Essential University Physics

Richard Wolfson

Chapter 23

Electrostatic Energy and Capacitors - all with Video Answers

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Chapter Questions

Problem 1

Two positive point charges are infinitely far apart. Is it possible, using a finite amount of work, to move them until they're a small distance $d$ apart?

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Problem 2

How does the energy density at a certain distance from a negative point charge compare with the energy density at the same distance from a positive point charge of equal magnitude?

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Problem 3

A dipole consists of two equal but opposite charges. Is the total energy stored in the dipole's electric field zero? Why or why not?

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Problem 4

Charge is spread over the surface of a balloon, which is then allowed to expand. What happens to the energy of the electric field?

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Problem 5

Does the superposition principle hold for electric-field energy densities? That is, if you double the field strength at some point. do you double the energy density as well?

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Problem 6

A student argues that the total energy associated with the electric field of a charged sphere must be infinite because its field extends throughout an infinite volume. Critique this argument.

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Problem 7

A capacitor is said to carry a charge $Q .$ What's the net charge on the entire capacitor?

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Problem 8

Does the capacitance describe the maximum amount of charge a capacitor can hold, in the same way that a bucket's capacity describes the maximum amount of water it can hold? Explain.

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Problem 9

Is a force needed to hold the plates of a charged capacitor in place? Explain.

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Problem 10

A solid conducting slab is inserted between the plates of a capacitor, not touching either plate. Does the capacitance increase, decrease, or remain the same?

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Problem 11

Two capacitors contain equal amounts of energy, yet one has twice the capacitance. How do their voltages compare?

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Problem 12

A parallel-plate capacitor is connected to a battery that imposes
a potential difference $V$ between its plates. If a dielectric slab is inserted between the plates, what happens to (a) the potential difference, (b) the capacitance $C$, and (c) the capacitor charge $Q$ ?

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Problem 13

Four 75 -\muC charges, initially far apart, are brought onto a line where they're spaced at 5.0 -cm intervals. How much work does it take to assemble this charge distribution?

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Problem 14

Three point charges $+q$ and a fourth, $-\frac{1}{2} q,$ are assembled to form a square of side $a$. Find an expression for the electrostatic energy of this charge distribution.

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Problem 15

Repeat Exercise 14 for the case when the fourth charge is $-q$

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Problem 16

If the three particles in Fig. 23.1 have identical charge $q$ and mass $m$ and if they're released from their positions on the triangle, what speed $v$ will they have when they're far away?

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Problem 17

A crude model of the water molecule has a negatively charged oxygen atom and two protons, as shown in Fig. $23.12 .$ Calculate the electrostatic energy of this configuration, which is therefore the magnitude of the energy released in forming this molecule. (Note: Your answer is an overestimate because electrons are actually "shared" among the three atoms, spending more time near the oxygen.)

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Problem 18

A capacitor consists of square conducting plates $25 \mathrm{cm}$ on a side and $5.0 \mathrm{mm}$ apart, carrying charges $\pm 1.1 \mu \mathrm{C}$. Find (a) the electric field, (b) the potential difference between the plates, and (c) the stored energy.

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Problem 19

An uncharged capacitor has parallel plates $5.0 \mathrm{cm}$ on a side, spaced $1.2 \mathrm{mm}$ apart. (a) How much work is required to transfer $7.2 \mu \mathrm{C}$ from one plate to the other? (b) How much work is required to transfer an additional $7.2 \mu \mathrm{C} ?$

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Problem 20

(a) How much charge must be transferred between the initially uncharged plates of the capacitor in Exercise 19 in order to store 15 mJ of energy? (b) What will be the resulting potential difference between the plates?

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Problem 21

A capacitor's plates hold $1.3 \mu \mathrm{C}$ when charged to $60 \mathrm{V}$. What's its capacitance?

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Problem 22

Show that the units of $\epsilon_{0}$ may be written as $\mathrm{F} / \mathrm{m}$

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Problem 23

Find the capacitance of a parallel-plate capacitor with circular plates $20 \mathrm{cm}$ in radius separated by $1.5 \mathrm{mm}$

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Problem 24

A parallel-plate capacitor with 1.1 -mm plate spacing has $\pm 2.3 \mu \mathrm{C}$ on its plates when charged to $150 \mathrm{V}$. What's the plate area?

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Problem 25

The power supply in a stereo receiver contains a 2500 - $\mu$ F capacitor charged to 35 V. How much energy does it store?

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Problem 26

Find the capacitance of a capacitor that stores $350 \mu \mathrm{J}$ when the potential difference across its plates is $100 \mathrm{V}$

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Problem 27

You have a 1.0 - $\mu \mathrm{F}$ and a 2.0 - $\mu \mathrm{F}$ capacitor. What capacitances can you get by connecting them in series or in parallel?

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Problem 28

Two capacitors are connected in series and the combination is charged to 100 V. If the voltage across each capacitor is $50 \mathrm{V}$ how do their capacitances compare?

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Problem 29

(a) Find the equivalent capacitance of the combination shown in Fig. $23.13 .$ Find (b) the charge and (c) the voltage on each capacitor when a $12.0-\mathrm{V}$ battery is connected across the combination.
(FIGURE CAN'T COPY)

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Problem 30

You're given three capacitors: $1.0 \mu \mathrm{F}, 2.0 \mu \mathrm{F},$ and $3.0 \mu \mathrm{F} .$ Find
(a) the maximum, (b) the minimum, and (c) two intermediate capacitances you could achieve using combinations of all three capacitors.

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Problem 31

The energy density in a uniform electric field is $3.0 \mathrm{J} / \mathrm{m}^{3} .$ What's the field strength?

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Problem 32

A car battery stores about 4 MJ of energy. If this energy were used to create a uniform $30-\mathrm{kV} / \mathrm{m}$ electric field, what volume would it occupy?

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Problem 33

Air undergoes dielectric breakdown at a field strength of $3 \mathrm{MV} / \mathrm{m} .$ Could you store energy in an electric field in air with the same energy density as gasoline? (Hint: See Appendix C.)

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Problem 34

Consider a proton to be a uniformly charged sphere 1 fm in radius. Find the electric energy density at the proton's surface.

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Problem 35

A charge $Q_{0}$ is at the origin. A second charge, $Q_{x}=2 Q_{0},$ is brought from infinity to the point $x=a, y=0 .$ Then a third charge $Q_{y}$ is brought from infinity to $x=0, y=a$. If it takes twice as much work to bring in $Q_{y}$ as it did $Q_{x}$, what's $Q_{y}$ in terms of $Q_{0} ?$

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Problem 36

A conducting sphere of radius $a$ is surrounded by a concentric spherical shell of radius $b$. Both are initially uncharged. How much work does it take to transfer charge from one to the other until they carry charges $\pm Q ?$

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Problem 37

Two closely spaced square conducting plates measure $10 \mathrm{cm}$ on a side. The electric-field energy density between them is $4.5 \mathrm{kJ} / \mathrm{m}^{3} .$ What's the charge on the plates?

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Problem 38

The potential difference across a cell membrane is $65 \mathrm{mV}$. On the outside are $1.5 \times 10^{6}$ singly ionized potassium atoms. Assuming an equal negative charge on the inside, find the membrane's capacitance.

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Problem 39

Which can store more energy: a 1.0 - $\mu$ F capacitor rated at $250 \mathrm{V}$ or a 470 -pF capacitor rated at $3 \mathrm{kV} ?$

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Problem 40

A $0.01-\mu \mathrm{F}, 300-\mathrm{V}$ capacitor costs $25 \alpha ;$ a $0.1-\mu \mathrm{F}, 100-\mathrm{V}$ capacitor costs $35 \%$, and a $30-\mu \mathrm{F}, 5-\mathrm{V}$ capacitor costs $88 \notin .$ (a) Which can store the most charge? (b) Which can store the most energy?
(c) Which is the most cost-effective energy-storage device, measured in $\mathrm{J} / \mathrm{g} ?$

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Problem 41

A medical defibrillator stores $950 \mathrm{J}$ in a $100-\mu \mathrm{F}$ capacitor.
(a) What is the voltage across the capacitor? (b) If the capacitor discharges 300 J of its stored energy in 2.5 ms, what's the power delivered during this time?

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Problem 42

A camera requires $5.0 \mathrm{J}$ of energy for a flash lasting $1.0 \mathrm{ms}$.
(a) What power does the flashtube use while it's flashing? (b) If the flashtube operates at $200 \mathrm{V},$ what size capacitor is needed to supply the flash energy? (c) If the flashtube is fired once every
$10 \mathrm{s},$ what's its average power consumption?

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Problem 43

Engineers testing an ultracapacitor (see Application on page 420 ) measure the capacitor's stored energy at different voltages. The table below gives the results. Determine a quantity that, when you plot stored energy against it, should give a straight line. Make your plot, establish a best-fit line, and use its slope to determine the capacitance.
$$\begin{array}{|l|l|l|l|l|l|l|l|}
\hline \text { Voltage (V) } & 12.2 & 20.1 & 31.8 & 37.9 & 45.7 & 50.2 & 56.0 \\
\hline \text { Energy (kJ) } & 9.25 & 27.2 & 62.5 & 94 & 139 & 158 & 203 \\
\hline
\end{array}$$

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Problem 44

Your company's purchasing department bought lots of cheap 2.0-\muF, 50-V capacitors. Your budget is maxed out and they won't let you buy additional capacitors for a circuit you're designing. You need $2.0-\mu \mathrm{F}, 100-\mathrm{V}$ capacitors and $0.5-\mu \mathrm{F}, 50-\mathrm{V}$ capacitors. How will you combine the available capacitors to make these?

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Problem 45

What's the equivalent capacitance measured between $A$ and $B$ in Fig. $23.14 ?$

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Problem 46

In Fig. $23.14,$ find the energy stored in the 1 - $\mu$ F capacitor when a 50-V battery is connected between $A$ and $B$.

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Problem 47

Capacitors $C_{1}$ and $C_{2}$ are in series, with voltage $V$ across the combination. Show that the voltages across the individual capacitors are $V_{1}=C_{2} V /\left(C_{1}+C_{2}\right)$ and $V_{2}=C_{1} V /\left(C_{1}+C_{2}\right)$

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Problem 48

You're evaluating a new hire in your company's engineering department. Together you're working on a circuit where a $0.1-\mu \mathrm{F}$ $50-\mathrm{V}$ capacitor is in series with a $0.2-\mu \mathrm{F}, 200-\mathrm{V}$ capacitor. The new engineer claims you can safely put $250 \mathrm{V}$ across the combination. What do you say?

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Problem 49

A parallel-plate capacitor has plates with area $50 \mathrm{cm}^{2}$ separated by $25 \mu \mathrm{m}$ of polyethylene. Find its (a) capacitance and (b) working voltage.

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Problem 50

A 470 -pF capacitor consists of two 15 -cm-radius circular plates, insulated with polystyrenc. Find (a) the thickness of the polystyrene and (b) the capacitor's working voltage.

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Problem 51

The first accurate estimate of cell membrane thickness used a capacitive technique, which determined the capacitance per unit area of cell membrane in a macroscopic suspension of cells; the result was about $1 \mu \mathrm{F} / \mathrm{cm}^{2} .$ Assuming a dielectric constant of about 3 for the membrane, find the membrane's thickness. (Note:
Your answer is the thickness of the bipolar lipid layer alone, and is lower by a factor of about 3 than values based on X-ray techniques.)

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Problem 52

Your company is still stuck with those 2 - $\mu$ F capacitors from Problem 44. They turn out to be so cheap that their capacitances are all too low, ranging from $1.7 \mu \mathrm{F}$ to $1.9 \mu \mathrm{F}$. A colleague suggests you put variable "trimmer" capacitors in parallel with the cheap capacitors and adjust the combination to precisely $2.00 \mu \mathrm{F}$ The available trimmers have variable capacitance from $25 \mathrm{nF}$ to $350 \mathrm{nF} .$ Will they work?

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Problem 53

A cubical region $1.0 \mathrm{m}$ on a side is located between $x=0$ and $x=1 \mathrm{m} .$ The region contains an electric field whose magnitude varies with $x$ but is independent of $y$ and $z: E=E_{0}\left(x / x_{0}\right),$ where $E_{0}=24 \mathrm{kV} / \mathrm{m}$ and $x_{0}=6.0 \mathrm{m} .$ Find the total energy in the region.

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Problem 54

A sphere of radius $R$ contains charge $Q$ spread uniformly throughout its volume. Find an expression for the electrostatic energy contained within the sphere itself. (Hint: Consult Example $21.3 .)$

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Problem 55

A sphere of radius $R$ carries total charge $Q$ distributed uniformly over its surface. Show that the energy stored in its electric field is $U=k Q^{2} / 2 R$

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Problem 56

A uranium-235 nucleus has diameter 6.6 fm and contains 92 protons and 143 neutrons. Assuming that charge is distributed uniformly throughout the nucleus, use the results of Problems 54 and
55 to calculate the total electrostatic energy of this configuration.

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Problem 57

Two widely separated 4.0 -mm-diameter water drops each carry
15 nC. Assuming all charge resides on the drops' surfaces, find the change in electrostatic potential energy if they're brought together to form a single spherical drop.

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Problem 58

A 2.1 -mm-diameter wire carries a uniform line charge density $\lambda=28 \mu \mathrm{C} / \mathrm{m} .$ Find the energy in a region $1.0 \mathrm{m}$ long within one wire diameter of the wire surface.

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Problem 59

A typical lightning flash transfers $30 \mathrm{C}$ across a potential difference of 30 MV. Assuming such flashes occur every 5 s in the thunderstorm of Example $23.4,$ roughly how long would the storm last if its electric energy were not replenished?

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Problem 60

A capacitor consists of two long concentric metal cylinders (Fig. 23,15 ). Find an expression for its capacitance in terms of the dimensions shown.
(FIGURE CAN'T COPY)

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Problem 61

A capacitor consists of a conducting sphere of radius $a$ surrounded by a concentric conducting shell of radius $b .$ Show that its capacitance is $C=a b / k(b-a)$

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Problem 62

Show that the result of Problem 61 reduces to that of a parallel-plate capacitor when the separation $b-a$ is much less than the radius $a$

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Problem 63

A solid sphere contains a uniform volume charge density. What fraction of the total electrostatic energy of this configuration is contained within the sphere?

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Problem 64

An air-insulated parallel-plate capacitor of capacitance $C_{0}$ is charged to voltage $V_{0}$ and then disconnected from the charging battery. A slab with dielectric constant $\kappa$ and thickness equal to the capacitor spacing is then inserted halfway into the capacitor (Fig. 23.16 ). Determine (a) the new capacitance, (b) the stored energy, and (c) the force on the slab in terms of $C_{0}, V_{0}, \kappa,$ and the plate length $L$

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Problem 65

Repeat parts (b) and (c) of Problem 64, now assuming the battery remains connected while the slab is inserted.

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Problem 66

A transmission line consists of two parallel wires, of radius $a$ and separation $b,$ carrying uniform line charge densities $\pm \lambda,$ respectively. With $a \ll b$, their electric field is the superposition of the fields from two long straight lines of charge. Find the capacitance per unit length for this transmission line.

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Problem 67

An infinitely long rod of radius $R$ carries uniform volume charge density $\rho .$ Find an expression for the electrostatic energy per unit length contained within the rod. (Hint: See Problem 21.56 .)

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Problem 68

(a) Write the electrostatic potential energy of a pair of oppositely charged, closely spaced parallel plates as a function of their separation $x$, their area $A$, and the charge magnitude $Q$. (b) Differentiate with respect to $x$ to find the magnitude of the attractive force between the plates. Why isn't the force equal to the charge on one plate times the electric field between the plates?

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Problem 69

An unknown capacitor $C$ is connected in series with a $3.0-\mu \mathrm{F}$ capacitor; this pair is placed in parallel with a 1.0 - $\mu$ F capacitor, and the entire combination is put in series with a 2.0 - $\mu$ F capacitor.
(a) Make a circuit diagram of this network. (b) When a potential difference of $100 \mathrm{V}$ is applied across the open ends of the network, the total energy stored in all the capacitors is $5.8 \mathrm{mJ}$. Find $C$.

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Problem 70

Nuclear fusion could provide humankind with limitless energy, making a gallon of seawater the energy equivalent of 300 gallons of gasoline. The National Ignition Facility (NIF) at Lawrence Livermore National Laboratory was designed for the "ignition" of nuclear fusion by bombarding a tiny deuterium-tritium pellet with energy from 192 converging laser beams. The NIF lasers deliver 2 MJ of energy in about 20 ns; Fig. 23.17 shows the target chamber where the laser beams converge. The energy is stored in capacitors that, because of conversion inefficiencies, have to store some 400 MJ. (Note: NIF is more complicated than described here, and the numbers and technical descriptions are only approximate.)
(FIGURE CAN'T COPY)
What total capacitance is required if the capacitor system is charged to $20 \mathrm{kV} ?$
a. $100 \mu \mathrm{F}$
b. $200 \mu F$
c. $1 \mathrm{F}$
d. $2 \mathrm{F}$

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Problem 71

Nuclear fusion could provide humankind with limitless energy, making a gallon of seawater the energy equivalent of 300 gallons of gasoline. The National Ignition Facility (NIF) at Lawrence Livermore National Laboratory was designed for the "ignition" of nuclear fusion by bombarding a tiny deuterium-tritium pellet with energy from 192 converging laser beams. The NIF lasers deliver 2 MJ of energy in about 20 ns; Fig. 23.17 shows the target chamber where the laser beams converge. The energy is stored in capacitors that, because of conversion inefficiencies, have to store some 400 MJ. (Note: NIF is more complicated than described here, and the numbers and technical descriptions are only approximate.)
(FIGURE CAN'T COPY)
If it were technically and economically feasible to double the voltage, how would the required capacitance change?
a. drop to one-quarter its original value
b. drop to one-half its original value
c. would not change
d. would double

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Problem 72

Nuclear fusion could provide humankind with limitless energy, making a gallon of seawater the energy equivalent of 300 gallons of gasoline. The National Ignition Facility (NIF) at Lawrence Livermore National Laboratory was designed for the "ignition" of nuclear fusion by bombarding a tiny deuterium-tritium pellet with energy from 192 converging laser beams. The NIF lasers deliver 2 MJ of energy in about 20 ns; Fig. 23.17 shows the target chamber where the laser beams converge. The energy is stored in capacitors that, because of conversion inefficiencies, have to store some 400 MJ. (Note: NIF is more complicated than described here, and the numbers and technical descriptions are only approximate.)
(FIGURE CAN'T COPY)
While they're firing, the average power delivered by the laser beams is
a. $100 \mathrm{KW}$
b. $100 \mathrm{MW}$
c. $100 \mathrm{GW}$
d. 100 TW.

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Problem 73

Among the capacitors that store energy at NIF are $1200300-\mu \mathrm{F}$ units charged to about $20 \mathrm{kV}$. The energy stored in each capacitor
is about
a. $3 \mathrm{J}$
b. $20 \mathrm{kJ}$
c. $60 \mathrm{kJ}$
d. $400 \mathrm{MJ}$

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