{"id":1076,"date":"2017-10-27T16:31:21","date_gmt":"2017-10-27T16:31:21","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/energy-stored-in-capacitors\/"},"modified":"2017-11-08T03:26:12","modified_gmt":"2017-11-08T03:26:12","slug":"energy-stored-in-capacitors","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/energy-stored-in-capacitors\/","title":{"raw":"Energy Stored in Capacitors","rendered":"Energy Stored in Capacitors"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>List some uses of capacitors.<\/li>\n<li>Express in equation form the energy stored in a capacitor.<\/li>\n<li>Explain the function of a defibrillator.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<p id=\"import-auto-id2597444\">Most of us have seen dramatizations in which medical personnel use a <span data-type=\"term\">defibrillator<\/span> to pass an electric current through a patient\u2019s heart to get it to beat normally. (Review <a href=\"#import-auto-id3096543\" class=\"autogenerated-content\">(Figure)<\/a>.) Often realistic in detail, the person applying the shock directs another person to \u201cmake it 400 joules this time.\u201d The energy delivered by the defibrillator is stored in a capacitor and can be adjusted to fit the situation. SI units of joules are often employed. Less dramatic is the use of capacitors in microelectronics, such as certain handheld calculators, to supply energy when batteries are charged. (See <a href=\"#import-auto-id3096543\" class=\"autogenerated-content\">(Figure)<\/a>.) Capacitors are also used to supply energy for flash lamps on cameras.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id3096543\">\n<div class=\"bc-figcaption figcaption\">Energy stored in the large capacitor is used to preserve the memory of an electronic calculator when its batteries are charged. (credit: Kucharek, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3154690\" data-alt=\"In an electronic calculator circuit the memory is preserved using large capacitors which store energy when the batteries are charged.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_07_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"In an electronic calculator circuit the memory is preserved using large capacitors which store energy when the batteries are charged.\" width=\"250\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2914172\">Energy stored in a capacitor is electrical potential energy, and it is thus related to the charge <em data-effect=\"italics\">[latex]Q[\/latex]<em data-effect=\"italics\"> and voltage [latex]V[\/latex] on the capacitor. We must be careful when applying the equation for electrical potential energy [latex]\\text{\u0394}\\text{PE}=q\\text{\u0394}V\\phantom{\\rule{0.25em}{0ex}}[\/latex] to a capacitor. Remember that [latex]\\text{\u0394}\\text{PE}[\/latex] is the potential energy of a charge <em data-effect=\"italics\">[latex]q[\/latex]<\/em> going through a voltage [latex]\\text{\u0394}V[\/latex]. But the capacitor starts with zero voltage and gradually comes up to its full voltage as it is charged. The first charge placed on a capacitor experiences a change in voltage [latex]\\text{\u0394}V=0[\/latex], since the capacitor has zero voltage when uncharged. The final charge placed on a capacitor experiences [latex]\\text{\u0394}V=V[\/latex], since the capacitor now has its full voltage [latex]V[\/latex] on it. The average voltage on the capacitor during the charging process is [latex]V\/2[\/latex], and so the average voltage experienced by the full charge <em data-effect=\"italics\">[latex]q[\/latex]<\/em> is [latex]V\/2[\/latex]. Thus the energy stored in a capacitor, [latex]{E}_{\\text{cap}}[\/latex], is <\/em><\/em><\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{E}_{\\text{cap}}=\\frac{QV}{2},[\/latex]<\/div>\n<p id=\"import-auto-id3179153\">where <em data-effect=\"italics\">[latex]Q[\/latex]<em data-effect=\"italics\"> is the charge on a capacitor with a voltage [latex]V[\/latex] applied. (Note that the energy is not [latex]\\text{QV}[\/latex], but [latex]\\text{QV}\/2[\/latex].) Charge and voltage  are related to the capacitance [latex]C[\/latex] of a capacitor by [latex]Q=\\text{CV}[\/latex], and so the expression for [latex]{E}_{\\text{cap}}[\/latex] can be algebraically manipulated into three equivalent expressions:<\/em><\/em><\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{E}_{\\text{cap}}=\\frac{\\text{QV}}{2}=\\frac{{\\text{CV}}^{2}}{2}=\\frac{{Q}^{2}}{2C},[\/latex]<\/div>\n<p id=\"import-auto-id1364149\">where <em data-effect=\"italics\">[latex]Q[\/latex]<em data-effect=\"italics\"> is the charge and [latex]V[\/latex] the voltage on a capacitor [latex]C[\/latex]. The energy is in joules for a charge in coulombs, voltage in volts, and capacitance in farads.<\/em><\/em><\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id3146816\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Energy Stored in Capacitors<\/div>\n<p id=\"import-auto-id2749767\">The energy stored in a capacitor can be expressed in three ways:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-91\">[latex]{E}_{\\text{cap}}=\\frac{\\text{QV}}{2}=\\frac{{\\text{CV}}^{2}}{2}=\\frac{{Q}^{2}}{2C},[\/latex]<\/div>\n<p id=\"import-auto-id1998705\">where <em data-effect=\"italics\">[latex]Q[\/latex]<em data-effect=\"italics\"> is the charge, [latex]V[\/latex] is the voltage, and [latex]C[\/latex] is the capacitance of the capacitor. The energy is in joules for a charge in coulombs, voltage in volts, and capacitance in farads.<\/em><\/em><\/p>\n<\/div>\n<p id=\"import-auto-id1505584\">In a defibrillator, the delivery of a large charge in a short burst to a set of paddles across a person\u2019s chest can be a lifesaver. The person\u2019s heart attack might have arisen from the onset of fast, irregular beating of the heart\u2014cardiac or ventricular fibrillation. The application of a large shock of electrical energy can terminate the arrhythmia and allow the body\u2019s pacemaker to resume normal patterns. Today it is common for ambulances to carry a defibrillator, which also uses an electrocardiogram to analyze the patient\u2019s heartbeat pattern. Automated external defibrillators (AED) are found in many public places (<a href=\"#import-auto-id1516605\" class=\"autogenerated-content\">(Figure)<\/a>). These are designed to be used by lay persons. The device automatically diagnoses the patient\u2019s heart condition and then applies the shock with appropriate energy and waveform. CPR is recommended in many cases before use of an AED.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1516605\">\n<div class=\"bc-figcaption figcaption\">Automated external defibrillators are found in many public places. These portable units provide verbal instructions for use in the important first few minutes for a person suffering a cardiac attack. (credit: Owain Davies, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1285606\" data-alt=\"Photograph of an automated external defibrillator.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_07_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"Photograph of an automated external defibrillator.\" width=\"250\"><\/span><\/p><\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2725068\">\n<div data-type=\"title\" class=\"title\">Capacitance in a Heart Defibrillator<\/div>\n<p id=\"import-auto-id2719922\">A heart defibrillator delivers [latex]4.00\u00d7{\\text{10}}^{\\text{2}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex] of energy by discharging a capacitor initially at [latex]1.00\u00d7{\\text{10}}^{\\text{4}}\\phantom{\\rule{0.25em}{0ex}}\\text{V}[\/latex]. What is its capacitance?<\/p>\n<p id=\"import-auto-id1665877\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2631974\">We are given [latex]{E}_{\\text{cap}}[\/latex] and [latex]V[\/latex], and we are asked to find the capacitance [latex]C[\/latex]. Of the three expressions in the equation for [latex]{E}_{\\text{cap}}[\/latex], the most convenient relationship is<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{E}_{\\text{cap}}=\\frac{{\\text{CV}}^{2}}{2}.[\/latex]<\/div>\n<p id=\"import-auto-id949088\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2635616\">Solving this expression for [latex]C[\/latex] and entering the given values yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-448\">[latex]\\begin{array}{lll}C&amp; =&amp; \\frac{2{E}_{\\text{cap}}}{{V}^{2}}=\\frac{2\\left(4\\text{.}\\text{00}\u00d7{\\text{10}}^{2}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\\right)}{\\left(1\\text{.}\\text{00}\u00d7{\\text{10}}^{4}\\phantom{\\rule{0.25em}{0ex}}\\text{V}{\\right)}^{2}}=8\\text{.}\\text{00}\u00d7{\\text{10}}^{\u20136}\\phantom{\\rule{0.25em}{0ex}}\\text{F}\\\\ &amp; =&amp; \\text{8}\\text{.}\\text{00 \u00b5F}\\text{.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id2946047\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id3181267\">This is a fairly large, but manageable, capacitance at [latex]1.00\u00d7{\\text{10}}^{\\text{4}}\\phantom{\\rule{0.25em}{0ex}}\\text{V}[\/latex].<\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1502616\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1010245\">\n<li id=\"import-auto-id1534698\">Capacitors are used in a variety of devices, including defibrillators, microelectronics such as calculators, and flash lamps, to supply energy.<\/li>\n<li id=\"import-auto-id1611426\">The energy stored in a capacitor can be expressed in three ways:\n<div data-type=\"equation\" class=\"equation\">[latex]{E}_{\\text{cap}}=\\frac{\\text{QV}}{2}=\\frac{{\\text{CV}}^{2}}{2}=\\frac{{Q}^{2}}{2C},[\/latex]<\/div>\n<p>where<br>\n[latex]Q[\/latex]<em data-effect=\"italics\"> is the charge, [latex]V[\/latex] is the voltage, and [latex]C[\/latex] is the capacitance of the capacitor. The energy is in joules when the charge is in coulombs, voltage is in volts, and capacitance is in farads.<\/em><\/p><\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1945775\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1579143\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2693553\">\n<p>How does the energy contained in a charged capacitor change when a dielectric is inserted, assuming the capacitor is isolated and its charge is constant? Does this imply that work was done? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"eip-482\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"eip-id1600770\">\n<p id=\"import-auto-id2502272\">What happens to the energy stored in a capacitor connected to a battery when a dielectric is inserted? Was work done in the process? <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2966465\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"eip-838\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p>(a) What is the energy stored in the<br>\n[latex]\\text{10.0 \u03bcF}[\/latex] capacitor of a heart defibrillator charged to<br>\n[latex]9.00\u00d7{\\text{10}}^{\\text{3}}\\phantom{\\rule{0.25em}{0ex}}\\text{V}[\/latex]? (b) Find the amount of stored charge.\n<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p>(a) [latex]\\text{405 J}[\/latex]\n  <\/p>\n<p id=\"eip-id1577977\">(b) [latex]\\text{90.0 mC}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2657760\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1935532\">\n<p id=\"fs-id2014471\">In open heart surgery, a much smaller amount of energy will defibrillate the heart. (a) What voltage is applied to the [latex]\\text{8.00 \u03bcF}[\/latex]  capacitor of a heart defibrillator that stores 40.0 J of energy? (b) Find the amount of stored charge. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3110340\">\n<p id=\"fs-id2727760\">(a) 3.16 kV<\/p>\n<p id=\"import-auto-id1665623\">(b) 25.3 mC<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1998705\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1346384\">\n<p id=\"import-auto-id952980\">A [latex]1\\text{65 \u00b5F}[\/latex] capacitor is used in conjunction with a motor. How much energy is stored in it when 119 V is applied?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p>Suppose you have a 9.00 V battery, a<br>\n[latex]\\text{2.00 \u03bcF}[\/latex]<br>\ncapacitor, and a [latex]\\text{7.40 \u03bcF}[\/latex]  capacitor. (a) Find the charge and energy stored if the capacitors are connected to the battery in series. (b) Do the same for a parallel connection.\n  <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p>(a)  [latex]1.42\u00d7{\\text{10}}^{\\text{\u22125}}\\phantom{\\rule{0.25em}{0ex}}\\text{C}[\/latex],  [latex]6.38\u00d7{\\text{10}}^{\\text{\u22125}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex]\n<\/p>\n<p>(b)  [latex]8.46\u00d7{\\text{10}}^{\\text{\u22125}}\\phantom{\\rule{0.25em}{0ex}}\\text{C}[\/latex],  [latex]3.81\u00d7{\\text{10}}^{\\text{\u22124}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id832930\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1261416\">\n<p id=\"import-auto-id2758865\">A nervous physicist worries that the two metal shelves of his wood frame bookcase might obtain a high voltage if charged by static electricity, perhaps produced by friction. (a) What is the capacitance of the empty shelves if they have area [latex]1.00\u00d7{\\text{10}}^{\\text{2}}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{\\text{2}}[\/latex] and are 0.200 m apart? (b) What is the voltage between them if opposite charges of magnitude 2.00 nC are placed on them? (c) To show that this voltage poses a small hazard, calculate the energy stored. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1285484\">\n<p id=\"import-auto-id1341804\">(a) [latex]4\\text{.}\\text{43}\u00d7{\\text{10}}^{\u2013\\text{12}}\\phantom{\\rule{0.25em}{0ex}}\\text{F}[\/latex]<\/p>\n<p>(b) [latex]\\text{452}\\phantom{\\rule{0.25em}{0ex}}\\text{V}[\/latex]<\/p>\n<p id=\"import-auto-id621629\">(c) [latex]4\\text{.}\\text{52}\u00d7{\\text{10}}^{\u20137}\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3152957\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id898212\">\n<p id=\"import-auto-id2512565\">Show that for a given dielectric material the maximum energy a parallel plate capacitor can store is directly proportional to the volume of dielectric ([latex]\\text{Volume =}\\phantom{\\rule{0.25em}{0ex}}A\u00b7d[\/latex]). Note that the applied voltage is limited by the dielectric strength. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1183288\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2705985\">\n<p id=\"eip-id1375405\"><strong>Construct Your Own Problem<\/strong><\/p>\n<p id=\"import-auto-id3204098\">Consider a heart defibrillator similar to that discussed in <a href=\"#fs-id2725068\" class=\"autogenerated-content\">(Figure)<\/a>. Construct a problem in which you examine the charge stored in the capacitor of a defibrillator as a function of stored energy. Among the things to be considered are the applied voltage and whether it should vary with energy to be delivered, the range of energies involved, and the capacitance of the defibrillator. You may also wish to consider the much smaller energy needed for defibrillation during open-heart surgery as a variation on this problem.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3088661\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1509224\">\n<p id=\"eip-id2912398\"><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"import-auto-id1513957\">(a) On a particular day, it takes [latex]9.60\u00d7{\\text{10}}^{\\text{3}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex] of electric energy to start a truck\u2019s engine. Calculate the capacitance of a capacitor that could store that amount of energy at 12.0 V. (b) What is unreasonable about this result? (c) Which assumptions are responsible?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3108768\">\n<p id=\"import-auto-id1489529\">(a) [latex]\\text{133}\\phantom{\\rule{0.25em}{0ex}}\\text{F}[\/latex]<\/p>\n<p id=\"import-auto-id2920416\">(b) Such a capacitor would be too large to carry with a truck. The size of the capacitor would be enormous.<\/p>\n<p id=\"import-auto-id2581572\">(c) It is unreasonable to assume that a capacitor can store the amount of energy needed.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"fs-id977377\">\n<dt>defibrillator<\/dt>\n<dd id=\"fs-id881521\">a machine used to provide an electrical shock to a heart attack victim's heart in order to restore the heart's normal rhythmic pattern<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>List some uses of capacitors.<\/li>\n<li>Express in equation form the energy stored in a capacitor.<\/li>\n<li>Explain the function of a defibrillator.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<p id=\"import-auto-id2597444\">Most of us have seen dramatizations in which medical personnel use a <span data-type=\"term\">defibrillator<\/span> to pass an electric current through a patient\u2019s heart to get it to beat normally. (Review <a href=\"#import-auto-id3096543\" class=\"autogenerated-content\">(Figure)<\/a>.) Often realistic in detail, the person applying the shock directs another person to \u201cmake it 400 joules this time.\u201d The energy delivered by the defibrillator is stored in a capacitor and can be adjusted to fit the situation. SI units of joules are often employed. Less dramatic is the use of capacitors in microelectronics, such as certain handheld calculators, to supply energy when batteries are charged. (See <a href=\"#import-auto-id3096543\" class=\"autogenerated-content\">(Figure)<\/a>.) Capacitors are also used to supply energy for flash lamps on cameras.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id3096543\">\n<div class=\"bc-figcaption figcaption\">Energy stored in the large capacitor is used to preserve the memory of an electronic calculator when its batteries are charged. (credit: Kucharek, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3154690\" data-alt=\"In an electronic calculator circuit the memory is preserved using large capacitors which store energy when the batteries are charged.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_07_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"In an electronic calculator circuit the memory is preserved using large capacitors which store energy when the batteries are charged.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2914172\">Energy stored in a capacitor is electrical potential energy, and it is thus related to the charge <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\"> and voltage <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> on the capacitor. We must be careful when applying the equation for electrical potential energy <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-40d1ebb8cae849aa252d90b1cc29683d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#69;&#125;&#61;&#113;&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#86;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -4px;\" \/> to a capacitor. Remember that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-08de69141512ad5611090977d33d93c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#69;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"24\" style=\"vertical-align: -1px;\" \/> is the potential energy of a charge <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/><\/em> going through a voltage <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-51a336c1f2199319ee1235dc1a1bbacb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>. But the capacitor starts with zero voltage and gradually comes up to its full voltage as it is charged. The first charge placed on a capacitor experiences a change in voltage <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1b0607def4f0b22ce6535fa884aff029_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#86;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\" \/>, since the capacitor has zero voltage when uncharged. The final charge placed on a capacitor experiences <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4ba2be2616f8323d3079c9e8dea200f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&Delta;&#125;&#86;&#61;&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"52\" style=\"vertical-align: 0px;\" \/>, since the capacitor now has its full voltage <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> on it. The average voltage on the capacitor during the charging process is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9817e52a6dafdf8c4dd015a44356113c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"29\" style=\"vertical-align: -5px;\" \/>, and so the average voltage experienced by the full charge <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ac7da57d7f507262338bb5168feb3e06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\" \/><\/em> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9817e52a6dafdf8c4dd015a44356113c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"29\" style=\"vertical-align: -5px;\" \/>. Thus the energy stored in a capacitor, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aa8150d69022790c218652cf911164bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#112;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -6px;\" \/>, is <\/em><\/em><\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b976c91b0767cfd7556ceb8d2e38dd89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#112;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#81;&#86;&#125;&#123;&#50;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id3179153\">where <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\"> is the charge on a capacitor with a voltage <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> applied. (Note that the energy is not <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-62291854a7f534e0955fb57e99ae80c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"27\" style=\"vertical-align: -3px;\" \/>, but <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2aacabce95625d6bdb706a96ba832e68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#86;&#125;&#47;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -5px;\" \/>.) Charge and voltage  are related to the capacitance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> of a capacitor by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-183a2c7eb806f2473e1a9c6822ce4ade_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"64\" style=\"vertical-align: -4px;\" \/>, and so the expression for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aa8150d69022790c218652cf911164bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#112;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -6px;\" \/> can be algebraically manipulated into three equivalent expressions:<\/em><\/em><\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-42aff791c406c8a61f34a92134a8afea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#112;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#86;&#125;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#86;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#81;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#67;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"187\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1364149\">where <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\"> is the charge and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> the voltage on a capacitor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>. The energy is in joules for a charge in coulombs, voltage in volts, and capacitance in farads.<\/em><\/em><\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id3146816\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Energy Stored in Capacitors<\/div>\n<p id=\"import-auto-id2749767\">The energy stored in a capacitor can be expressed in three ways:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-91\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-42aff791c406c8a61f34a92134a8afea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#112;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#86;&#125;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#86;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#81;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#67;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"187\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1998705\">where <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\"> is the charge, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the voltage, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the capacitance of the capacitor. The energy is in joules for a charge in coulombs, voltage in volts, and capacitance in farads.<\/em><\/em><\/p>\n<\/div>\n<p id=\"import-auto-id1505584\">In a defibrillator, the delivery of a large charge in a short burst to a set of paddles across a person\u2019s chest can be a lifesaver. The person\u2019s heart attack might have arisen from the onset of fast, irregular beating of the heart\u2014cardiac or ventricular fibrillation. The application of a large shock of electrical energy can terminate the arrhythmia and allow the body\u2019s pacemaker to resume normal patterns. Today it is common for ambulances to carry a defibrillator, which also uses an electrocardiogram to analyze the patient\u2019s heartbeat pattern. Automated external defibrillators (AED) are found in many public places (<a href=\"#import-auto-id1516605\" class=\"autogenerated-content\">(Figure)<\/a>). These are designed to be used by lay persons. The device automatically diagnoses the patient\u2019s heart condition and then applies the shock with appropriate energy and waveform. CPR is recommended in many cases before use of an AED.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1516605\">\n<div class=\"bc-figcaption figcaption\">Automated external defibrillators are found in many public places. These portable units provide verbal instructions for use in the important first few minutes for a person suffering a cardiac attack. (credit: Owain Davies, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1285606\" data-alt=\"Photograph of an automated external defibrillator.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_20_07_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"Photograph of an automated external defibrillator.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2725068\">\n<div data-type=\"title\" class=\"title\">Capacitance in a Heart Defibrillator<\/div>\n<p id=\"import-auto-id2719922\">A heart defibrillator delivers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0f64bcc56fae739f5b43f852d90a9111_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -1px;\" \/> of energy by discharging a capacitor initially at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-311b8347f4d26b69dbe49468f66df966_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -1px;\" \/>. What is its capacitance?<\/p>\n<p id=\"import-auto-id1665877\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2631974\">We are given <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aa8150d69022790c218652cf911164bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#112;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, and we are asked to find the capacitance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>. Of the three expressions in the equation for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aa8150d69022790c218652cf911164bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#112;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"34\" style=\"vertical-align: -6px;\" \/>, the most convenient relationship is<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-31873559b6fb6483621ac39caa2e0c02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#112;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#86;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id949088\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2635616\">Solving this expression for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> and entering the given values yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-448\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3ebc6bd5013d0af13e6c3475ffb04596_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#67;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#123;&#69;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#112;&#125;&#125;&#125;&#123;&#123;&#86;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#52;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#125;&#123;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&micro;&#70;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"319\" style=\"vertical-align: -21px;\" \/><\/div>\n<p id=\"import-auto-id2946047\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id3181267\">This is a fairly large, but manageable, capacitance at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-311b8347f4d26b69dbe49468f66df966_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -1px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1502616\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id1010245\">\n<li id=\"import-auto-id1534698\">Capacitors are used in a variety of devices, including defibrillators, microelectronics such as calculators, and flash lamps, to supply energy.<\/li>\n<li id=\"import-auto-id1611426\">The energy stored in a capacitor can be expressed in three ways:\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-42aff791c406c8a61f34a92134a8afea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#112;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#81;&#86;&#125;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#86;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#81;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#67;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"187\" style=\"vertical-align: -6px;\" \/><\/div>\n<p>where<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c758bec4c272382411b95fc0e7ee250_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#81;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\"> is the charge, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the voltage, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f34f74d98915e33f37a086f8cbfb996a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the capacitance of the capacitor. The energy is in joules when the charge is in coulombs, voltage is in volts, and capacitance is in farads.<\/em><\/p>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1945775\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1579143\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2693553\">\n<p>How does the energy contained in a charged capacitor change when a dielectric is inserted, assuming the capacitor is isolated and its charge is constant? Does this imply that work was done? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"eip-482\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"eip-id1600770\">\n<p id=\"import-auto-id2502272\">What happens to the energy stored in a capacitor connected to a battery when a dielectric is inserted? Was work done in the process? <\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2966465\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"eip-838\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p>(a) What is the energy stored in the<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c30e17a03b553bc5726accf9f841376e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#46;&#48;&#32;&mu;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\" \/> capacitor of a heart defibrillator charged to<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0c72e630e532592b8995c3959646de7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -1px;\" \/>? (b) Find the amount of stored charge.\n<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p>(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43a4dd5ffef3ecf480e0411771960df6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#53;&#32;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"42\" style=\"vertical-align: -1px;\" \/>\n  <\/p>\n<p id=\"eip-id1577977\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1970b54d0dc5acce61680a5586ff27db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#46;&#48;&#32;&#109;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"64\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2657760\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1935532\">\n<p id=\"fs-id2014471\">In open heart surgery, a much smaller amount of energy will defibrillate the heart. (a) What voltage is applied to the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5748e0f4733dbcfe7ade88dadb28300d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#46;&#48;&#48;&#32;&mu;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: -1px;\" \/>  capacitor of a heart defibrillator that stores 40.0 J of energy? (b) Find the amount of stored charge. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3110340\">\n<p id=\"fs-id2727760\">(a) 3.16 kV<\/p>\n<p id=\"import-auto-id1665623\">(b) 25.3 mC<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1998705\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1346384\">\n<p id=\"import-auto-id952980\">A <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-20896eb31cc43e68d770bd8ca280f2e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#53;&#32;&micro;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"43\" style=\"vertical-align: -1px;\" \/> capacitor is used in conjunction with a motor. How much energy is stored in it when 119 V is applied?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p>Suppose you have a 9.00 V battery, a<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-08f7d34d18ae2752bca4c16d4b85953b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#46;&#48;&#48;&#32;&mu;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"48\" style=\"vertical-align: -1px;\" \/><br \/>\ncapacitor, and a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0e50b47aa4ec378023f9fda19cea6655_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#46;&#52;&#48;&#32;&mu;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"48\" style=\"vertical-align: -1px;\" \/>  capacitor. (a) Find the charge and energy stored if the capacitors are connected to the battery in series. (b) Do the same for a parallel connection.\n  <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p>(a)  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3f6de4c44d06cf9c5ae83bb71311cefa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#52;&#50;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#53;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"72\" style=\"vertical-align: -1px;\" \/>,  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-758f73b580cdae40fdb01a760ef7f15a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#46;&#51;&#56;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#53;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -1px;\" \/>\n<\/p>\n<p>(b)  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-92cd319d9a49bdf0a613a794042e380f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#46;&#52;&#54;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#53;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -1px;\" \/>,  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e84cd2cabbd11b9363c8299e7d254ddb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#46;&#56;&#49;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#52;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id832930\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1261416\">\n<p id=\"import-auto-id2758865\">A nervous physicist worries that the two metal shelves of his wood frame bookcase might obtain a high voltage if charged by static electricity, perhaps produced by friction. (a) What is the capacitance of the empty shelves if they have area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-022540d1cba1bb41e89f17c08bc9baf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -1px;\" \/> and are 0.200 m apart? (b) What is the voltage between them if opposite charges of magnitude 2.00 nC are placed on them? (c) To show that this voltage poses a small hazard, calculate the energy stored. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1285484\">\n<p id=\"import-auto-id1341804\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1f9046762e4b274819951259832dc19f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#51;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"90\" style=\"vertical-align: -1px;\" \/><\/p>\n<p>(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-817c98b857d9481ec0867a4e1d3c262b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"44\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id621629\">(c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-363fd666de27af21ead9221c990bbe97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"81\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3152957\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id898212\">\n<p id=\"import-auto-id2512565\">Show that for a given dielectric material the maximum energy a parallel plate capacitor can store is directly proportional to the volume of dielectric (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e6352f8fac01b446f40b56823c1dc0b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#86;&#111;&#108;&#117;&#109;&#101;&#32;&#61;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#65;&middot;&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"105\" style=\"vertical-align: 0px;\" \/>). Note that the applied voltage is limited by the dielectric strength. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1183288\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2705985\">\n<p id=\"eip-id1375405\"><strong>Construct Your Own Problem<\/strong><\/p>\n<p id=\"import-auto-id3204098\">Consider a heart defibrillator similar to that discussed in <a href=\"#fs-id2725068\" class=\"autogenerated-content\">(Figure)<\/a>. Construct a problem in which you examine the charge stored in the capacitor of a defibrillator as a function of stored energy. Among the things to be considered are the applied voltage and whether it should vary with energy to be delivered, the range of energies involved, and the capacitance of the defibrillator. You may also wish to consider the much smaller energy needed for defibrillation during open-heart surgery as a variation on this problem.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3088661\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1509224\">\n<p id=\"eip-id2912398\"><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"import-auto-id1513957\">(a) On a particular day, it takes <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-39400ce41793a36df383d4f43a21c53f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#46;&#54;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -1px;\" \/> of electric energy to start a truck\u2019s engine. Calculate the capacitance of a capacitor that could store that amount of energy at 12.0 V. (b) What is unreasonable about this result? (c) Which assumptions are responsible?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3108768\">\n<p id=\"import-auto-id1489529\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-484a8edffbfe3256489dc22c5fba4925_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id2920416\">(b) Such a capacitor would be too large to carry with a truck. The size of the capacitor would be enormous.<\/p>\n<p id=\"import-auto-id2581572\">(c) It is unreasonable to assume that a capacitor can store the amount of energy needed.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"fs-id977377\">\n<dt>defibrillator<\/dt>\n<dd id=\"fs-id881521\">a machine used to provide an electrical shock to a heart attack victim&#8217;s heart in order to restore the heart&#8217;s normal rhythmic pattern<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1076","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":1030,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1076","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1076\/revisions"}],"predecessor-version":[{"id":1077,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1076\/revisions\/1077"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/1030"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1076\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=1076"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1076"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=1076"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=1076"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}