Essential University Physics Global Edition

Richard Wolfson

Chapter 23

Electrostatic Energy and Capacitors - all with Video Answers

Educators

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

Suppose there is a system of charges comprising both the signs. Does the energy density of the system depend on the signs of the charges involved or on the configuration of the charges? Explain.

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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. Does the energy stored in the balloon's electric field increase or decrease? Speculate on where the energy comes from if it increases, or where it goes if it decreases.

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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

In any kind of capacitor, what is the net charge? How can we find out the potential difference between the plates of the capacitor?

Prabhat Tyagi

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Problem 7

If someone accidentally increases the distance between the plates of a parallel plate capacitor, what would be the easiest way to restore the original value of the capacitance without touching the plates?

Manish Kumar

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Problem 8

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

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Problem 9

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

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Problem 10

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 11

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

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Problem 12

Three point charges $+Q$, and a fourth, $-Q$, lie at the corners of a square. Find the electrostatic energy of this charge distribution.

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Problem 13

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 14

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 15

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 $7.2 \mu \mathrm{C}$ ?

Ajay Singhal

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Problem 16

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

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Problem 17

A capacitor's plates hold $1.0 \mu \mathrm{C}$ when charged to $72 \mathrm{~V}$. What's its capacitance?

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Problem 18

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

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Problem 19

Find the capacitance of a parallel-plate capacitor with circular plates $18 \mathrm{~cm}$ in radius separated by $2.5 \mathrm{~mm}$.

Melissa T

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Problem 20

A parallel-plate capacitor with $1.2-\mathrm{mm}$ plate spacing has $\pm 2.9 \mu \mathrm{C}$ on its plates when charged to $230 \mathrm{~V}$. What's the plate area?

Melissa T

Numerade Educator

Problem 21

FastCAP Systems is a cutting-edge ultracapacitor manufacturer spun off from MIT. Their $35-\mathrm{F}$ model EE-125-35 capacitor is the size and voltage of a standard AA battery. How much energy does this model store when it's fully charged to $1.5 \mathrm{~V} ?$

Melissa T

Numerade Educator

Problem 22

You have a $1.4-\mu \mathrm{F}$ and a $2.6-\mu \mathrm{F}$ capacitor. What capacitances can you get by connecting them in series or in parallel?

Melissa T

Numerade Educator

Problem 23

(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 $12.0$-V battery is connected across the combination.

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Problem 24

You're given three capacitors: $1.3 \mu \mathrm{F}, 2.3 \mu \mathrm{F}$, and $3.3 \mu \mathrm{F}$. 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 25

The energy density in a uniform electric field is $4.0 \mathrm{~J} / \mathrm{m}^{3}$. What's the field strength?

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Problem 26

A car battery stores about $3.6 \mathrm{MJ}$ of energy. If this energy were used to create a uniform $32-\mathrm{kV} / \mathrm{m}$ electric field, what volume would it occupy?

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Problem 27

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 28

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

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Problem 29

Find the equivalent capacitance in the circuit of Fig. $23.9 a$, now taking $C_{1}=6.8 \mu \mathrm{F}, C_{2}=4.7 \mu \mathrm{F}$, and $C_{3}=2.2 \mu \mathrm{F}$ (these are common commercially available capacitances).

Luis Rios

Numerade Educator

Problem 30

What voltage applied between points $A$ and $B$ in Fig. $23.9 a$ will result in a potential difference of $48 \mathrm{~V}$ across $C_{2}$ ? (Use the capacitances given in the figure, not in the preceding problem.)

Farhanul Hasan

Numerade Educator

Problem 31

Find the equivalent capacitance measured between points $A$ and $B$ in Fig. 23.14.

Vishal Gupta

Numerade Educator

Problem 32

In the circuit of Fig. 23.14, how much energy is stored in the $2.7-\mu \mathrm{F}$ capacitor when a potential difference of $75 \mathrm{~V}$ is applied between points $A$ and $B$ ?

James Kiss

Numerade Educator

Problem 33

A spherical shell of radius $R$ carries a charge $Q$ spread uniformly over its surface. Find an expression for the work required to shrink the shell to half its original radius.

Rodger Claar

Numerade Educator

Problem 34

The energy stored in the electric field of the proton is approximately $130 \mathrm{fJ}\left(130 \times 10^{-15} \mathrm{~J}\right)$. Assuming the proton to be a spherical shell with charge spread uniformly over its surface, estimate its radius to one significant figure. (This is an unrealistic model for the proton, but your answer is close to the accepted value for the proton radius.)

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Problem 35

A sphere of radius $R$ contains charge $Q$ spread uniformly throughout its volume. Use the result of Example $21.3$ to show that the electrostatic energy contained within the sphere is $Q^{2} / 40 \pi \epsilon_{0} R$ (equivalently, $k Q^{2} / 10 R$ ).

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 36

Use the result of the preceding problem to repeat Problem 34 , now assuming that the proton's charge is spread uniformly throughout its volume and that the energy given in that problem includes electrostatic energy both inside and outside the proton.

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 37

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 38

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 39

Two closely spaced square conducting plates measure $12 \mathrm{~cm}$ on a side. The electric-field energy density between them is $2.8 \mathrm{~kJ} / \mathrm{m}^{3}$. What's the charge on the plates?

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Problem 40

The potential difference across a cell membrane is $65 \mathrm{mV}$. On the outside are $1.8 \times 10^{6}$ singly ionized potassium atoms. Assuming an equal negative charge on the inside, find the membrane's capacitance.

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Problem 41

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

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Problem 42

A $0.01-\mu \mathrm{F}, 300-\mathrm{V}$ capacitor costs $25 \psi ;$ a $0.1-\mu \mathrm{F}, 100-\mathrm{V}$ capacitor costs $35 \psi ;$ and a $30-\mu \mathrm{F}, 5-\mathrm{V}$ capacitor costs $88 \not$. (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 $\mathrm{J} / \mathcal{\text { ? }}$

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Problem 43

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 $280 \mathrm{~J}$ of its stored energy in $3.1 \mathrm{~ms}$, what's the power delivered during this time?

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Problem 44

A camera requires $5.5 \mathrm{~J}$ 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 $210 \mathrm{~V}$, 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 45

Engineers testing an ultracapacitor (see Application on page 452) 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.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 46

Your company's purchasing department bought lots of cheap $2.0-\mu \mathrm{F}, 50-\mathrm{V}$ 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 $0.5-\mu \mathrm{F}, 200-\mathrm{V}$ capacitors. How will you combine the available capacitors to make these?

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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 -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 $53 \mathrm{~cm}^{2}$ separated by $30 \mu \mathrm{m}$ of polyethylene. Find its (a) capacitance and (b) working voltage.

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Problem 50

A $560-\mathrm{pF}$ capacitor consists of two $15-\mathrm{cm}$-radius circular plates, insulated with polystyrene. 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 $2-\mu \mathrm{F}$ capacitors from Problem 46 . 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 $E_{0}=16 \mathrm{kV} / \mathrm{m}$ and $x_{0}=7.0 \mathrm{~m}$. Find the total energy in the region.

Timothy James

Numerade Educator

Problem 54

The electric field within a spherical region of radius $R$ is inversely proportional to the distance $r$ from the center of the region: $E=E_{0} R / r$, where $E_{0}$ is a constant. Find an expression for the electrostatic energy stored within the region.

Timothy James

Numerade Educator

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

We live inside a giant capacitor! Its plates are Earth's surface and the ionosphere, a conducting layer of the atmosphere beginning about $60 \mathrm{~km}$ up. (a) Find the capacitance of this system, approximating it as a parallel-plate capacitor. (This is justified because the atmosphere is so thin compared with Earth's radius; however, your answer is an underestimate because the atmospheric electric field isn't uniform.) (b) The potential difference between Earth's surface and the ionosphere is about $400 \mathrm{kV}$ (this is maintained by the action of thunderstorms). Estimate the total energy stored in the planetary capacitor.

Sarah Mccrumb

Numerade Educator

Problem 57

Two widely separated $4.7$-mm-diameter water drops each carry $22 \mathrm{nC}$. 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.

Prashant Bana

Numerade Educator

Problem 58

A 3.9-mm-diameter wire carries a uniform line charge density $\lambda=32 \mu \mathrm{C} / \mathrm{m}$. 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 $30 \mathrm{MV}$. Assuming such flashes occur every $5 \mathrm{~s}$ 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.

Prashant Bana

Numerade Educator

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 $L .$

Ajay Singhal

Numerade Educator

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 Chapter 20, Problem 42.)

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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 $1.0-\mu \mathrm{F}$ capacitor, and the entire combination is put in series with a $2.0-\mu \mathrm{F}$ 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

What total capacitance is required if the capacitor system is charged to $20 \mathrm{kV}$ ?
a. $100 \mu \mathrm{F}$
b. $200 \mu \mathrm{F}$
c. $1 \mathrm{~F}$
d. $2 \mathrm{~F}$

Mirza Aslam Beig

Mirza Aslam Beig

Numerade Educator

Problem 71

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

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 72

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. $100 \mathrm{TW}$.

Amit Srivastava

Amit Srivastava

Numerade Educator

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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