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Energy Stored in Capacitors

Module by: OpenStax College. E-mail the author

Summary:

  • 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 defibrillator to pass an electric current through a patient’s heart to get it to beat normally. (Review Figure 1.) 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. (See Figure 1.) Capacitors are also used to supply energy for flash lamps on cameras.

Figure 1: 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)
In an electronic calculator circuit the memory is preserved using large capacitors which store energy when the batteries are charged.

Energy stored in a capacitor is electrical potential energy, and it is thus related to the charge QQ size 12{Q} {} and voltage VV size 12{V} {} on the capacitor. We must be careful when applying the equation for electrical potential energy ΔPE=qΔVΔPE=qΔV size 12{?"PE"=q?V} {} to a capacitor. Remember that ΔPEΔPE size 12{?"PE"} {} is the potential energy of a charge qq size 12{q} {} going through a voltage ΔVΔV size 12{?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 ΔV=0ΔV=0 size 12{?V=0} {}, since the capacitor has zero voltage when uncharged. The final charge placed on a capacitor experiences ΔV=VΔV=V size 12{?V=V} {}, since the capacitor now has its full voltage VV size 12{V} {} on it. The average voltage on the capacitor during the charging process is V/2V/2 size 12{V/2} {}, and so the average voltage experienced by the full charge qq size 12{q} {} is V/2V/2 size 12{V/2} {}. Thus the energy stored in a capacitor, EcapEcap size 12{E rSub { size 8{"cap"} } } {}, is

Ecap=QV2,Ecap=QV2, size 12{E rSub { size 8{"cap"} } =Q { {V} over {2} } } {}
(1)

where QQ size 12{Q} {} is the charge on a capacitor with a voltage VV size 12{V} {} applied. (Note that the energy is not QVQV size 12{ ital "QV"} {}, but QV/2QV/2 size 12{ ital "QV"/2} {}.) Charge and voltage are related to the capacitance C C of a capacitor by Q=CVQ=CV size 12{Q= ital "CV"} {}, and so the expression for EcapEcap size 12{E rSub { size 8{"cap"} } } {} can be algebraically manipulated into three equivalent expressions:

E cap = QV 2 = CV 2 2 = Q 2 2C , E cap = QV 2 = CV 2 2 = Q 2 2C , size 12{E rSub { size 8{"cap"} } = { { ital "QV"} over {2} } = { { ital "CV" rSup { size 8{2} } } over {2} } = { {Q rSup { size 8{2} } } over {2C} } } {}
(2)

where QQ size 12{Q} {} is the charge and VV size 12{V} {} the voltage on a capacitor CC size 12{C} {}. The energy is in joules for a charge in coulombs, voltage in volts, and capacitance in farads.

Energy Stored in Capacitors:

The energy stored in a capacitor can be expressed in three ways:

E cap = QV 2 = CV 2 2 = Q 2 2C , E cap = QV 2 = CV 2 2 = Q 2 2C , size 12{E rSub { size 8{"cap"} } = { { ital "QV"} over {2} } = { { ital "CV" rSup { size 8{2} } } over {2} } = { {Q rSup { size 8{2} } } over {2C} } } {}
(3)

where QQ size 12{Q} {} is the charge, VV size 12{V} {} is the voltage, and CC size 12{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 (Figure 2). 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 2: 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)
Photograph of an automated external defibrillator.

Example 1: Capacitance in a Heart Defibrillator

A heart defibrillator delivers 4.00 × 10 2 J 4.00 × 10 2 J of energy by discharging a capacitor initially at 1.00 × 10 4 V 1.00 × 10 4 V . What is its capacitance?

Strategy

We are given EcapEcap size 12{E rSub { size 8{"cap"} } } {} and VV size 12{V} {}, and we are asked to find the capacitance CC size 12{C} {}. Of the three expressions in the equation for EcapEcap size 12{E rSub { size 8{"cap"} } } {}, the most convenient relationship is

Ecap=CV22.Ecap=CV22. size 12{E rSub { size 8{"cap"} } = { { ital "CV" rSup { size 8{2} } } over {2} } } {}
(4)

Solution

Solving this expression for CC size 12{C} {} and entering the given values yields

C = 2 E cap V 2 = 2 ( 4 . 00 × 10 2 J ) ( 1 . 00 × 10 4 V ) 2 = 8 . 00 × 10 6 F = 8 . 00 µF . C = 2 E cap V 2 = 2 ( 4 . 00 × 10 2 J ) ( 1 . 00 × 10 4 V ) 2 = 8 . 00 × 10 6 F = 8 . 00 µF . alignl { stack { size 12{C= { {2E rSub { size 8{"cap"} } } over {V rSup { size 8{2} } } } = { {"800"" J"} over { \( 1 "." "00"´"10" rSup { size 8{4} } " V" \) rSup { size 8{2} } } } =8 "." "00"´"10" rSup { size 8{-6} } " F"} {} # "=8" "." "00 "mF "." {} } } {}
(5)

Discussion

This is a fairly large, but manageable, capacitance at 1.00 × 10 4 V 1.00 × 10 4 V .

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:
    Ecap=QV2=CV22=Q22C,Ecap=QV2=CV22=Q22C, size 12{E rSub { size 8{"cap"} } = { { ital "QV"} over {2} } = { { ital "CV" rSup { size 8{2} } } over {2} } = { {Q rSup { size 8{2} } } over {2C} } } {}
    (6)
    where QQ size 12{Q} {} is the charge, VV size 12{V} {} is the voltage, and CC size 12{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

Exercise 1

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?

Exercise 2

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

Exercise 1

(a) What is the energy stored in the 10.0 μF 10.0 μF capacitor of a heart defibrillator charged to 9.00 × 10 3 V 9.00 × 10 3 V ? (b) Find the amount of stored charge.

Solution

(a) 405 J 405 J

(b) 90.0 mC 90.0 mC

Exercise 2

In open heart surgery, a much smaller amount of energy will defibrillate the heart. (a) What voltage is applied to the 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.

Solution

(a) 3.16 kV

(b) 25.3 mC

Exercise 3

A 165 µF165 µF size 12{1"65 "mF} {} capacitor is used in conjunction with a motor. How much energy is stored in it when 119 V is applied?

Exercise 4

Suppose you have a 9.00 V battery, a 2.00 μF 2.00 μF capacitor, and a 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.

Solution

(a) 1.42 × 10 −5 C 1.42 × 10 −5 C , 6.38 × 10 −5 J 6.38 × 10 −5 J

(b) 8.46 × 10 −5 C 8.46 × 10 −5 C , 3.81 × 10 −4 J 3.81 × 10 −4 J

Exercise 5

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 1.00 × 10 2 m 2 1.00 × 10 2 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.

Solution

(a) 4.43×1012 F4.43×1012 F size 12{4 "." "43" times "10" rSup { size 8{ - "12"} } " F"} {}

(b) 452 V452 V size 12{"452"" V"} {}

(c) 4.52×107 J4.52×107 J size 12{4 "." "52" times "10" rSup { size 8{ - 7} } " J"} {}

Exercise 6

Show that for a given dielectric material the maximum energy a parallel plate capacitor can store is directly proportional to the volume of dielectric (Volume =A·dVolume =A·d size 12{"Volume="A cdot d} {}). Note that the applied voltage is limited by the dielectric strength.

Exercise 7

Construct Your Own Problem

Consider a heart defibrillator similar to that discussed in Example 1. 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.

Exercise 8

Unreasonable Results

(a) On a particular day, it takes 9.60 × 10 3 J 9.60 × 10 3 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?

Solution

(a) 133F133F size 12{"133"" 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.

Glossary

defibrillator:
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

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