#set document(title: "19.7 Energy Stored in Capacitors", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 19.7#h(0.6em)Energy Stored in Capacitors === Learning Objectives By the end of this section, you will be able to: - List some uses of capacitors. - Express in equation form the energy stored in a capacitor. - Explain the function of a defibrillator. Most of us have seen dramatizations in which medical personnel use a #strong[defibrillator] to pass an electric current through a patient’s heart to get it to beat normally. (Review .) Often realistic in detail, the person applying the shock directs another person to “make it 400 joules this time.” 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. () Capacitors are also used to supply energy for flash lamps on cameras. #figure(figph[In an electronic calculator circuit the memory is preserved using large capacitors which store energy when the batteries are charged.], alt: "In an electronic calculator circuit the memory is preserved using large capacitors which store energy when the batteries are charged.", caption: [Energy stored in the large capacitor is used to preserve the memory of an electronic calculator when its batteries are charged.]) Energy stored in a capacitor is electrical potential energy, and it is thus related to the charge #math.equation(block: false, alt: "Q")[$Q$] and voltage #math.equation(block: false, alt: "V")[$V$] on the capacitor. We must be careful when applying the equation for electrical potential energy #math.equation(block: false, alt: "Δ PE equals q Δ V")[$"Δ" "PE" = q "Δ" V #h(0.25em)$] to a capacitor. Remember that #math.equation(block: false, alt: "Δ PE")[$"Δ" "PE"$] is the potential energy of a charge #emph[#math.equation(block: false, alt: "q")[$q$]] going through a voltage #math.equation(block: false, alt: "Δ V")[$"Δ" V$]. 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 #math.equation(block: false, alt: "Δ V equals 0")[$"Δ" V = 0$], since the capacitor has zero voltage when uncharged. The final charge placed on a capacitor experiences #math.equation(block: false, alt: "Δ V equals V")[$"Δ" V = V$], since the capacitor now has its full voltage #math.equation(block: false, alt: "V")[$V$] on it. The average voltage on the capacitor during the charging process is #math.equation(block: false, alt: "V / 2")[$V / 2$], and so the average voltage experienced by the full charge #emph[#math.equation(block: false, alt: "q")[$q$]] is #math.equation(block: false, alt: "V / 2")[$V / 2$]. Thus the energy stored in a capacitor, #math.equation(block: false, alt: "E sub cap")[$E_("cap")$], is #math.equation(block: true, alt: "E sub cap equals the fraction Q V over 2 ,")[$E_("cap") = frac(Q V, 2) ,$] where #math.equation(block: false, alt: "Q")[$Q$] is the charge on a capacitor with a voltage #math.equation(block: false, alt: "V")[$V$] applied. (Note that the energy is not #math.equation(block: false, alt: "QV")[$"QV"$], but #math.equation(block: false, alt: "QV / 2")[$"QV" / 2$].) Charge and voltage are related to the capacitance #math.equation(block: false, alt: "C")[$C$] of a capacitor by #math.equation(block: false, alt: "Q equals CV")[$Q = "CV"$], and so the expression for #math.equation(block: false, alt: "E sub cap")[$E_("cap")$] can be algebraically manipulated into three equivalent expressions: #math.equation(block: true, alt: "E sub cap equals the fraction QV over 2 equals the fraction CV squared over 2 equals the fraction Q squared over 2 C ,")[$E_("cap") = frac("QV", 2) = frac("CV"^(2), 2) = frac(Q^(2), 2 C) ,$] where #math.equation(block: false, alt: "Q")[$Q$] is the charge and #math.equation(block: false, alt: "V")[$V$] the voltage on a capacitor #math.equation(block: false, alt: "C")[$C$]. The energy is in joules for a charge in coulombs, voltage in volts, and capacitance in farads. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Energy Stored in Capacitors] The energy stored in a capacitor can be expressed in three ways: #math.equation(block: true, alt: "E sub cap equals the fraction QV over 2 equals the fraction CV squared over 2 equals the fraction Q squared over 2 C ,")[$E_("cap") = frac("QV", 2) = frac("CV"^(2), 2) = frac(Q^(2), 2 C) ,$] where #math.equation(block: false, alt: "Q")[$Q$] is the charge, #math.equation(block: false, alt: "V")[$V$] is the voltage, and #math.equation(block: false, alt: "C")[$C$] is the capacitance of the capacitor. The energy is in joules for a charge in coulombs, voltage in volts, and capacitance in farads. ] In a defibrillator, the delivery of a large charge in a short burst to a set of paddles across a person’s chest can be a lifesaver. The person’s heart attack might have arisen from the onset of fast, irregular beating of the heart—cardiac or ventricular fibrillation. The application of a large shock of electrical energy can terminate the arrhythmia and allow the body’s pacemaker to resume normal patterns. Today it is common for ambulances to carry a defibrillator, which also uses an electrocardiogram to analyze the patient’s heartbeat pattern. Automated external defibrillators (AED) are found in many public places . These are designed to be used by lay persons. The device automatically diagnoses the patient’s heart condition and then applies the shock with appropriate energy and waveform. CPR is recommended in many cases before use of an AED. #figure(figph[Photograph of an automated external defibrillator.], alt: "Photograph of an automated external defibrillator.", caption: [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.]) #examplebox("Example 1")[Capacitance in a Heart Defibrillator][ A heart defibrillator delivers #math.equation(block: false, alt: "4.00 times 10 squared J")[$4.00 × "10"^("2") #h(0.25em) "J"$] of energy by discharging a capacitor initially at #math.equation(block: false, alt: "1.00 times 10 to the power 4 V")[$1.00 × "10"^("4") #h(0.25em) "V"$]. What is its capacitance? Strategy We are given #math.equation(block: false, alt: "E sub cap")[$E_("cap")$] and #math.equation(block: false, alt: "V")[$V$], and we are asked to find the capacitance #math.equation(block: false, alt: "C")[$C$]. Of the three expressions in the equation for #math.equation(block: false, alt: "E sub cap")[$E_("cap")$], the most convenient relationship is #math.equation(block: true, alt: "E sub cap equals the fraction CV squared over 2 .")[$E_("cap") = frac("CV"^(2), 2) .$] Solution Solving this expression for #math.equation(block: false, alt: "C")[$C$] and entering the given values yields #math.equation(block: true, alt: "C, equals, the fraction 2 E sub cap over V squared equals the fraction 2 open parenthesis 4 . 00 × 10 squared J close parenthesis over open parenthesis 1 . 00 × 10 to the power 4 V close parenthesis squared equals 8 . 00 × 10 to the power minus 6 F; equals, 8.00 µF.")[$C & = & frac(2 E_("cap"), V^(2)) = frac(2 ( 4 "." "00" × "10"^(2) #h(0.25em) "J" ), ( 1 "." "00" × "10"^(4) #h(0.25em) "V" )^(2)) = 8 "." "00" × "10"^(− 6) #h(0.25em) "F" \ & = & "8.00 µF."$] Discussion This is a fairly large, but manageable, capacitance at #math.equation(block: false, alt: "1.00 times 10 to the power 4 V")[$1.00 × "10"^("4") #h(0.25em) "V"$]. ] === Test Prep for AP Courses Consider a parallel plate capacitor, with no dielectric material, attached to a battery with a fixed voltage. What happens when a dielectric is inserted into the capacitor? + Nothing changes, except now there is a dielectric in the capacitor. + The energy in the system decreases, making it very easy to move the dielectric in. + You have to do work to move the dielectric, increasing the energy in the system. + The reversed polarity destroys the battery. #solutionbox[ (c) ] Consider a parallel plate capacitor with no dielectric material. It was attached to a battery with a fixed voltage to charge up, but now the battery has been disconnected. What happens to the energy of the system and the dielectric material when a dielectric is inserted into the capacitor? What happens to the energy stored in a circuit as you increase the number of capacitors connected in parallel? Series? + increases, increases + increases, decreases + decreases, increases + decreases, decreases #solutionbox[ (b) ] What would the capacitance of a capacitor with the same total internal energy as the car battery in Example 19.1 have to be? Can you explain why we use batteries instead of capacitors for this application? Consider a parallel plate capacitor with metal plates, each of square shape of 1.00 m on a side, separated by 1.00 mm. What is the energy of this capacitor with 3.00×10#super[3] V applied to it? + 3.98×10#super[-2] J + 5.08×10#super[14] J + 1.33×10#super[-5] J + 1.69×10#super[11] J #solutionbox[ (a) ] Consider a parallel plate capacitor with metal plates, each of square shape of 1.00 m on a side, separated by 1.00 mm. What is the internal energy stored in this system if the charge on the capacitor is 30.0 µC? Consider a parallel plate capacitor with metal plates, each of square shape of 1.00 m on a side, separated by 1.00 mm. If the plates grow in area while the voltage is held fixed, the capacitance \_\_\_ and the stored energy \_\_\_. + decreases, decreases + decreases, increases + increases, decreases + increases, increases #solutionbox[ (d) ] Consider a parallel plate capacitor with metal plates, each of square shape of 1.00 m on a side, separated by 1.00 mm. What happens to the energy of this system if the area of the plates increases while the charge remains fixed? === Section Summary - Capacitors are used in a variety of devices, including defibrillators, microelectronics such as calculators, and flash lamps, to supply energy. - The energy stored in a capacitor can be expressed in three ways: #math.equation(block: true, alt: "E sub cap equals the fraction QV over 2 equals the fraction CV squared over 2 equals the fraction Q squared over 2 C ,")[$E_("cap") = frac("QV", 2) = frac("CV"^(2), 2) = frac(Q^(2), 2 C) ,$] where #math.equation(block: false, alt: "Q")[$Q$] is the charge, #math.equation(block: false, alt: "V")[$V$] is the voltage, and #math.equation(block: false, alt: "C")[$C$] 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. === Conceptual Questions 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? What happens to the energy stored in a capacitor connected to a battery when a dielectric is inserted? Was work done in the process? === Problems & Exercises (a) What is the energy stored in the #math.equation(block: false, alt: "10.0 μF")[$"10.0 μF"$] capacitor of a heart defibrillator charged to #math.equation(block: false, alt: "9.00 times 10 cubed V")[$9.00 × "10"^("3") #h(0.25em) "V"$]? (b) Find the amount of stored charge. #solutionbox[ (a) #math.equation(block: false, alt: "405 J")[$"405 J"$] (b) #math.equation(block: false, alt: "90.0 mC")[$"90.0 mC"$] ] In open heart surgery, a much smaller amount of energy will defibrillate the heart. (a) What voltage is applied to the #math.equation(block: false, alt: "8.00 μF")[$"8.00 μF"$] capacitor of a heart defibrillator that stores 40.0 J of energy? (b) Find the amount of stored charge. #solutionbox[ (a) 3.16 kV (b) 25.3 mC ] A #math.equation(block: false, alt: "1 65 µF")[$1 "65 µF"$] capacitor is used in conjunction with a motor. How much energy is stored in it when 119 V is applied? Suppose you have a 9.00 V battery, a #math.equation(block: false, alt: "2.00 μF")[$"2.00 μF"$] capacitor, and a #math.equation(block: false, alt: "7.40 μF")[$"7.40 μF"$] 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. #solutionbox[ (a) #math.equation(block: false, alt: "1.42 times 10 to the power −5 C")[$1.42 × "10"^("−5") #h(0.25em) "C"$], #math.equation(block: false, alt: "6.38 times 10 to the power −5 J")[$6.38 × "10"^("−5") #h(0.25em) "J"$] (b) #math.equation(block: false, alt: "8.46 times 10 to the power −5 C")[$8.46 × "10"^("−5") #h(0.25em) "C"$], #math.equation(block: false, alt: "3.81 times 10 to the power −4 J")[$3.81 × "10"^("−4") #h(0.25em) "J"$] ] 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 #math.equation(block: false, alt: "1.00 times 10 squared m squared")[$1.00 × "10"^("2") #h(0.25em) "m"^("2")$] 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. #solutionbox[ (a) #math.equation(block: false, alt: "4.43 times 10 to the power minus 9 F")[$4.43 × "10"^(− "9") #h(0.25em) " F"$] (b) #math.equation(block: false, alt: "0.452 V")[$0.452 #h(0.25em) "V"$] (c) #math.equation(block: false, alt: "4.52 times 10 to the power minus 10 J")[$4.52 × "10"^(− 10) #h(0.25em) "J"$] ] Show that for a given dielectric material the maximum energy a parallel plate capacitor can store is directly proportional to the volume of dielectric (#math.equation(block: false, alt: "Volume = A times d")[$"Volume =" #h(0.25em) A · d$]). Note that the applied voltage is limited by the dielectric strength. Construct Your Own Problem Consider a heart defibrillator similar to that discussed in . 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. Unreasonable Results (a) On a particular day, it takes #math.equation(block: false, alt: "9.60 times 10 cubed J")[$9.60 × "10"^("3") #h(0.25em) "J"$] of electric energy to start a truck’s 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? #solutionbox[ (a) #math.equation(block: false, alt: "133 F")[$"133" #h(0.25em) "F"$] (b) Such a capacitor would be too large to carry with a truck. The size of the capacitor would be enormous. (c) It is unreasonable to assume that a capacitor can store the amount of energy needed. ]