" href="#measuring-the-wavelength-of-light-using-a-diffraction-grating" title="measuring-the-wavelength-of-light-using-a-diffraction-grating"><span class="octicon octicon-link"></span></a>Measuring the wavelength of light using a diffraction grating</h3><p><img src="http://www.pstcc.edu/departments/natural_behavioral_sciences/Web%20Physics/E2020D1205.gif" alt=""></p><p>The wavelength is found using <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-202-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>&#x03BB;</mi><mo>=</mo><mi>d</mi><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1786" style="width: 6.253em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.126em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1005.07em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1787"><span class="mi" id="MathJax-Span-1788" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mi" id="MathJax-Span-1789" style="font-family: STIXGeneral; font-style: italic;">λ</span><span class="mo" id="MathJax-Span-1790" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1791" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="mi" id="MathJax-Span-1792" style="font-family: STIXGeneral; font-style: italic;">s</span><span class="mi" id="MathJax-Span-1793" style="font-family: STIXGeneral; font-style: italic;">i</span><span class="mi" id="MathJax-Span-1794" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1795" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1796" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1797" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mi>λ</mi><mo>=</mo><mi>d</mi><mi>s</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-202">n\lambda = dsin(\theta)</script></span>. If the grating has <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-203-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1798" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1799"><span class="mi" id="MathJax-Span-1800" style="font-family: STIXGeneral; font-style: italic;">X<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math></span></span><script type="math/tex" id="MathJax-Element-203">X</script></span> slits per metre the slit spacing, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-204-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1801" style="width: 0.72em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.57em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1802"><span class="mi" id="MathJax-Span-1803" style="font-family: STIXGeneral; font-style: italic;">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span><script type="math/tex" id="MathJax-Element-204">d</script></span>, is the reciprocal of this value.<br>
The angle <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-205-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1804" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1805"><span class="mi" id="MathJax-Span-1806" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-205">\theta</script></span> is found with a protractor. The percentage uncertainty in <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-206-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1807" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1808"><span class="mi" id="MathJax-Span-1809" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-206">\theta</script></span> is decreased if <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-207-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1810" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1811"><span class="mi" id="MathJax-Span-1812" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-207">\theta</script></span> is as large as possible. This occurs for smaller slit separations and also for higher order fringes, however the intensity of the higher order fringes is lower, making them difficult to locate.</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-208-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03B8;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1813" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1814"><span class="mi" id="MathJax-Span-1815" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math></span></span><script type="math/tex" id="MathJax-Element-208">\theta</script></span> could also be found geometrically, using <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-209-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>a</mi><mi>n</mi><mo stretchy="false">(</mo><mi>&#x03B8;</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>x</mi><mi>D</mi></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1816" style="width: 5.587em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.562em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.693em 1004.56em 3.076em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1817"><span class="mi" id="MathJax-Span-1818" style="font-family: STIXGeneral; font-style: italic;">t<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mi" id="MathJax-Span-1819" style="font-family: STIXGeneral; font-style: italic;">a</span><span class="mi" id="MathJax-Span-1820" style="font-family: STIXGeneral; font-style: italic;">n</span><span class="mo" id="MathJax-Span-1821" style="font-family: STIXGeneral;">(</span><span class="mi" id="MathJax-Span-1822" style="font-family: STIXGeneral; font-style: italic;">θ<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1823" style="font-family: STIXGeneral;">)</span><span class="mo" id="MathJax-Span-1824" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1825" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.537em 1000.31em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-1826" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.51em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-1827" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">D</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.441em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mi>a</mi><mi>n</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>x</mi><mi>D</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-209">tan(\theta) = \frac{x}{D}</script></span>, where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-210-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1828" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1829"><span class="mi" id="MathJax-Span-1830" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math></span></span><script type="math/tex" id="MathJax-Element-210">x</script></span> is the distance from the central maxima and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-211-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1831" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1832"><span class="mi" id="MathJax-Span-1833" style="font-family: STIXGeneral; font-style: italic;">D</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></span></span><script type="math/tex" id="MathJax-Element-211">D</script></span> is the distance of the grating from the screen.</p><h2 id="stationary-waves"><a class="anchor hidden-xs" href="#stationary-waves" title="stationary-waves"><span class="octicon octicon-link"></span></a>Stationary waves</h2><p>It is possible to confine a wave, and its energy to a fixed position. These waves are called standing or stationary waves.<br>
Examples include waves on strings, such as stringed instruments, and air columns in tubes or pipes.</p><h3 id="formation-of-stationary-waves"><a class="anchor hidden-xs" href="#formation-of-stationary-waves" title="formation-of-stationary-waves"><span class="octicon octicon-link"></span></a>Formation of stationary waves</h3><p>Stationary waves are produced by interference in accordance with the principle of superposition.<br>
In order for a stationary wave to be produced on a string or in a pipe, the two waves that overlap must be travelling in opposite directions, have the same frequency, and approximately equal amplitudes.</p><p>The process</p><ul>
<li>Initially there are two progressive waves moving in opposite directions</li>
<li>The two waves are in antiphase, with a phase difference of 180° or π rad</li>
<li>This leads to a resultant wave of amplitude zero</li>
<li>As the waves move, they move into phase and they create constructive interference to produce a wave with an amplitude equal to the sum of the amplitudes of the individual waves</li>
<li>The wave pattern for a whole cycle returns the waves to their original states</li>
</ul><p><strong>Node</strong> A node in a stationary wave is a point at which there is no displacement of the particles at any time</p><p><strong>Antinode</strong> An antinode in a stationary wave is a point at which the displacement of the particles varies by the maximum amount</p><p><img src="http://www.mrsciguy.com/Physics/Waves/images/standingwaves3.gif" alt=""></p><h2 id="stationary-wave-experiments"><a class="anchor hidden-xs" href="#stationary-wave-experiments" title="stationary-wave-experiments"><span class="octicon octicon-link"></span></a>Stationary wave experiments</h2><h3 id="stationary-waves-on-strings"><a class="anchor hidden-xs" href="#stationary-waves-on-strings" title="stationary-waves-on-strings"><span class="octicon octicon-link"></span></a>Stationary waves on strings</h3><p>The frequency of vibration of a string is governed by</p><ul>
<li>Its mass per units length</li>
<li>Its tension</li>
<li>The length of the string</li>
</ul><p>The effect of bowing a violin string is to create progressive transverse waves that travel in opposite directions along the string away from the bow. When these waves reach the points where the string is fixed to the body of the violin, they are reflected back along the string. They then interfere with each other producing a stationary wave.</p><p>A stationary wave in a guitar string obeys the equation</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-212-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>L</mi><msqrt><mfrac><mi>T</mi><mi>&#x03BC;</mi></mfrac></msqrt></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1834" style="width: 6.457em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.279em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.976em 1005.28em 3.742em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1835"><span class="mi" id="MathJax-Span-1836" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-1837" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1838" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-1839" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-1840" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-1841" style="font-family: STIXGeneral; font-style: italic;">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msqrt" id="MathJax-Span-1842"><span style="display: inline-block; position: relative; width: 2.154em; height: 0px;"><span style="position: absolute; clip: rect(2.513em 1000.98em 5.023em -999.997em); top: -3.993em; left: 1.13em;"><span class="mrow" id="MathJax-Span-1843"><span class="mfrac" id="MathJax-Span-1844"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.305em;"><span class="mi" id="MathJax-Span-1845" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.435em 1000.46em 4.357em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-1846" style="font-family: STIXGeneral; font-style: italic;">μ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.77em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.771em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1001.03em 3.384em -999.997em); top: -4.557em; left: 1.13em;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0.515em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.259em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(1.796em 1001.18em 4.562em -999.997em); top: -3.379em; left: 0em;"><span style="font-family: STIXSizeTwoSym;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.309em; border-left: 0px solid; width: 0px; height: 3.128em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>f</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>L</mi><msqrt><mfrac><mi>T</mi><mi>μ</mi></mfrac></msqrt></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-212">f = \frac{1}{2}L \sqrt{\frac{T}{\mu}}</script></span></p><p>where<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-213-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1847" style="width: 0.515em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1848"><span class="mi" id="MathJax-Span-1849" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-213">f</script></span> is the frequency of vibration (<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-214-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Hz</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1850" style="width: 1.437em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.181em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.13em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1851"><span class="mtext" id="MathJax-Span-1852" style="font-family: STIXGeneral;">Hz</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Hz</mtext></math></span></span><script type="math/tex" id="MathJax-Element-214">\text{Hz}</script></span>)<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-215-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1853" style="width: 0.72em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.57em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1854"><span class="mi" id="MathJax-Span-1855" style="font-family: STIXGeneral; font-style: italic;">L<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math></span></span><script type="math/tex" id="MathJax-Element-215">L</script></span> is the length of the string (<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-216-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>m</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1856" style="width: 0.976em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.77em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1857"><span class="mtext" id="MathJax-Span-1858" style="font-family: STIXGeneral;">m</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>m</mtext></math></span></span><script type="math/tex" id="MathJax-Element-216">\text{m}</script></span>)<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-217-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1859" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.67em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1860"><span class="mi" id="MathJax-Span-1861" style="font-family: STIXGeneral; font-style: italic;">T<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math></span></span><script type="math/tex" id="MathJax-Element-217">T</script></span> is the tension in the string (<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-218-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>N</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1862" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1863"><span class="mtext" id="MathJax-Span-1864" style="font-family: STIXGeneral;">N</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>N</mtext></math></span></span><script type="math/tex" id="MathJax-Element-218">\text{N}</script></span>)<br>
<span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-219-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BC;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1865" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.001em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1866"><span class="mi" id="MathJax-Span-1867" style="font-family: STIXGeneral; font-style: italic;">μ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.878em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math></span></span><script type="math/tex" id="MathJax-Element-219">\mu</script></span> is the mass per unit length of the string <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-220-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>kg m</mtext><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1868" style="width: 4.408em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.589em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1003.54em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1869"><span class="mo" id="MathJax-Span-1870" style="font-family: STIXGeneral;">(</span><span class="msubsup" id="MathJax-Span-1871"><span style="display: inline-block; position: relative; width: 2.923em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1002.05em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mtext" id="MathJax-Span-1872" style="font-family: STIXGeneral;">kg m</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 2.052em;"><span class="texatom" id="MathJax-Span-1873"><span class="mrow" id="MathJax-Span-1874"><span class="mo" id="MathJax-Span-1875" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1876" style="font-size: 70.7%; font-family: STIXGeneral;">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-1877" style="font-family: STIXGeneral;">)</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.503em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mtext>kg m</mtext><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-220">(\text{kg m}^{-1})</script></span></p><h3 id="fundamental-mode-of-vibration-and-harmonics"><a class="anchor hidden-xs" href="#fundamental-mode-of-vibration-and-harmonics" title="fundamental-mode-of-vibration-and-harmonics"><span class="octicon octicon-link"></span></a>Fundamental mode of vibration and harmonics</h3><p>The simplest type of stationary wave on a string is a wave with a wavelength equal to twice the length of the string.<br>
This type of oscillation is called the fundamental mode of vibration. The fundamental frequency is the lowest frequency, highest wavelength that can be produced on a string. However other notes of higher frequency can also be produced, and these are called harmonics.</p><p>The frequency of a harmonic is always an integer multiple of the fundamental frequency.</p><p><img src="http://study.com/cimages/multimages/16/standingwaves.png" alt=""></p><h2 id="stationary-longitudinal-waves"><a class="anchor hidden-xs" href="#stationary-longitudinal-waves" title="stationary-longitudinal-waves"><span class="octicon octicon-link"></span></a>Stationary longitudinal waves</h2><h3 id="stationary-waves-in-closed-tubes"><a class="anchor hidden-xs" href="#stationary-waves-in-closed-tubes" title="stationary-waves-in-closed-tubes"><span class="octicon octicon-link"></span></a>Stationary waves in closed tubes</h3><p>A stationary longitudinal wave is a specific type of stationary wave, created by blowing across one end of a tube.<br>
As with the stationary wave produced on a string, the stationary wave in the tube is produced when progressive waves travel in opposite directions through one another.</p><p>A progressive wave can be started at one end of a closed tube by blowing across the open end. This wave travels down the tube and is reflected at the closed end. This produces two progressive waves travelling in opposite directions, which then interfere to produce a stationary wave.<br>
However, to produce a stationary wave the length of the tube must be such that a node is formed at the closed end, and an antinode at the open end.</p><p><img src="http://hep.physics.indiana.edu/~rickv/pipe_3.gif" alt=""></p><h3 id="stationary-waves-in-closed-tubes1"><a class="anchor hidden-xs" href="#stationary-waves-in-closed-tubes1" title="stationary-waves-in-closed-tubes1"><span class="octicon octicon-link"></span></a>Stationary waves in closed tubes</h3><p>The sound wave will also be reflected at the open ends of a pipe. Since the tube is open at both ends, each end is an antinode.</p><h2 id="measuring-the-speed-of-sound-in-air"><a class="anchor hidden-xs" href="#measuring-the-speed-of-sound-in-air" title="measuring-the-speed-of-sound-in-air"><span class="octicon octicon-link"></span></a>Measuring the speed of sound in air</h2><p>The speed of sound in air can be measured using a tuning fork and a tube of water.</p><p>The tube is held by a clamp and is movable so that its length can be altered. Because of the water in the measure cylinder, the tube is effectively closed at one end.</p><p>When a tuning fork of known frequency is struck and held at the open end, air molecules in the tube will vibrate and a stationary wave will be set up in the table.<br>
By listening carefully, the fundamental frequency can be obtained, when the sound is loudest at the minimum length. This is achieved when the length of the tube is equal to one quarter <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-221-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1878" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1879"><span class="mi" id="MathJax-Span-1880" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-221">\lambda</script></span>.</p><p>The tube can then be lengthened by loosening the clamp, and the loudness of the sound will reduce initially before increasing again to a second maximum loudness when the length of the tube is equal to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-222-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>&#x03BB;</mi></mrow><mn>4</mn></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1881" style="width: 1.283em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.13em 1001.03em 2.718em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1882"><span class="mfrac" id="MathJax-Span-1883"><span style="display: inline-block; position: relative; width: 0.822em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.67em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.356em;"><span class="mrow" id="MathJax-Span-1884"><span class="mn" id="MathJax-Span-1885" style="font-size: 70.7%; font-family: STIXGeneral;">3</span><span class="mi" id="MathJax-Span-1886" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-1887" style="font-size: 70.7%; font-family: STIXGeneral;">4</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.82em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.822em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>λ</mi></mrow><mn>4</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-222">\frac{3\lambda}{4}</script></span>. The difference between these two lengths is equal to half the wavelength of the sound, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-223-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>&#x03BB;</mi><mn>2</mn></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1888" style="width: 0.822em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.669em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.13em 1000.67em 2.718em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-1889"><span class="mfrac" id="MathJax-Span-1890"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.31em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-1891" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.36em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-1892" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>λ</mi><mn>2</mn></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-223">\frac{\lambda}{2}</script></span>, and the speed of sound can be determined by multiplying this value by two, and then by the frequency of the tuning fork.</p><h1 id="quantum-physics"><a class="anchor hidden-xs" href="#quantum-physics" title="quantum-physics"><span class="octicon octicon-link"></span></a>Quantum physics</h1><h2 id="the-photon"><a class="anchor hidden-xs" href="#the-photon" title="the-photon"><span class="octicon octicon-link"></span></a>The photon</h2><p><strong>Quantum</strong> A small discrete unit of energy</p><p><strong>Photon</strong> A quantum associated with electromagnetic radiation</p><p><strong>Planck constant</strong> The Planck constant, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-224-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1893" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1894"><span class="mi" id="MathJax-Span-1895" style="font-family: STIXGeneral; font-style: italic;">h</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></span></span><script type="math/tex" id="MathJax-Element-224">h</script></span>, has a value of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-225-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6.626</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>34</mn></mrow></msup><mtext>&#xA0;J s</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1896" style="width: 8.455em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.919em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1006.87em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1897"><span class="mn" id="MathJax-Span-1898" style="font-family: STIXGeneral;">6.626</span><span class="mo" id="MathJax-Span-1899" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-1900" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-1901" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-1902"><span class="mrow" id="MathJax-Span-1903"><span class="mo" id="MathJax-Span-1904" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-1905" style="font-size: 70.7%; font-family: STIXGeneral;">34</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mtext" id="MathJax-Span-1906" style="font-family: STIXGeneral;"> J s</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6.626</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>34</mn></mrow></msup><mtext> J s</mtext></math></span></span><script type="math/tex" id="MathJax-Element-225">6.626\times10^{-34}\text{ J s}</script></span>. Photon energies are always emitted in multiples of this value</p><p>The equation relating the speed of light, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-226-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1907" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1908"><span class="mi" id="MathJax-Span-1909" style="font-family: STIXGeneral; font-style: italic;">c</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math></span></span><script type="math/tex" id="MathJax-Element-226">c</script></span>, the frequency of the wave, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-227-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1910" style="width: 0.515em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1911"><span class="mi" id="MathJax-Span-1912" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-227">f</script></span>, and its wavelength, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-228-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1913" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1914"><span class="mi" id="MathJax-Span-1915" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-228">\lambda</script></span>, is</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display"><span class="MathJax" id="MathJax-Element-229-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><merror><mtext>\text[wave&#xA0;speed}&#xA0;=&#xA0;\text{frequency}&#xA0;\times&#xA0;\text{wavelength}</mtext></merror></math>" role="presentation" style="position: relative;"><span class="math" id="MathJax-Span-1916" style="" aria-hidden="true"><span class="noError" id="MathJax-Span-1917" style="display: inline-block;">\text[wave speed} = \text{frequency} \times \text{wavelength}</span></span><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><merror><mtext>\text[wave speed} = \text{frequency} \times \text{wavelength}</mtext></merror></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-229">\text[wave speed} = \text{frequency} \times \text{wavelength}</script></span></p><p>or</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-230-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>c</mi><mo>=</mo><mi>f</mi><mi>&#x03BB;</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1918" style="width: 3.23em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.615em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.56em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1919"><span class="mi" id="MathJax-Span-1920" style="font-family: STIXGeneral; font-style: italic;">c</span><span class="mo" id="MathJax-Span-1921" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1922" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mi" id="MathJax-Span-1923" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>c</mi><mo>=</mo><mi>f</mi><mi>λ</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-230">c = f\lambda</script></span></p><p>Rearranging this equation and substituting <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-231-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mi>h</mi><mi>f</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1924" style="width: 3.537em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.871em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.87em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1925"><span class="mi" id="MathJax-Span-1926" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1927" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-1928" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">h</span><span class="mi" id="MathJax-Span-1929" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mi>h</mi><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-231">E = hf</script></span> gives an alternative equation for calculating the energy of a photon</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-232-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo>=</mo><mfrac><mrow><mi>h</mi><mi>c</mi></mrow><mi>&#x03BB;</mi></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1930" style="width: 3.896em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.179em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1003.18em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1931"><span class="mi" id="MathJax-Span-1932" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-1933" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1934" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.92em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.459em;"><span class="mrow" id="MathJax-Span-1935"><span class="mi" id="MathJax-Span-1936" style="font-family: STIXGeneral; font-style: italic;">h</span><span class="mi" id="MathJax-Span-1937" style="font-family: STIXGeneral; font-style: italic;">c</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1938" style="font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo>=</mo><mfrac><mrow><mi>h</mi><mi>c</mi></mrow><mi>λ</mi></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-232">E = \frac{hc}{\lambda}</script></span></p><h3 id="using-leds-to-determine-a-value-for-the-planck-constant"><a class="anchor hidden-xs" href="#using-leds-to-determine-a-value-for-the-planck-constant" title="using-leds-to-determine-a-value-for-the-planck-constant"><span class="octicon octicon-link"></span></a>Using LEDs to determine a value for the Planck constant</h3><p>LEDs come in a range of colours. Because the colour or wavelength of the light being emitted is related to photon energy, we can use different LEDs to determine a value for the Planck constant.<br>
We need the following equipment</p><ul>
<li>Four LEDs of different colours and known wavelenghts</li>
<li>A <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-233-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mtext>&#xA0;V</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1939" style="width: 1.847em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.488em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.49em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1940"><span class="mn" id="MathJax-Span-1941" style="font-family: STIXGeneral;">6</span><span class="mtext" id="MathJax-Span-1942" style="font-family: STIXGeneral;"> V</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mtext> V</mtext></math></span></span><script type="math/tex" id="MathJax-Element-233">6\text{ V}</script></span> DC power supply</li>
<li>An ammeter and a voltmeter</li>
<li>A <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-234-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mtext>&#xA0;k</mtext><mi mathvariant="normal">&#x03A9;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1943" style="width: 2.462em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.001em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.95em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1944"><span class="mn" id="MathJax-Span-1945" style="font-family: STIXGeneral;">1</span><span class="mtext" id="MathJax-Span-1946" style="font-family: STIXGeneral;"> k</span><span class="mi" id="MathJax-Span-1947" style="font-family: STIXGeneral;">Ω</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mtext> k</mtext><mi mathvariant="normal">Ω</mi></math></span></span><script type="math/tex" id="MathJax-Element-234">1\text{ k}\Omega</script></span> resistor</li>
</ul><p>The apparatus is set up with the LED, power supply, ammeter and resistor in series, and the voltmeter in parallel across the LED.</p><p>We can then measure the current flowing as we increase the p.d across the LED by adjusting the variable resistor. Obtain at least six values for <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-235-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>V</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1948" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1949"><span class="mtext" id="MathJax-Span-1950" style="font-family: STIXGeneral;">V</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>V</mtext></math></span></span><script type="math/tex" id="MathJax-Element-235">\text{V}</script></span> and <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-236-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>I</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1951" style="width: 0.412em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.31em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.31em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1952"><span class="mtext" id="MathJax-Span-1953" style="font-family: STIXGeneral;">I</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>I</mtext></math></span></span><script type="math/tex" id="MathJax-Element-236">\text{I}</script></span>, in steps of <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-237-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.05</mn><mtext>&#xA0;V</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1954" style="width: 3.332em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.718em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1955"><span class="mn" id="MathJax-Span-1956" style="font-family: STIXGeneral;">0.05</span><span class="mtext" id="MathJax-Span-1957" style="font-family: STIXGeneral;"> V</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0.05</mn><mtext> V</mtext></math></span></span><script type="math/tex" id="MathJax-Element-237">0.05\text{ V}</script></span> over the range where the LED starts to emit light.</p><p>We then determine the activation voltage for each LED.<br>
Using these values to plot a graph of activation voltage against the reciprocal of the respective wavelength, we achieve a graph with a gradient of</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-238-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mi mathvariant="normal">&#x0394;</mi><mi>V</mi></mrow><mrow><mi mathvariant="normal">&#x0394;</mi><mfrac><mn>1</mn><mi>&#x03BB;</mi></mfrac></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>h</mi><mi>c</mi></mrow><mi>e</mi></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1958" style="width: 5.33em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.357em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1004.36em 3.947em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1959"><span class="mfrac" id="MathJax-Span-1960"><span style="display: inline-block; position: relative; width: 1.539em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1001.44em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.715em;"><span class="mrow" id="MathJax-Span-1961"><span class="mi" id="MathJax-Span-1962" style="font-family: STIXGeneral;">Δ</span><span class="mi" id="MathJax-Span-1963" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.974em 1001.44em 4.562em -999.997em); top: -3.122em; left: 50%; margin-left: -0.715em;"><span class="mrow" id="MathJax-Span-1964"><span class="mi" id="MathJax-Span-1965" style="font-family: STIXGeneral;">Δ</span><span class="mfrac" id="MathJax-Span-1966"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.26em 4.152em -999.997em); top: -4.403em; left: 50%; margin-left: -0.151em;"><span class="mn" id="MathJax-Span-1967" style="font-size: 70.7%; font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.31em 4.152em -999.997em); top: -3.584em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-1968" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.46em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.464em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.54em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.539em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-1969" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1970" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.92em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.459em;"><span class="mrow" id="MathJax-Span-1971"><span class="mi" id="MathJax-Span-1972" style="font-family: STIXGeneral; font-style: italic;">h</span><span class="mi" id="MathJax-Span-1973" style="font-family: STIXGeneral; font-style: italic;">c</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1974" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.559em; border-left: 0px solid; width: 0px; height: 3.316em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>V</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mfrac><mn>1</mn><mi>λ</mi></mfrac></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>h</mi><mi>c</mi></mrow><mi>e</mi></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-238">\frac{\Delta V}{\Delta \frac{1}{\lambda}} = \frac{hc}{e}</script></span></p><p>The energy of a photon is equal to the energy of the electrons that are excited in the semiconductor material of the LED</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-239-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><mi>V</mi><mo>=</mo><mfrac><mrow><mi>h</mi><mi>c</mi></mrow><mi>&#x03BB;</mi></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1975" style="width: 4.562em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.742em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1003.74em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1976"><span class="mi" id="MathJax-Span-1977" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="mi" id="MathJax-Span-1978" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-1979" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1980" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.92em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.459em;"><span class="mrow" id="MathJax-Span-1981"><span class="mi" id="MathJax-Span-1982" style="font-family: STIXGeneral; font-style: italic;">h</span><span class="mi" id="MathJax-Span-1983" style="font-family: STIXGeneral; font-style: italic;">c</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1984" style="font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><mi>V</mi><mo>=</mo><mfrac><mrow><mi>h</mi><mi>c</mi></mrow><mi>λ</mi></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-239">eV = \frac{hc}{\lambda}</script></span></p><p>which can be rearranged to</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-240-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo>=</mo><mfrac><mrow><mi>h</mi><mi>c</mi></mrow><mi>e</mi></mfrac><mfrac><mn>1</mn><mi>&#x03BB;</mi></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1985" style="width: 5.074em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.152em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1004.15em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1986"><span class="mi" id="MathJax-Span-1987" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-1988" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-1989" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.078em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.92em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.459em;"><span class="mrow" id="MathJax-Span-1990"><span class="mi" id="MathJax-Span-1991" style="font-family: STIXGeneral; font-style: italic;">h</span><span class="mi" id="MathJax-Span-1992" style="font-family: STIXGeneral; font-style: italic;">c</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1993" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.08em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.078em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mfrac" id="MathJax-Span-1994"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-1995" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-1996" style="font-family: STIXGeneral; font-style: italic;">λ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>V</mi><mo>=</mo><mfrac><mrow><mi>h</mi><mi>c</mi></mrow><mi>e</mi></mfrac><mfrac><mn>1</mn><mi>λ</mi></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-240">V = \frac{hc}{e} \frac{1}{\lambda}</script></span></p><p>Comparing this to the form of a straight line graph, we can see that <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-241-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-1997" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-1998"><span class="mi" id="MathJax-Span-1999" style="font-family: STIXGeneral; font-style: italic;">h</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></span></span><script type="math/tex" id="MathJax-Element-241">h</script></span> is the gradient of the graph multiple by a constant</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-242-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>e</mi><mi>c</mi></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2000" style="width: 0.976em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.771em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(0.925em 1000.77em 3.076em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-2001"><span class="mfrac" id="MathJax-Span-2002"><span style="display: inline-block; position: relative; width: 0.566em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-2003" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-2004" style="font-family: STIXGeneral; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.57em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.566em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.316em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mfrac><mi>e</mi><mi>c</mi></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-242">\frac{e}{c}</script></span></p><p>If we measure the gradient of the graph of activation voltage against the reciprocal of wavelength, and then multiply by the charge on an electron before dividing by the speed of light, we find a value for <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-243-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2005" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2006"><span class="mi" id="MathJax-Span-2007" style="font-family: STIXGeneral; font-style: italic;">h</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></span></span><script type="math/tex" id="MathJax-Element-243">h</script></span>.</p><h2 id="the-electronvolt"><a class="anchor hidden-xs" href="#the-electronvolt" title="the-electronvolt"><span class="octicon octicon-link"></span></a>The electronvolt</h2><p><strong>Electronvolt</strong> The kinetic energy gained by an electron when it is accelerated through a potential difference of 1 volt</p><p>We calculate the energy gained by an electron using</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-244-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>energy gained</mtext><mo>=</mo><mtext>charge</mtext><mo>&#x00D7;</mo><mtext>potential difference</mtext></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2008" style="width: 22.441em; display: inline-block;"><span style="display: inline-block; position: relative; width: 18.394em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1018.39em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2009"><span class="mtext" id="MathJax-Span-2010" style="font-family: STIXGeneral;">energy gained</span><span class="mo" id="MathJax-Span-2011" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mtext" id="MathJax-Span-2012" style="font-family: STIXGeneral; padding-left: 0.31em;">charge</span><span class="mo" id="MathJax-Span-2013" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="mtext" id="MathJax-Span-2014" style="font-family: STIXGeneral; padding-left: 0.259em;">potential difference</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtext>energy gained</mtext><mo>=</mo><mtext>charge</mtext><mo>×</mo><mtext>potential difference</mtext></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-244">\text{energy gained} = \text{charge} \times \text{potential difference}</script></span></p><p>or</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-245-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo>=</mo><mi>Q</mi><mi>V</mi></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2015" style="width: 4.101em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.332em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.33em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2016"><span class="mi" id="MathJax-Span-2017" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2018" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2019" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">Q</span><span class="mi" id="MathJax-Span-2020" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>E</mi><mo>=</mo><mi>Q</mi><mi>V</mi></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-245">E = QV</script></span></p><p>The charge on an electron is <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-246-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1.602</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>19</mn></mrow></msup><mtext>&#xA0;C</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2021" style="width: 7.994em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.56em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1006.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2022"><span class="mn" id="MathJax-Span-2023" style="font-family: STIXGeneral;">1.602</span><span class="mo" id="MathJax-Span-2024" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-2025" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-2026" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-2027"><span class="mrow" id="MathJax-Span-2028"><span class="mo" id="MathJax-Span-2029" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-2030" style="font-size: 70.7%; font-family: STIXGeneral;">19</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mtext" id="MathJax-Span-2031" style="font-family: STIXGeneral;"> C</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1.602</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>19</mn></mrow></msup><mtext> C</mtext></math></span></span><script type="math/tex" id="MathJax-Element-246">1.602\times10^{-19}\text{ C}</script></span> and the potential difference is <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-247-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mi>V</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2032" style="width: 1.539em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1001.23em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2033"><span class="mn" id="MathJax-Span-2034" style="font-family: STIXGeneral;">1</span><span class="mi" id="MathJax-Span-2035" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mi>V</mi></math></span></span><script type="math/tex" id="MathJax-Element-247">1V</script></span>, so <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-248-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mn>1.602</mn><mo>&#x00D7;</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>19</mn></mrow></msup><mtext>&#xA0;J</mtext></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2036" style="width: 10.044em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.199em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1008.2em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2037"><span class="mi" id="MathJax-Span-2038" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2039" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-2040" style="font-family: STIXGeneral; padding-left: 0.31em;">1.602</span><span class="mo" id="MathJax-Span-2041" style="font-family: STIXGeneral; padding-left: 0.259em;">×</span><span class="msubsup" id="MathJax-Span-2042" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-2043" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-2044"><span class="mrow" id="MathJax-Span-2045"><span class="mo" id="MathJax-Span-2046" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-2047" style="font-size: 70.7%; font-family: STIXGeneral;">19</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mtext" id="MathJax-Span-2048" style="font-family: STIXGeneral;"> J</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mn>1.602</mn><mo>×</mo><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>19</mn></mrow></msup><mtext> J</mtext></math></span></span><script type="math/tex" id="MathJax-Element-248">E = 1.602\times10^{-19}\text{ J}</script></span>.</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-249-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>1</mn><mtext>&#xA0;eV</mtext><mo>=</mo><mn>1.602</mn><mi>x</mi><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>&#x2212;</mo><mn>19</mn></mrow></msup><mtext>&#xA0;J</mtext></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2049" style="width: 10.761em; display: inline-block;"><span style="display: inline-block; position: relative; width: 8.814em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.539em 1008.81em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2050"><span class="mn" id="MathJax-Span-2051" style="font-family: STIXGeneral;">1</span><span class="mtext" id="MathJax-Span-2052" style="font-family: STIXGeneral;"> eV</span><span class="mo" id="MathJax-Span-2053" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mn" id="MathJax-Span-2054" style="font-family: STIXGeneral; padding-left: 0.31em;">1.602</span><span class="mi" id="MathJax-Span-2055" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="msubsup" id="MathJax-Span-2056"><span style="display: inline-block; position: relative; width: 2.257em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.98em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mn" id="MathJax-Span-2057" style="font-family: STIXGeneral;">10</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 1.027em;"><span class="texatom" id="MathJax-Span-2058"><span class="mrow" id="MathJax-Span-2059"><span class="mo" id="MathJax-Span-2060" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mn" id="MathJax-Span-2061" style="font-size: 70.7%; font-family: STIXGeneral;">19</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mtext" id="MathJax-Span-2062" style="font-family: STIXGeneral;"> J</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.253em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mn>1</mn><mtext> eV</mtext><mo>=</mo><mn>1.602</mn><mi>x</mi><msup><mn>10</mn><mrow class="MJX-TeXAtom-ORD"><mo>−</mo><mn>19</mn></mrow></msup><mtext> J</mtext></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-249">1\text{ eV} = 1.602x10^{-19}\text{ J}</script></span></p><h2 id="the-photoelectric-effect"><a class="anchor hidden-xs" href="#the-photoelectric-effect" title="the-photoelectric-effect"><span class="octicon octicon-link"></span></a>The photoelectric effect</h2><p>When electromagnetic radiation of a particular frequency is shone on the surface of a metal, electrons are emitted from its surface.<br>
This phenomenon is known as the photoelectric effect, and electrons that are released are called photoelectrons.</p><h3 id="demonstrating-the-photoelectric-effect-with-a-gold-leaf-electroscope"><a class="anchor hidden-xs" href="#demonstrating-the-photoelectric-effect-with-a-gold-leaf-electroscope" title="demonstrating-the-photoelectric-effect-with-a-gold-leaf-electroscope"><span class="octicon octicon-link"></span></a>Demonstrating the photoelectric effect with a gold-leaf electroscope</h3><p>The gold-leaf electroscope is composed of a brass stem to which a thin gold leaf is attached. There is a metal cap attached to the top of the stem and the metal to be irradiated with electromagnetic radiation is placed on the metal cap.</p><p>A metal plate (usually zinc is placed on the metal and is then charged negatively by touching it with a negatively charged polythene rod, or by electrostatic induction.<br>
When this is done the metal stem and gold leaf will also become negatively charged, meaning that the stem and the leaf will repel each other.</p><p>It is also possible to make the zinc plate, metal stem, and the gold leaf positively charged.</p><h3 id="observations-of-the-photoelectric-effect"><a class="anchor hidden-xs" href="#observations-of-the-photoelectric-effect" title="observations-of-the-photoelectric-effect"><span class="octicon octicon-link"></span></a>Observations of the photoelectric effect</h3><p>When visible light is incident on a positively charged metal plate, there is no movement of the gold leaf. The same is also true when UV light is shone on the positively charged plate.</p><p>Shining visible light on the negatively charged zinc plate also causes no movements in the gold leaf. No matter how bright or intense the visible light is, no movement of the gold leaf is observed.</p><p>However, when UV light is shone on the negatively charged zinc plate, the gold leaf falls.</p><p>This shows that the metal plate loses its negative charge through the emission of electrons, which are repelled by the negative charge on the electroscope. The discharge of the electroscope cannot be caused by ions in the air, because the electroscope is in a sealed vacuum.</p><p>Further experiments show that</p><ul>
<li>Electrons will be emitted from the surface of a metal only if the incident radiation is above a minimum frequency called the threshold frequency. This is determined by the metal itself. Below this frequency, no electrons are emitted.</li>
<li>Emission of electrons starts the instant the surface starts to be irradiated, provided that the incident radiation exceeds the threshold frequency.</li>
</ul><p><strong>Threshold frequency</strong> The lowest frequency of radiation that will result in the emission of electrons from a particular metal surface</p><h2 id="explaining-the-photoelectric-effect"><a class="anchor hidden-xs" href="#explaining-the-photoelectric-effect" title="explaining-the-photoelectric-effect"><span class="octicon octicon-link"></span></a>Explaining the photoelectric effect</h2><h3 id="the-work-function"><a class="anchor hidden-xs" href="#the-work-function" title="the-work-function"><span class="octicon octicon-link"></span></a>The work function</h3><p>In order for electrons to be released from the metal, the frequency of the incident radiation must exceed the threshold frequency. This is needed to provide at least the minimum energy required to release an electron from the surface.</p><p><strong>Work function</strong> The work function of a metal is the minimum energy required to release an electron from its surface, overcoming the electrostatic attraction between the electron and the positive metal ions</p><p>When an incident photon collides with the metal</p><ul>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-250-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mi>f</mi><mo>&lt;</mo><mi>&#x03D5;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2063" style="width: 3.537em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.871em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.82em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2064"><span class="mi" id="MathJax-Span-2065" style="font-family: STIXGeneral; font-style: italic;">h</span><span class="mi" id="MathJax-Span-2066" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-2067" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mi" id="MathJax-Span-2068" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mi>f</mi><mo><</mo><mi>ϕ</mi></math></span></span><script type="math/tex" id="MathJax-Element-250">hf < \phi</script></span> there is no emission of electrons</li>
<li>If <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-251-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><msub><mi>f</mi><mn>0</mn></msub><mo>&lt;</mo><mi>&#x03D5;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2069" style="width: 3.845em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.128em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1003.08em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2070"><span class="mi" id="MathJax-Span-2071" style="font-family: STIXGeneral; font-style: italic;">h</span><span class="msubsup" id="MathJax-Span-2072"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.357em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2073" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.259em;"><span class="mn" id="MathJax-Span-2074" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2075" style="font-family: STIXGeneral; padding-left: 0.31em;"><</span><span class="mi" id="MathJax-Span-2076" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><msub><mi>f</mi><mn>0</mn></msub><mo><</mo><mi>ϕ</mi></math></span></span><script type="math/tex" id="MathJax-Element-251">hf_0 < \phi</script></span> an electron is released, but it has zero kinetic energy</li>
<li><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-252-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mi>f</mi><mo>&gt;</mo><mi>&#x03D5;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2077" style="width: 3.537em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.871em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.82em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2078"><span class="mi" id="MathJax-Span-2079" style="font-family: STIXGeneral; font-style: italic;">h</span><span class="mi" id="MathJax-Span-2080" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-2081" style="font-family: STIXGeneral; padding-left: 0.31em;">></span><span class="mi" id="MathJax-Span-2082" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">ϕ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mi>f</mi><mo>></mo><mi>ϕ</mi></math></span></span><script type="math/tex" id="MathJax-Element-252">hf > \phi</script></span> an electron with <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-253-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>k</mi></msub><mo>&gt;</mo><mn>0</mn></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2083" style="width: 3.435em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.82em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1002.82em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2084"><span class="msubsup" id="MathJax-Span-2085"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2086" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-2087" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2088" style="font-family: STIXGeneral; padding-left: 0.31em;">></span><span class="mn" id="MathJax-Span-2089" style="font-family: STIXGeneral; padding-left: 0.31em;">0</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mi>k</mi></msub><mo>></mo><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-253">E_k > 0</script></span> is released from the metal surface</li>
</ul><p>As the intensity of the radiation is increased, more electrons are released, but their kinetic energy does not increase.</p><h2 id="calculations-with-the-photoelectric-effect"><a class="anchor hidden-xs" href="#calculations-with-the-photoelectric-effect" title="calculations-with-the-photoelectric-effect"><span class="octicon octicon-link"></span></a>Calculations with the photoelectric effect</h2><h3 id="determining-the-maximum-kinetic-energy-of-photoelectrons"><a class="anchor hidden-xs" href="#determining-the-maximum-kinetic-energy-of-photoelectrons" title="determining-the-maximum-kinetic-energy-of-photoelectrons"><span class="octicon octicon-link"></span></a>Determining the maximum kinetic energy of photoelectrons</h3><p>We can equate the work done to accelerate or decelerate charged particles with the kinetic energy transferred to the electrons</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-254-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2090" style="width: 5.945em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.869em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1004.87em 3.384em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2091"><span class="mi" id="MathJax-Span-2092" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="mi" id="MathJax-Span-2093" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-2094" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2095" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-2096" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-2097" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-2098" style="font-family: STIXGeneral; font-style: italic;">m</span><span class="msubsup" id="MathJax-Span-2099"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2100" style="font-family: STIXGeneral; font-style: italic;">v</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.464em;"><span class="mn" id="MathJax-Span-2101" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.872em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-254">eV = \frac{1}{2}mv^2</script></span></p><p>At the stopping potential, all the emitted electrons have been brought to rest, so we obtain a value for the maximum kinetic energy of the photoelectrons by multiplying the stopping potential <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-255-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>0</mn></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2102" style="width: 1.283em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1001.03em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-2103"><span class="msubsup" id="MathJax-Span-2104"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2105" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mn" id="MathJax-Span-2106" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>0</mn></msub></math></span></span><script type="math/tex" id="MathJax-Element-255">V_0</script></span> by the charge on an electron, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-256-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2107" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2108"><span class="mi" id="MathJax-Span-2109" style="font-family: STIXGeneral; font-style: italic;">e</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></span></span><script type="math/tex" id="MathJax-Element-256">e</script></span>.</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-257-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><msub><mi>V</mi><mn>0</mn></msub><mo>=</mo><mi>K</mi><msub><mi>E</mi><mrow class="MJX-TeXAtom-ORD"><mtext>max</mtext></mrow></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2110" style="width: 6.56em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.382em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1005.38em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2111"><span class="mi" id="MathJax-Span-2112" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="msubsup" id="MathJax-Span-2113"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2114" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mn" id="MathJax-Span-2115" style="font-size: 70.7%; font-family: STIXGeneral;">0</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2116" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2117" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="msubsup" id="MathJax-Span-2118"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2119" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="texatom" id="MathJax-Span-2120"><span class="mrow" id="MathJax-Span-2121"><span class="mtext" id="MathJax-Span-2122" style="font-size: 70.7%; font-family: STIXGeneral;">max</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><msub><mi>V</mi><mn>0</mn></msub><mo>=</mo><mi>K</mi><msub><mi>E</mi><mrow class="MJX-TeXAtom-ORD"><mtext>max</mtext></mrow></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-257">eV_0 = KE_{\text{max}}</script></span></p><h3 id="rate-of-emission-of-electrons"><a class="anchor hidden-xs" href="#rate-of-emission-of-electrons" title="rate-of-emission-of-electrons"><span class="octicon octicon-link"></span></a>Rate of emission of electrons</h3><ul>
<li>The number of emitted photoelectrons is directly proportional to the intensity of the incident radiation, provided that the radiation is of a single frequency and is above the threshold frequency</li>
<li>The kinetic energy of the emitted photoelectrons is not affected by the intensity of the incident radiation</li>
<li>The kinetic energy of the emitted photoelectrons is affected by the frequency of the incident photons. If the frequency of the incident radiation is increased, the kinetic energy of the photoelectrons also increases</li>
</ul><h3 id="millikan’s-photoelectric-experiment"><a class="anchor hidden-xs" href="#millikan’s-photoelectric-experiment" title="millikan’s-photoelectric-experiment"><span class="octicon octicon-link"></span></a>Millikan’s photoelectric experiment</h3><p>Millikan irradiated the metals sodium, potassium, and lithium with monochromatic light. By applying a positive potential to the target metal, he could decelerate the electrons.<br>
He increased the size of the potential until the most energetic electrons were unable to reach the cathode, causing the current to fall to zero.<br>
This enabled him to determine the stopping potential, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-258-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>s</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2123" style="width: 1.181em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1000.98em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-2124"><span class="msubsup" id="MathJax-Span-2125"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2126" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-2127" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>s</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-258">V_s</script></span>, of each metal, and show that this depends on the frequency of the radiation.</p><p>Not only did Millikan’s experiment obtain accurate values for the work functions of the metals, it also obtained an accurate value for Planck’s constant.</p><p>We know that</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-259-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><msub><mi>V</mi><mi>s</mi></msub><mo>=</mo><mi>K</mi><msub><mi>E</mi><mrow class="MJX-TeXAtom-ORD"><mtext>max</mtext></mrow></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2128" style="width: 6.509em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.33em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1005.33em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2129"><span class="mi" id="MathJax-Span-2130" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="msubsup" id="MathJax-Span-2131"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2132" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-2133" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2134" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2135" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="msubsup" id="MathJax-Span-2136"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2137" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="texatom" id="MathJax-Span-2138"><span class="mrow" id="MathJax-Span-2139"><span class="mtext" id="MathJax-Span-2140" style="font-size: 70.7%; font-family: STIXGeneral;">max</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><msub><mi>V</mi><mi>s</mi></msub><mo>=</mo><mi>K</mi><msub><mi>E</mi><mrow class="MJX-TeXAtom-ORD"><mtext>max</mtext></mrow></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-259">eV_s = KE_{\text{max}}</script></span></p><p>and</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-260-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>h</mi><mi>f</mi><mo>=</mo><mi>&#x03C6;</mi><mo>+</mo><mi>K</mi><msub><mi>E</mi><mrow class="MJX-TeXAtom-ORD"><mtext>max</mtext></mrow></msub></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2141" style="width: 8.148em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.662em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1006.66em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2142"><span class="mi" id="MathJax-Span-2143" style="font-family: STIXGeneral; font-style: italic;">h</span><span class="mi" id="MathJax-Span-2144" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-2145" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2146" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">φ</span><span class="mo" id="MathJax-Span-2147" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-2148" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">K<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.054em;"></span></span><span class="msubsup" id="MathJax-Span-2149"><span style="display: inline-block; position: relative; width: 1.898em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.62em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2150" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="texatom" id="MathJax-Span-2151"><span class="mrow" id="MathJax-Span-2152"><span class="mtext" id="MathJax-Span-2153" style="font-size: 70.7%; font-family: STIXGeneral;">max</span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>h</mi><mi>f</mi><mo>=</mo><mi>φ</mi><mo>+</mo><mi>K</mi><msub><mi>E</mi><mrow class="MJX-TeXAtom-ORD"><mtext>max</mtext></mrow></msub></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-260">hf = \varphi + KE_{\text{max}}</script></span></p><p>If we divide both sides by the electron charge, e, and rearrange for <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-261-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>s</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2154" style="width: 1.181em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1000.98em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-2155"><span class="msubsup" id="MathJax-Span-2156"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2157" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-2158" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>s</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-261">V_s</script></span> we have</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-262-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>V</mi><mi>s</mi></msub><mo>=</mo><mrow><mo>(</mo><mfrac><mi>h</mi><mi>e</mi></mfrac><mo>)</mo></mrow><mi>f</mi><mo>&#x2212;</mo><mfrac><mi>&#x03C6;</mi><mi>e</mi></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2159" style="width: 8.763em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.175em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(2.257em 1007.17em 4.716em -999.997em); top: -3.788em; left: 0em;"><span class="mrow" id="MathJax-Span-2160"><span class="msubsup" id="MathJax-Span-2161"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2162" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-2163" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span class="mo" id="MathJax-Span-2164" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mrow" id="MathJax-Span-2165" style="padding-left: 0.31em;"><span class="mo" id="MathJax-Span-2166" style="vertical-align: -0.459em;"><span><span style="font-size: 111%; font-family: STIXSizeTwoSym;">(</span></span></span><span class="mfrac" id="MathJax-Span-2167"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-2168" style="font-family: STIXGeneral; font-style: italic;">h</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-2169" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mo" id="MathJax-Span-2170" style="vertical-align: -0.459em;"><span><span style="font-size: 111%; font-family: STIXSizeTwoSym;">)</span></span></span></span><span class="mi" id="MathJax-Span-2171" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.207em;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span><span class="mo" id="MathJax-Span-2172" style="font-family: STIXGeneral; padding-left: 0.259em;">−</span><span class="mfrac" id="MathJax-Span-2173" style="padding-left: 0.259em;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.384em 1000.62em 4.357em -999.997em); top: -4.659em; left: 50%; margin-left: -0.305em;"><span class="mi" id="MathJax-Span-2174" style="font-family: STIXGeneral; font-style: italic;">φ</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.202em;"><span class="mi" id="MathJax-Span-2175" style="font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.72em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.72em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.794em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.997em; border-left: 0px solid; width: 0px; height: 2.753em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>V</mi><mi>s</mi></msub><mo>=</mo><mrow><mo>(</mo><mfrac><mi>h</mi><mi>e</mi></mfrac><mo>)</mo></mrow><mi>f</mi><mo>−</mo><mfrac><mi>φ</mi><mi>e</mi></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-262">V_s = \left(\frac{h}{e}\right)f - \frac{\varphi}{e}</script></span></p><p>This equation is of the form <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-263-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>c</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2176" style="width: 5.587em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.562em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.847em 1004.56em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2177"><span class="mi" id="MathJax-Span-2178" style="font-family: STIXGeneral; font-style: italic;">y</span><span class="mo" id="MathJax-Span-2179" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2180" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">m</span><span class="mi" id="MathJax-Span-2181" style="font-family: STIXGeneral; font-style: italic;">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2182" style="font-family: STIXGeneral; padding-left: 0.259em;">+</span><span class="mi" id="MathJax-Span-2183" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.259em;">c</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.066em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>c</mi></math></span></span><script type="math/tex" id="MathJax-Element-263">y = mx + c</script></span>, meaning that if we plot a graph of stopping potential, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-264-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>s</mi></msub></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2184" style="width: 1.181em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.386em 1000.98em 2.513em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-2185"><span class="msubsup" id="MathJax-Span-2186"><span style="display: inline-block; position: relative; width: 0.976em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1000.67em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2187" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.617em;"><span class="mi" id="MathJax-Span-2188" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">s</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 1.128em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>s</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-264">V_s</script></span>, against frequency, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-265-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2189" style="width: 0.515em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.412em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2190"><span class="mi" id="MathJax-Span-2191" style="font-family: STIXGeneral; font-style: italic;">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.156em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math></span></span><script type="math/tex" id="MathJax-Element-265">f</script></span>, we will have a graph of gradient <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-266-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2192" style="width: 0.669em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.515em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.51em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2193"><span class="mi" id="MathJax-Span-2194" style="font-family: STIXGeneral; font-style: italic;">h</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></span></span><script type="math/tex" id="MathJax-Element-266">h</script></span> over <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-267-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2195" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2196"><span class="mi" id="MathJax-Span-2197" style="font-family: STIXGeneral; font-style: italic;">e</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></span></span><script type="math/tex" id="MathJax-Element-267">e</script></span>, and a <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-268-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2198" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2199"><span class="mi" id="MathJax-Span-2200" style="font-family: STIXGeneral; font-style: italic;">y</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></span></span><script type="math/tex" id="MathJax-Element-268">y</script></span> intercept equal to <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-269-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>&#x2212;</mo><mi>&#x03C6;</mi></mrow><mi>e</mi></mfrac></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2201" style="width: 1.539em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.232em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.13em 1001.23em 2.718em -999.997em); top: -2.2em; left: 0em;"><span class="mrow" id="MathJax-Span-2202"><span class="mfrac" id="MathJax-Span-2203"><span style="display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.435em 1000.92em 4.306em -999.997em); top: -4.557em; left: 50%; margin-left: -0.459em;"><span class="mrow" id="MathJax-Span-2204"><span class="mo" id="MathJax-Span-2205" style="font-size: 70.7%; font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-2206" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">φ</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.537em 1000.31em 4.152em -999.997em); top: -3.635em; left: 50%; margin-left: -0.151em;"><span class="mi" id="MathJax-Span-2207" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">e</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.03em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.027em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.205em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.497em; border-left: 0px solid; width: 0px; height: 1.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>−</mo><mi>φ</mi></mrow><mi>e</mi></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-269">\frac{-\varphi}{e}</script></span>.<br>
If we measure the gradient and multiply it by <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-270-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2208" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2209"><span class="mi" id="MathJax-Span-2210" style="font-family: STIXGeneral; font-style: italic;">e</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math></span></span><script type="math/tex" id="MathJax-Element-270">e</script></span>, we get a value for Planck’s constant.<br>
If we read the <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-271-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2211" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.46em 2.923em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2212"><span class="mi" id="MathJax-Span-2213" style="font-family: STIXGeneral; font-style: italic;">y</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.309em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math></span></span><script type="math/tex" id="MathJax-Element-271">y</script></span> intercept and multiply by <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-272-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>&#x2212;</mo><mi>e</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2214" style="width: 1.386em; display: inline-block;"><span style="display: inline-block; position: relative; width: 1.13em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.847em 1001.08em 2.871em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2215"><span class="mo" id="MathJax-Span-2216" style="font-family: STIXGeneral;">−</span><span class="mi" id="MathJax-Span-2217" style="font-family: STIXGeneral; font-style: italic;">e</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.247em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mi>e</mi></math></span></span><script type="math/tex" id="MathJax-Element-272">-e</script></span>, we get a value for the work function of the metal.</p><h2 id="wave-particle-duality"><a class="anchor hidden-xs" href="#wave-particle-duality" title="wave-particle-duality"><span class="octicon octicon-link"></span></a>Wave particle duality</h2><h3 id="the-wavelength-of-electrons"><a class="anchor hidden-xs" href="#the-wavelength-of-electrons" title="the-wavelength-of-electrons"><span class="octicon octicon-link"></span></a>The wavelength of electrons</h3><p>De Broglie proposed that if electrons and other particles travel through space as a wave, they have an associated wavelength. By combining the idea of an energy quantum with <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-273-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2218" style="width: 4.255em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.486em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.591em 1003.49em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2219"><span class="mi" id="MathJax-Span-2220" style="font-family: STIXGeneral; font-style: italic;">E<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span class="mo" id="MathJax-Span-2221" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mi" id="MathJax-Span-2222" style="font-family: STIXGeneral; font-style: italic; padding-left: 0.31em;">m</span><span class="msubsup" id="MathJax-Span-2223"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2224" style="font-family: STIXGeneral; font-style: italic;">c</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.352em; left: 0.464em;"><span class="mn" id="MathJax-Span-2225" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.191em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mi>m</mi><msup><mi>c</mi><mn>2</mn></msup></math></span></span><script type="math/tex" id="MathJax-Element-273">E = mc^2</script></span> he derived a formula for the wavelength, <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-274-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>&#x03BB;</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2226" style="width: 0.566em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.464em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.41em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2227"><span class="mi" id="MathJax-Span-2228" style="font-family: STIXGeneral; font-style: italic;">λ</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 1.003em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math></span></span><script type="math/tex" id="MathJax-Element-274">\lambda</script></span>, of a particle</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-275-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>&#x03BB;</mi><mo>=</mo><mfrac><mi>h</mi><mrow><mi>m</mi><mi>v</mi></mrow></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2229" style="width: 3.947em; display: inline-block;"><span style="display: inline-block; position: relative; width: 3.23em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1003.23em 3.435em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2230"><span class="mi" id="MathJax-Span-2231" style="font-family: STIXGeneral; font-style: italic;">λ</span><span class="mo" id="MathJax-Span-2232" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2233" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 1.283em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-2234" style="font-family: STIXGeneral; font-style: italic;">h</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.384em 1001.13em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.561em;"><span class="mrow" id="MathJax-Span-2235"><span class="mi" id="MathJax-Span-2236" style="font-family: STIXGeneral; font-style: italic;">m</span><span class="mi" id="MathJax-Span-2237" style="font-family: STIXGeneral; font-style: italic;">v</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1001.28em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 1.283em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.934em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>λ</mi><mo>=</mo><mfrac><mi>h</mi><mrow><mi>m</mi><mi>v</mi></mrow></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-275">\lambda = \frac{h}{mv}</script></span></p><p>where <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-276-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2238" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.949em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2239"><span class="mi" id="MathJax-Span-2240" style="font-family: STIXGeneral; font-style: italic;">m</span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.691em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></span></span><script type="math/tex" id="MathJax-Element-276">m</script></span> is the mass of the particle, and h is the Planck constant.</p><p>This equation was confirmed by observing the behaviour of electrons that had been diffracted from the surface of a nickel crystal.<br>
By accelerating electrons, of charge e, through a potential difference <span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><span class="MathJax" id="MathJax-Element-277-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math>" role="presentation" style="position: relative;"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2241" style="width: 0.873em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.72em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.744em 1000.72em 2.718em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2242"><span class="mi" id="MathJax-Span-2243" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.059em; border-left: 0px solid; width: 0px; height: 0.941em;"></span></span></nobr><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math></span></span><script type="math/tex" id="MathJax-Element-277">V</script></span>, they observed a pattern of electron diffraction from which they could calculate the electron’s wavelength.</p><p>The predicted wavelength is found by equating the work done to accelerate the electrons with the kinetic energy transferred into the electrons</p><p>Substituting</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-278-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2244" style="width: 5.945em; display: inline-block;"><span style="display: inline-block; position: relative; width: 4.869em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.078em 1004.87em 3.384em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2245"><span class="mi" id="MathJax-Span-2246" style="font-family: STIXGeneral; font-style: italic;">e</span><span class="mi" id="MathJax-Span-2247" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mo" id="MathJax-Span-2248" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2249" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 0.617em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.41em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-2250" style="font-family: STIXGeneral;">1</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -3.327em; left: 50%; margin-left: -0.254em;"><span class="mn" id="MathJax-Span-2251" style="font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1000.62em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.617em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span><span class="mi" id="MathJax-Span-2252" style="font-family: STIXGeneral; font-style: italic;">m</span><span class="msubsup" id="MathJax-Span-2253"><span style="display: inline-block; position: relative; width: 0.873em; height: 0px;"><span style="position: absolute; clip: rect(3.384em 1000.41em 4.152em -999.997em); top: -3.993em; left: 0em;"><span class="mi" id="MathJax-Span-2254" style="font-family: STIXGeneral; font-style: italic;">v</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; top: -4.403em; left: 0.464em;"><span class="mn" id="MathJax-Span-2255" style="font-size: 70.7%; font-family: STIXGeneral;">2</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.872em; border-left: 0px solid; width: 0px; height: 2.628em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>e</mi><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>m</mi><msup><mi>v</mi><mn>2</mn></msup></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-278">eV = \frac{1}{2}mv^2</script></span></p><p>into the de Broglie equation gives</p><p><span class="mathjax"><span class="MathJax_Preview" style="color: inherit; display: none;"></span><div class="MathJax_Display" style="text-align: center;"><span class="MathJax" id="MathJax-Element-279-Frame" tabindex="0" style="text-align: center; position: relative;" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>&#x03BB;</mi><mo>=</mo><mfrac><mi>h</mi><msqrt><mn>2</mn><mi>m</mi><mi>V</mi><mi>e</mi></msqrt></mfrac></math>" role="presentation"><nobr aria-hidden="true"><span class="math" id="MathJax-Span-2256" style="width: 6.355em; display: inline-block;"><span style="display: inline-block; position: relative; width: 5.177em; height: 0px; font-size: 122%;"><span style="position: absolute; clip: rect(1.027em 1005.18em 3.64em -999.997em); top: -2.559em; left: 0em;"><span class="mrow" id="MathJax-Span-2257"><span class="mi" id="MathJax-Span-2258" style="font-family: STIXGeneral; font-style: italic;">λ</span><span class="mo" id="MathJax-Span-2259" style="font-family: STIXGeneral; padding-left: 0.31em;">=</span><span class="mfrac" id="MathJax-Span-2260" style="padding-left: 0.31em;"><span style="display: inline-block; position: relative; width: 3.23em; height: 0px; margin-right: 0.105em; margin-left: 0.105em;"><span style="position: absolute; clip: rect(3.179em 1000.46em 4.152em -999.997em); top: -4.659em; left: 50%; margin-left: -0.254em;"><span class="mi" id="MathJax-Span-2261" style="font-family: STIXGeneral; font-style: italic;">h</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1003.13em 4.306em -999.997em); top: -3.225em; left: 50%; margin-left: -1.534em;"><span class="msqrt" id="MathJax-Span-2262"><span style="display: inline-block; position: relative; width: 3.128em; height: 0px;"><span style="position: absolute; clip: rect(3.179em 1002.31em 4.152em -999.997em); top: -3.993em; left: 0.72em;"><span class="mrow" id="MathJax-Span-2263"><span class="mn" id="MathJax-Span-2264" style="font-family: STIXGeneral;">2</span><span class="mi" id="MathJax-Span-2265" style="font-family: STIXGeneral; font-style: italic;">m</span><span class="mi" id="MathJax-Span-2266" style="font-family: STIXGeneral; font-style: italic;">V<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.105em;"></span></span><span class="mi" id="MathJax-Span-2267" style="font-family: STIXGeneral; font-style: italic;">e</span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(3.025em 1002.36em 3.384em -999.997em); top: -3.993em; left: 0.72em;"><span style="display: inline-block; position: relative; width: 2.359em; height: 0px;"><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 0em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; font-family: STIXGeneral; top: -3.993em; left: 1.898em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.361em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 0.72em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.13em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="font-family: STIXGeneral; position: absolute; top: -3.993em; left: 1.488em;">‾<span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(2.923em 1000.77em 4.152em -999.997em); top: -3.891em; left: 0em;"><span style="font-family: STIXVariants;">√</span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.998em;"></span></span><span style="position: absolute; clip: rect(0.873em 1003.23em 1.232em -999.997em); top: -1.278em; left: 0em;"><span style="display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 3.23em; height: 0px;"></span><span style="display: inline-block; width: 0px; height: 1.078em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.564em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.184em; border-left: 0px solid; width: 0px; height: 2.941em;"></span></span></nobr><span class="MJX_Assistive_MathML MJX_Assistive_MathML_Block" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>λ</mi><mo>=</mo><mfrac><mi>h</mi><msqrt><mn>2</mn><mi>m</mi><mi>V</mi><mi>e</mi></msqrt></mfrac></math></span></span></div><script type="math/tex; mode=display" id="MathJax-Element-279">\lambda = \frac{h}{\sqrt{2mVe}}</script></span></p><p><img src="http://s3-ap-southeast-1.amazonaws.com/subscriber.images/physics/2016/10/03090849/Davisson-Germer-experiment.png" alt=""></p><h3 id="wave-particle-duality-for-matter"><a class="anchor hidden-xs" href="#wave-particle-duality-for-matter" title="wave-particle-duality-for-matter"><span class="octicon octicon-link"></span></a>Wave-particle duality for matter</h3><p>The experiment confirming De Broglie’s results shows that matter could exhibit wave properties. According to De Broglie’s equation, all moving objects have a wavelength, including solids.<br>
The greater the mass of the object, the shorter the wavelength, and macroscopic objects have such small wavelengths that their effects are negligible.</p><p>In modern physics, the wave nature of a particle is considered a probability wave. It represents the probability of an observer measuring a particle at a given place and time.</p></div>
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<div class="toc"><ul class="nav"><li class=""><a href="#forces-and-motion">Forces and motion</a><ul class="nav"><li><a href="#constant-acceleration-equations">Constant acceleration equations</a></li><li><a href="#free-fall-and-projectile-motion">Free fall and projectile motion</a></li><li><a href="#measurement-of-g">Measurement of g</a><ul class="nav"><li><a href="#investigation-use-of-a-trap-door-and-an-electromagnet-to-determine-g">Investigation- Use of a trap door and an electromagnet to determine g</a></li></ul></li><li><a href="#car-stopping-distances">Car stopping distances</a></li><li><a href="#force-and-the-newton">Force and the Newton</a></li><li><a href="#dynamics">Dynamics</a><ul class="nav"><li><a href="#free-body-diagrams">Free body diagrams</a></li></ul></li><li><a href="#drag-and-terminal-velocity">Drag and terminal velocity</a><ul class="nav"><li><a href="#terminal-velocity">Terminal velocity</a></li></ul></li><li><a href="#equilibrium">Equilibrium</a></li><li><a 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X-Rays</a></li></ul></li><li><a href="#polarisation">Polarisation</a></li><li><a href="#rotating-the-plane-of-polarisation">Rotating the plane of polarisation</a><ul class="nav"><li><a href="#uses-of-polarisation">Uses of polarisation</a></li></ul></li><li><a href="#refraction-of-light">Refraction of light</a><ul class="nav"><li><a href="#snell’s-law">Snell’s law</a></li></ul></li><li><a href="#total-internal-reflection">Total internal reflection</a><ul class="nav"><li><a href="#the-critical-angle">The critical angle</a></li><li><a href="#determining-the-critical-angle-for-a-transport-material">Determining the critical angle for a transport material</a></li></ul></li><li><a href="#interference1">Interference</a><ul class="nav"><li><a href="#the-principle-of-superposition">The principle of superposition</a></li><li><a href="#interference2">Interference</a></li><li><a href="#coherence">Coherence</a></li><li><a href="#interference-using-sound-waves">Interference using sound waves</a></li></ul></li><li><a href="#the-young-double-split-experiment">The Young double-split experiment</a><ul class="nav"><li><a href="#the-experiment">The experiment</a></li><li><a href="#finding-the-wavelength-of-the-light-source">Finding the wavelength of the light source</a></li></ul></li><li><a href="#the-diffraction-grating">The diffraction grating</a><ul class="nav"><li><a href="#multiple-slits">Multiple slits</a></li><li><a href="#measuring-the-wavelength-of-light-using-a-diffraction-grating">Measuring the wavelength of light using a diffraction grating</a></li></ul></li><li><a href="#stationary-waves">Stationary waves</a><ul class="nav"><li><a href="#formation-of-stationary-waves">Formation of stationary waves</a></li></ul></li><li><a href="#stationary-wave-experiments">Stationary wave experiments</a><ul class="nav"><li><a href="#stationary-waves-on-strings">Stationary waves on strings</a></li><li><a href="#fundamental-mode-of-vibration-and-harmonics">Fundamental mode of vibration and harmonics</a></li></ul></li><li><a href="#stationary-longitudinal-waves">Stationary longitudinal waves</a><ul class="nav"><li><a href="#stationary-waves-in-closed-tubes">Stationary waves in closed tubes</a></li><li><a href="#stationary-waves-in-closed-tubes1">Stationary waves in closed tubes</a></li></ul></li><li><a href="#measuring-the-speed-of-sound-in-air">Measuring the speed of sound in air</a></li></ul></li><li><a href="#quantum-physics">Quantum physics</a><ul class="nav"><li><a href="#the-photon">The photon</a><ul class="nav"><li><a href="#using-leds-to-determine-a-value-for-the-planck-constant">Using LEDs to determine a value for the Planck constant</a></li></ul></li><li><a href="#the-electronvolt">The electronvolt</a></li><li><a href="#the-photoelectric-effect">The photoelectric effect</a><ul class="nav"><li><a href="#demonstrating-the-photoelectric-effect-with-a-gold-leaf-electroscope">Demonstrating the photoelectric effect with a gold-leaf 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<div class="toc"><ul class="nav"><li class=""><a href="#forces-and-motion">Forces and motion</a><ul class="nav"><li class=""><a href="#constant-acceleration-equations">Constant acceleration equations</a></li><li class=""><a href="#free-fall-and-projectile-motion">Free fall and projectile motion</a></li><li class=""><a href="#measurement-of-g">Measurement of g</a><ul class="nav"><li class=""><a href="#investigation-use-of-a-trap-door-and-an-electromagnet-to-determine-g">Investigation- Use of a trap door and an electromagnet to determine g</a></li></ul></li><li class=""><a href="#car-stopping-distances">Car stopping distances</a></li><li class=""><a href="#force-and-the-newton">Force and the Newton</a></li><li class=""><a href="#dynamics">Dynamics</a><ul class="nav"><li class=""><a href="#free-body-diagrams">Free body diagrams</a></li></ul></li><li class=""><a href="#drag-and-terminal-velocity">Drag and terminal velocity</a><ul class="nav"><li class=""><a href="#terminal-velocity">Terminal velocity</a></li></ul></li><li class=""><a href="#equilibrium">Equilibrium</a></li><li class=""><a href="#turning-forces">Turning forces</a><ul class="nav"><li class=""><a href="#principle-of-moments">Principle of moments</a></li></ul></li><li class=""><a href="#centre-of-mass">Centre of mass</a><ul class="nav"><li class=""><a href="#method-for-determining-the-mass-of-irregular-shapes">Method for determining the mass of irregular shapes</a></li></ul></li><li><a href="#density">Density</a></li><li class=""><a href="#pressure">Pressure</a></li><li class=""><a href="#work-and-the-joule">Work and the joule</a></li><li class=""><a href="#the-conservation-of-energy">The conservation of energy</a></li><li class=""><a href="#potential-and-kinetic-energy">Potential and kinetic energy</a></li><li class=""><a href="#power-and-the-watt">Power and the watt</a></li><li><a href="#efficiency">Efficiency</a></li><li><a href="#deformation-of-materials">Deformation of materials</a><ul class="nav"><li><a href="#tensile-forces">Tensile forces</a></li><li><a href="#measuring-the-stretch-of-a-wire">Measuring the stretch of a wire</a></li><li><a href="#elastic-limit">Elastic limit</a></li></ul></li><li><a href="#hooke’s-law">Hooke’s law</a><ul class="nav"><li><a href="#springs-in-series-and-combination">Springs in series and combination</a></li><li><a href="#force-extension-graphs-and-work-done">Force extension graphs and work done</a></li></ul></li><li><a href="#the-young-modulus">The Young modulus</a><ul class="nav"><li><a href="#investigating-young-modulus">Investigating Young modulus</a></li></ul></li><li><a href="#categorisation-of-materials">Categorisation of materials</a></li><li><a href="#three-laws-of-motion">Three laws of motion</a><ul class="nav"><li><a href="#newton’s-first-law">Newton’s first law</a></li><li><a href="#newton’s-second-law">Newton’s second law</a></li><li><a href="#newton’s-third-law">Newton’s third law</a></li></ul></li><li><a href="#momentum">Momentum</a></li><li><a href="#force-and-impulse">Force and impulse</a><ul class="nav"><li><a href="#elastic-and-inelastic-collisions">Elastic and inelastic collisions</a></li></ul></li><li><a href="#electrons-waves-and-photons">Electrons, Waves, and Photons</a></li><li><a href="#electric-circuit-components">Electric circuit components</a></li><li><a href="#electric-current-and-charge">Electric current and charge</a><ul class="nav"><li><a href="#kirchhoff’s-first-law0">Kirchhoff’s first law</a></li></ul></li><li><a href="#electron-drift-velocity">Electron drift velocity</a><ul class="nav"><li><a href="#calculating-electron-drift-velocity">Calculating electron drift velocity</a></li></ul></li><li><a href="#potential-difference-and-emf">Potential difference and E.M.F</a><ul class="nav"><li><a href="#the-difference-between-potential-difference-and-electromotive-force">The difference between potential difference and electromotive force</a></li></ul></li><li><a href="#resistance-and-ohm’s-law">Resistance and Ohm’s law</a><ul class="nav"><li><a href="#ohm’s-law">Ohm’s law</a></li><li><a href="#resistance">Resistance</a></li><li><a href="#resistivity">Resistivity</a></li><li><a href="#the-effect-of-temperature-on-resistivity">The effect of temperature on resistivity</a></li></ul></li><li><a href="#electrical-power">Electrical power</a><ul class="nav"><li><a href="#the-cost-of-electrical-energy">The cost of electrical energy</a></li></ul></li><li><a href="#kirchhoff’s-first-and-second-laws">Kirchhoff’s first and second laws</a><ul class="nav"><li><a href="#kirchhoff’s-first-law">Kirchhoff’s first law</a></li><li><a href="#kirchhoff’s-second-law">Kirchhoff’s second law</a></li></ul></li><li><a href="#series-circuits">Series circuits</a><ul class="nav"><li><a href="#resistance-in-series-circuits">Resistance in series circuits</a></li><li><a href="#current-in-series-circuits">Current in series circuits</a></li><li><a href="#voltage-in-series-circuits">Voltage in series circuits</a></li></ul></li><li><a href="#parallel-circuits">Parallel circuits</a><ul class="nav"><li><a href="#resistance-in-parallel-circuits">Resistance in parallel circuits</a></li><li><a href="#current-in-parallel-circuits">Current in parallel circuits</a></li><li><a href="#voltage-in-parallel-circuits">Voltage in parallel circuits</a></li></ul></li><li><a href="#ammeters-and-voltmeters-in-circuits">Ammeters and voltmeters in circuits</a><ul class="nav"><li><a href="#ammeters">Ammeters</a></li><li><a href="#voltmeters">Voltmeters</a></li></ul></li><li><a href="#the-potential-divider">The potential divider</a><ul class="nav"><li><a href="#using-a-variable-resistor">Using a variable resistor</a></li><li><a href="#using-a-thermistor-in-a-temperature-control-circuit">Using a thermistor in a temperature control circuit</a></li><li><a href="#using-a-light-dependent-resistor-in-an-illumination-control-circuit">Using a light-dependent resistor in an illumination control circuit</a></li></ul></li><li><a href="#internal-resistance">Internal resistance</a><ul class="nav"><li><a href="#finding-the-internal-resistance-of-a-cell">Finding the internal resistance of a cell</a></li></ul></li><li><a href="#circuit-analysis">Circuit analysis</a><ul class="nav"><li><a href="#analysis-of-series-and-parallel-hybrid-circuits">Analysis of series and parallel hybrid circuits</a></li><li><a href="#series-circuits-involving-two-sources-of-emf">Series circuits involving two sources of e.m.f</a></li><li><a href="#parallel-circuits-involving-two-sources-of-emf">Parallel circuits involving two sources of e.m.f</a></li></ul></li><li><a href="#wave-motion">Wave motion</a></li><li><a href="#wave-terminology">Wave terminology</a></li><li><a href="#using-an-oscilloscope">Using an oscilloscope</a><ul class="nav"><li><a href="#oscilloscope-controls">Oscilloscope controls</a></li></ul></li><li><a href="#wave-speed-and-the-wave-equation">Wave speed and the wave equation</a><ul class="nav"><li><a href="#the-wave-equation">The wave equation</a></li><li><a href="#energy-transfer-by-waves">Energy transfer by waves</a></li><li><a href="#intensity-and-amplitude">Intensity and amplitude</a></li></ul></li><li><a href="#common-properties-of-waves">Common properties of waves</a><ul class="nav"><li><a href="#reflection">Reflection</a></li><li><a href="#refraction">Refraction</a></li><li><a href="#investigating-refraction-and-reflection-in-a-ripple-tank">Investigating refraction and reflection in a ripple tank</a></li><li><a href="#diffraction">Diffraction</a></li><li><a href="#interference">Interference</a></li></ul></li><li><a href="#electromagnetic-waves">Electromagnetic waves</a><ul class="nav"><li><a href="#common-properties-of-electromagnetic-waves">Common properties of electromagnetic waves</a></li><li><a href="#non-ionising-electromagnetic-radiation">Non-ionising electromagnetic radiation</a></li><li><a href="#ionising-electromagnetic-radiation">Ionising electromagnetic radiation</a></li><li><a href="#production-of-x-rays">Production of X-Rays</a></li></ul></li><li><a href="#polarisation">Polarisation</a></li><li><a href="#rotating-the-plane-of-polarisation">Rotating the plane of polarisation</a><ul class="nav"><li><a href="#uses-of-polarisation">Uses of polarisation</a></li></ul></li><li><a href="#refraction-of-light">Refraction of light</a><ul class="nav"><li><a href="#snell’s-law">Snell’s law</a></li></ul></li><li><a href="#total-internal-reflection">Total internal reflection</a><ul class="nav"><li><a href="#the-critical-angle">The critical angle</a></li><li><a href="#determining-the-critical-angle-for-a-transport-material">Determining the critical angle for a transport material</a></li></ul></li><li><a href="#interference1">Interference</a><ul class="nav"><li><a href="#the-principle-of-superposition">The principle of superposition</a></li><li><a href="#interference2">Interference</a></li><li><a href="#coherence">Coherence</a></li><li><a href="#interference-using-sound-waves">Interference using sound waves</a></li></ul></li><li><a href="#the-young-double-split-experiment">The Young double-split experiment</a><ul class="nav"><li><a href="#the-experiment">The experiment</a></li><li><a href="#finding-the-wavelength-of-the-light-source">Finding the wavelength of the light source</a></li></ul></li><li><a href="#the-diffraction-grating">The diffraction grating</a><ul class="nav"><li><a href="#multiple-slits">Multiple slits</a></li><li><a href="#measuring-the-wavelength-of-light-using-a-diffraction-grating">Measuring the wavelength of light using a diffraction grating</a></li></ul></li><li><a href="#stationary-waves">Stationary waves</a><ul class="nav"><li><a href="#formation-of-stationary-waves">Formation of stationary waves</a></li></ul></li><li><a href="#stationary-wave-experiments">Stationary wave experiments</a><ul class="nav"><li><a href="#stationary-waves-on-strings">Stationary waves on strings</a></li><li><a href="#fundamental-mode-of-vibration-and-harmonics">Fundamental mode of vibration and harmonics</a></li></ul></li><li><a href="#stationary-longitudinal-waves">Stationary longitudinal waves</a><ul class="nav"><li><a href="#stationary-waves-in-closed-tubes">Stationary waves in closed tubes</a></li><li><a href="#stationary-waves-in-closed-tubes1">Stationary waves in closed tubes</a></li></ul></li><li><a href="#measuring-the-speed-of-sound-in-air">Measuring the speed of sound in air</a></li></ul></li><li><a href="#quantum-physics">Quantum physics</a><ul class="nav"><li><a href="#the-photon">The photon</a><ul class="nav"><li><a href="#using-leds-to-determine-a-value-for-the-planck-constant">Using LEDs to determine a value for the Planck constant</a></li></ul></li><li><a href="#the-electronvolt">The electronvolt</a></li><li><a href="#the-photoelectric-effect">The photoelectric effect</a><ul class="nav"><li><a href="#demonstrating-the-photoelectric-effect-with-a-gold-leaf-electroscope">Demonstrating the photoelectric effect with a gold-leaf electroscope</a></li><li><a href="#observations-of-the-photoelectric-effect">Observations of the photoelectric effect</a></li></ul></li><li><a href="#explaining-the-photoelectric-effect">Explaining the photoelectric effect</a><ul class="nav"><li><a href="#the-work-function">The work function</a></li></ul></li><li><a href="#calculations-with-the-photoelectric-effect">Calculations with the photoelectric effect</a><ul class="nav"><li><a href="#determining-the-maximum-kinetic-energy-of-photoelectrons">Determining the maximum kinetic energy of photoelectrons</a></li><li><a href="#rate-of-emission-of-electrons">Rate of emission of electrons</a></li><li><a href="#millikan’s-photoelectric-experiment">Millikan’s photoelectric experiment</a></li></ul></li><li><a href="#wave-particle-duality">Wave particle duality</a><ul class="nav"><li><a href="#the-wavelength-of-electrons">The wavelength of electrons</a></li><li><a href="#wave-particle-duality-for-matter">Wave-particle duality for matter</a></li></ul></li></ul></li></ul></div><div class="toc-menu"><a class="expand-toggle" href="#">Expand all</a><a class="back-to-top" href="#">Back to top</a><a class="go-to-bottom" href="#">Go to bottom</a></div>
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