{"id":1029,"date":"2017-10-24T19:31:48","date_gmt":"2017-10-24T23:31:48","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/test3\/chapter\/energy-stored-in-capacitors\/"},"modified":"2017-10-24T19:31:48","modified_gmt":"2017-10-24T23:31:48","slug":"energy-stored-in-capacitors","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/test3\/chapter\/energy-stored-in-capacitors\/","title":{"raw":"Energy Stored in Capacitors","rendered":"Energy Stored in Capacitors"},"content":{"raw":"<div class=\"textbox learning-objectives\"><h3>Learning Objectives<\/h3><ul><li>List some uses of capacitors.<\/li><li>Express in equation form the energy stored in a capacitor.<\/li><li>Explain the function of a defibrillator.<\/li><\/ul><\/div><div class=\"bc-section section\"><p id=\"import-auto-id2597444\">Most of us have seen dramatizations in which medical personnel use a <span>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\">[link]<\/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\">[link]<\/a>.) Capacitors are also used to supply energy for flash lamps on cameras.<\/p><div class=\"bc-figure figure\" id=\"import-auto-id3096543\"><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><span id=\"import-auto-id3154690\"><img src=\"https:\/\/pressbooks.bccampus.ca\/test3\/wp-content\/uploads\/sites\/27\/2017\/10\/Figure_20_07_01a.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><\/div><p id=\"import-auto-id2914172\">Energy stored in a capacitor is electrical potential energy, and it is thus related to the charge <em>[latex]Q[\/latex]<em> 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>[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>[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><div class=\"equation\">[latex]{E}_{\\text{cap}}=\\frac{QV}{2},[\/latex]<\/div><p id=\"import-auto-id3179153\">where <em>[latex]Q[\/latex]<em> 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><div class=\"equation\">[latex]{E}_{\\text{cap}}=\\frac{\\text{QV}}{2}=\\frac{{\\text{CV}}^{2}}{2}=\\frac{{Q}^{2}}{2C},[\/latex]<\/div><p id=\"import-auto-id1364149\">where <em>[latex]Q[\/latex]<em> 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><div class=\"note\" id=\"fs-id3146816\"><div class=\"title\">Energy Stored in Capacitors<\/div><p id=\"import-auto-id2749767\">The energy stored in a capacitor can be expressed in three ways:<\/p><div class=\"equation\" id=\"eip-91\">[latex]{E}_{\\text{cap}}=\\frac{\\text{QV}}{2}=\\frac{{\\text{CV}}^{2}}{2}=\\frac{{Q}^{2}}{2C},[\/latex]<\/div><p id=\"import-auto-id1998705\">where <em>[latex]Q[\/latex]<em> 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><\/div><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\">[link]<\/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><div class=\"bc-figure figure\" id=\"import-auto-id1516605\"><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><span id=\"import-auto-id1285606\"><img src=\"https:\/\/pressbooks.bccampus.ca\/test3\/wp-content\/uploads\/sites\/27\/2017\/10\/Figure_20_07_02a.jpg\" alt=\"Photograph of an automated external defibrillator.\" width=\"250\" \/><\/span><\/div><div class=\"textbox examples\" id=\"fs-id2725068\"><div class=\"title\">Capacitance in a Heart Defibrillator<\/div><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><p id=\"import-auto-id1665877\"><strong>Strategy<\/strong><\/p><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><div class=\"equation\">[latex]{E}_{\\text{cap}}=\\frac{{\\text{CV}}^{2}}{2}.[\/latex]<\/div><p id=\"import-auto-id949088\"><strong>Solution<\/strong><\/p><p id=\"import-auto-id2635616\">Solving this expression for [latex]C[\/latex] and entering the given values yields<\/p><div 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><p id=\"import-auto-id2946047\"><strong>Discussion<\/strong><\/p><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><\/div><\/div><div class=\"section-summary\" id=\"fs-id1502616\"><h1>Section Summary<\/h1><ul id=\"fs-id1010245\"><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><li id=\"import-auto-id1611426\">The energy stored in a capacitor can be expressed in three ways:\n    <div class=\"equation\">[latex]{E}_{\\text{cap}}=\\frac{\\text{QV}}{2}=\\frac{{\\text{CV}}^{2}}{2}=\\frac{{Q}^{2}}{2C},[\/latex]<\/div>\nwhere\n[latex]Q[\/latex]<em> 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><\/li><\/ul><\/div><div class=\"conceptual-questions\" id=\"fs-id1945775\"><h1>Conceptual Questions<\/h1><div class=\"exercise\" id=\"fs-id1579143\"><div class=\"problem\" id=\"fs-id2693553\"><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><\/div><\/div><div class=\"exercise\" id=\"eip-482\"><div class=\"problem\" id=\"eip-id1600770\"><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><\/div><\/div><\/div><div class=\"problems-exercises\" id=\"fs-id2966465\"><h1>Problems &amp; Exercises<\/h1><div class=\"exercise\" id=\"eip-838\"><div class=\"problem\"><p>(a) What is the energy stored in the \n[latex]\\text{10.0 \u03bcF}[\/latex] capacitor of a heart defibrillator charged to \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><\/div><div class=\"solution\"><p>(a) [latex]\\text{405 J}[\/latex]\n  <\/p><p id=\"eip-id1577977\">(b) [latex]\\text{90.0 mC}[\/latex]<\/p><\/div><\/div><div class=\"exercise\" id=\"fs-id2657760\"><div class=\"problem\" id=\"fs-id1935532\"><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><\/div><div class=\"solution\" id=\"fs-id3110340\"><p id=\"fs-id2727760\">(a) 3.16 kV<\/p><p id=\"import-auto-id1665623\">(b) 25.3 mC<\/p><\/div><\/div><div class=\"exercise\" id=\"fs-id1998705\"><div class=\"problem\" id=\"fs-id1346384\"><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><\/div><\/div><div class=\"exercise\"><div class=\"problem\"><p>Suppose you have a 9.00 V battery, a  \n[latex]\\text{2.00 \u03bcF}[\/latex] \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><\/div><div class=\"solution\"><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><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><\/div><\/div><div class=\"exercise\" id=\"fs-id832930\"><div class=\"problem\" id=\"fs-id1261416\"><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><\/div><div class=\"solution\" id=\"fs-id1285484\"><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><p>(b) [latex]\\text{452}\\phantom{\\rule{0.25em}{0ex}}\\text{V}[\/latex]<\/p><p id=\"import-auto-id621629\">(c) [latex]4\\text{.}\\text{52}\u00d7{\\text{10}}^{\u20137}\\phantom{\\rule{0.25em}{0ex}}\\text{J}[\/latex]<\/p><\/div><\/div><div class=\"exercise\" id=\"fs-id3152957\"><div class=\"problem\" id=\"fs-id898212\"><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><\/div><\/div><div class=\"exercise\" id=\"fs-id1183288\"><div class=\"problem\" id=\"fs-id2705985\"><p id=\"eip-id1375405\"><strong>Construct Your Own Problem<\/strong><\/p><p id=\"import-auto-id3204098\">Consider a heart defibrillator similar to that discussed in <a href=\"#fs-id2725068\" class=\"autogenerated-content\">[link]<\/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><\/div><\/div><div class=\"exercise\" id=\"fs-id3088661\"><div class=\"problem\" id=\"fs-id1509224\"><p id=\"eip-id2912398\"><strong>Unreasonable Results<\/strong><\/p><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><\/div><div class=\"solution\" id=\"fs-id3108768\"><p id=\"import-auto-id1489529\">(a) [latex]\\text{133}\\phantom{\\rule{0.25em}{0ex}}\\text{F}[\/latex]<\/p><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><p id=\"import-auto-id2581572\">(c) It is unreasonable to assume that a capacitor can store the amount of energy needed.<\/p><\/div><\/div><\/div><div class=\"textbox shaded\"><h2>Glossary<\/h2><dl class=\"definition\" id=\"fs-id977377\"><dt>defibrillator<\/dt><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><\/dl><\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>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\">\n<p id=\"import-auto-id2597444\">Most of us have seen dramatizations in which medical personnel use a <span>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\">[link]<\/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\">[link]<\/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 id=\"import-auto-id3154690\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/test3\/wp-content\/uploads\/sites\/27\/2017\/10\/Figure_20_07_01a.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><\/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>[latex]Q[\/latex]<em> 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>[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>[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 class=\"equation\">[latex]{E}_{\\text{cap}}=\\frac{QV}{2},[\/latex]<\/div>\n<p id=\"import-auto-id3179153\">where <em>[latex]Q[\/latex]<em> 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 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>[latex]Q[\/latex]<em> 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 class=\"note\" id=\"fs-id3146816\">\n<div 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 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>[latex]Q[\/latex]<em> 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\">[link]<\/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 id=\"import-auto-id1285606\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/test3\/wp-content\/uploads\/sites\/27\/2017\/10\/Figure_20_07_02a.jpg\" alt=\"Photograph of an automated external defibrillator.\" width=\"250\" \/><\/span><\/div>\n<div class=\"textbox examples\" id=\"fs-id2725068\">\n<div 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 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 class=\"equation\" id=\"eip-448\">[latex]\\begin{array}{lll}C& =& \\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}\\\\ & =& \\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\" id=\"fs-id1502616\">\n<h1>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 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> 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><\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" id=\"fs-id1945775\">\n<h1>Conceptual Questions<\/h1>\n<div class=\"exercise\" id=\"fs-id1579143\">\n<div 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 class=\"exercise\" id=\"eip-482\">\n<div 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\" id=\"fs-id2966465\">\n<h1>Problems &amp; Exercises<\/h1>\n<div class=\"exercise\" id=\"eip-838\">\n<div 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 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 class=\"exercise\" id=\"fs-id2657760\">\n<div 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 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 class=\"exercise\" id=\"fs-id1998705\">\n<div 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 class=\"exercise\">\n<div 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 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 class=\"exercise\" id=\"fs-id832930\">\n<div 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 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 class=\"exercise\" id=\"fs-id3152957\">\n<div 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 class=\"exercise\" id=\"fs-id1183288\">\n<div 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\">[link]<\/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 class=\"exercise\" id=\"fs-id3088661\">\n<div 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 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 class=\"textbox shaded\">\n<h2>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":8,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["openstaxcollege","openstaxcollege"],"pb_section_license":"cc-by"},"chapter-type":[],"contributor":[53],"license":[59],"class_list":["post-1029","chapter","type-chapter","status-publish","hentry","contributor-openstaxcollege","license-cc-by"],"part":990,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/test3\/wp-json\/pressbooks\/v2\/chapters\/1029","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/test3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/test3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/test3\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/test3\/wp-json\/pressbooks\/v2\/chapters\/1029\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/test3\/wp-json\/pressbooks\/v2\/parts\/990"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/test3\/wp-json\/pressbooks\/v2\/chapters\/1029\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/test3\/wp-json\/wp\/v2\/media?parent=1029"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/test3\/wp-json\/pressbooks\/v2\/chapter-type?post=1029"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/test3\/wp-json\/wp\/v2\/contributor?post=1029"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/test3\/wp-json\/wp\/v2\/license?post=1029"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}