Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 24

Capacitance, Dielectrics, Electric Energy Storage - all with Video Answers

Educators

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Chapter Questions

Problem 1

The Problems in this Section are ranked I, II, or III according to estimated difficulty, with (I) Problems being easiest. Level (III) Problems are meant mainly as a challenge for the best students, for
"extra credit." The Problems are arranged by Sections, meaning that the reader should have read up to and including that Section, but this Chapter has a group of General Problems that are not
arranged by Section and not ranked.] (1) The two plates of a capacitor hold $+2800 \mu C$ and
$-2800 \mu C$ of charge, respectively, when the potential difference is 930 V. What is the capacitance?

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Karl Jacob

Numerade Educator

Problem 2

(1) How much charge flows from a $12.0-\mathrm{V}$ battery when it is connected to a $12.6-\mu \mathrm{F}$ capacitor?

Kai Chen

Princeton University

Problem 3

(I) The potential difference between two short sections of parallel wire in air is 24.0 $\mathrm{V}$ . They carry equal and opposite charge of magnitude 75 $\mathrm{pC}$ . What is the capacitance of the two wires?

kj

Karl Jacob

Numerade Educator

Problem 4

(I) The charge on a capacitor increases by 26$\mu C$ when the voltage across it increases from 28 $\mathrm{V}$ to 78 $\mathrm{V}$ . What is the capacitance of the capacitor?

Kai Chen

Princeton University

Problem 5

(II) A $7.7-\mu \mathrm{F}$ capacitor is charged by a $125-\mathrm{V}$ battery (Fig. 20 $\mathrm{a} )$ and then is disconnected from the battery. When this capacitor $\left(C_{1}\right)$ is then connected (Fig. 20 $\mathrm{b} )$ to a second (initially uncharged) capacitor, $C_{2},$ the final voltage on each
capacitor is 15 $\mathrm{V}$ . What is the value of $C_{2} ?[$Hint. . Charge is conserved.]

kj

Karl Jacob

Numerade Educator

Problem 6

(II) An isolated capacitor $C_{1}$ carries a charge $Q_{0} .$ Its wires are then connected to those of a second capacitor $C_{2},$ previously uncharged. What charge will each carry now? What will be the potential difference across each?

Kai Chen

Princeton University

Problem 7

(II) It takes 15 $\mathrm{J}$ of energy to move a 0.20 $\mathrm{-mC}$ charge from one plate of a $15-\mu \mathrm{F}$ capacitor to the other. How much charge is on each plate?

kj

Karl Jacob

Numerade Educator

Problem 8

(II) $\mathrm{A} 2.70-\mu \mathrm{F}$ capacitor is charged to 475 $\mathrm{V}$ and a $4.00-\mu \mathrm{F}$ capacitor is charged to 525 $\mathrm{V}$ . (a) These capacitors are then disconnected from their batteries, and the positive plates are now connected to each other and the negative plates
are connected to each other. What will be the potential difference across each capacitor and the charge on each? (b) What is the voltage and charge for each capacitor if plates of opposite sign are connected?

Kai Chen

Princeton University

Problem 9

(II) Compact "ultracapacitors" with capacitance values up to several thousand farads are now commercially available. One application for ultracapacitors is in providing power for electrical circuits when other sources (such as a battery) are turned off. To get an idea of how much charge can be stored in such a component, assume a $1200-\mathrm{F}$ ultracapacitor is initially charged to 12.0 $\mathrm{V}$ by a battery and is then disconnected from the battery. If charge is then drawn off the
plates of this capacitor at a rate of $1.0 \mathrm{mC} / \mathrm{s},$ say, to power the backup memory of some electrical gadget, how long (in days) will it take for the potential difference across this capacitor to drop to 6.0 $\mathrm{V}$ ?

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Karl Jacob

Numerade Educator

Problem 10

(II) In a dynamic random access memory (DRAM) computer chip, each memory cell chiefly consists of a capacitor for charge storage. Each of these cells represents a single binary-bit value of 1 when its 35 -fF capacitor $\left(1 \mathrm{fF}=10^{-15} \mathrm{F}\right)$ is charged at $1.5 \mathrm{V},$ or 0 when uncharged at 0.V. (a) When it is fully charged, how many excess electrons are on a cell capacitor's negative plate? (b) After charge has been placed on a cell capacitor's plate, it slowly "leaks" off (through a variety of mechanisms) at a constant rate of 0.30 $\mathrm{fC} / \mathrm{s}$ . How long does it take for the potential difference across this capacitor to decrease by 1.0$\%$ from its fully charged value? (Because of this leakage effect, the charge on a DRAM capacitor is "refreshed" many times per second.)

Kai Chen

Princeton University

Problem 11

(1) To make a $0.40-\mu$ F capacitor, what area must the plates
have if they are to be separated by a 2.8 -mm air gap?

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Karl Jacob

Numerade Educator

Problem 12

(I) What is the capacitance per unit length (F/m) of a coaxial cable whose inner conductor has a 1.0 -mm diameter and the outer cylindrical sheath has a $5.0-\mathrm{mm}$ diameter? Assume the space between is filled with air.

Kai Chen

Princeton University

Problem 13

(1) Determine the capacitance of the Earth, assuming it to
be a spherical conductor.

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Karl Jacob

Numerade Educator

Problem 14

(II) Use Gauss's law to show that $\vec{\mathbf{E}}=0$ inside the inner
conductor of a cylindrical capacitor (see Fig, 6 and Example 2
of "Capacitance, Dielectrics, Electric Energy Storage") as
well as outside the outer cylinder.

Kai Chen

Princeton University

Problem 15

(II) Dry air will break down if the electric field exceeds about $3.0 \times 10^{6} \mathrm{V} / \mathrm{m}$ . What amount of charge can be placed on a capacitor if the area of each plate is 6.8 $\mathrm{cm}^{2} ?$

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Karl Jacob

Numerade Educator

Problem 16

(II) An electric field of $4.80 \times 10^{5} \mathrm{V} / \mathrm{m}$ is desired between two parallel plates, each of area 21.0 $\mathrm{cm}^{2}$ and separated by 0.250 $\mathrm{cm}$ of air. What charge must be on each plate?

Kai Chen

Princeton University

Problem 17

(II) How strong is the electric field between the plates of a $0.80-\mu \mathrm{F}$ air-gap capacitor if they are 2.0 $\mathrm{mm}$ apart and each has a charge of 92$\mu \mathrm{C}$ ?

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Karl Jacob

Numerade Educator

Problem 18

(1I) A large metal sheet of thickness $\ell$ is placed between, and parallel to, the plates of the parallel-plate capacitor of Fig. $4 .$ It does not touch the plates, and extends beyond their edges. (a) What is now the net capacitance in terms of $A, d,$ and $\ell ?(b)$ If $\ell=0.40 d,$ by what factor does the capacitance change when the sheet is inserted?

Kai Chen

Princeton University

Problem 19

(III) Small distances are commonly measured capacitively. Consider an air-filled parallel-plate capacitor with fixed plate area $A=25 \mathrm{mm}^{2}$ and a variable plate-separation distance $x$ Assume this capacitor is attached to a capacitance-measuring instrument which can measure capacitance $C$ in the range $\left(x_{\min } \leq x \leq x_{\max }\right)$ can the plate-separation distance $($ in $\mu \mathrm{m})$ be determined by this setup? (b) Define $\Delta x$ to be the accuracy (magnitude) to which $x$ can be determined, and determine a formula for $\Delta x$ (c) Determine the percent accuracy to which $x_{\min }$ and $x_{\text { max }}$ can be measured.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 20

(III) In an electrostatic air cleaner ("precipitator"), the strong nonuniform electric field in the central region of a cylindrical capacitor (with outer and inner cylindrical radii $R_{\mathrm{a}}$ and $R_{\mathrm{b}} )$ is used to create ionized air molecules for use in charging dust and soot particles (Fig. $21 ) .$ Under standard atmospheric conditions, if air is subjected to an electric
field magnitude that exceeds its dielectric strength $E_{\mathrm{S}}=2.7 \times 10^{6} \mathrm{N} / \mathrm{C}$ , air molecules will dissociate into positively charged ions and free electrons. In a precipitator, the region within which air is ionized (the corona discharge region) occupies a cylindrical volume of radius $R$ that is typically five times that of the inner cylinder. Assume a
particular precipitator is constructed with $R_{\mathrm{b}}=0.10 \mathrm{mm}$ and $R_{\mathrm{a}}=10.0 \mathrm{cm} .$ In order to create a corona discharge region with radius $R=5.0 R_{\mathrm{b}}$ , what potential difference $V$ should be applied between the precipitator's inner and outer conducting cylinders? [Besides dissociating air, the charged inner cylinder repels the resulting positive ions from the corona discharge region, where they are put to use in charging dust particles, which are then "collected" on the negatively charged outer cylinder.

Kai Chen

Princeton University

Problem 21

(1) The capacitance of a portion of a circuit is to be reduced from 2900 pF to 1600 pF. What capacitance can be added to the circuit to produce this effect without removing existing circuit elements? Must any existing connections be broken to accomplish this?

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Karl Jacob

Numerade Educator

Problem 22

(I) $(a)$ Six $3.8-\mu$ F capacitors are connected in parallel. What is the equivalent capacitance? (b) What is their equivalent capacitance if connected in series?

Kai Chen

Princeton University

Problem 23

(1I) Given three capacitors, $C_{1}=2.0 \mu \mathrm{F}, C_{2}=1.5 \mu \mathrm{F},$ and $C_{3}=3.0 \mu \mathrm{F}, \quad$ what arrangement of parallel and series connections with a $12-\mathrm{V}$ battery will give the minimum voltage drop across the $2.0-\mu \mathrm{F}$ capacitor? What is the minimum voltage drop?

kj

Karl Jacob

Numerade Educator

Problem 24

(1I) Suppose three parallel-plate capacitors, whose plates have areas $A_{1}, A_{2},$ and $A_{3}$ and separations $d_{1}, d_{2},$ and $d_{3}$ are connected in parallel. Show, using only $\mathrm{Eq} .2,$ that Eq. 3 is valid.
$$C=\frac{Q}{V}=\epsilon_{0} \frac{A}{d} \cdot \quad[\text { parallel-plate capacitor }] (2)$$
$$C_{\mathrm{eq}}=C_{1}+C_{2}+C_{3} . \quad \quad \text { [parallell } ] (3)$$

Kai Chen

Princeton University

Problem 25

(II) An electric circuit was accidentally constructed using a $5.0-\mu \mathrm{F}$ capacitor instead of the required $16-\mu \mathrm{F}$ value. Without removing the $5.0-\mu \mathrm{F}$ capacitor, what can a technician add to correct this circuit?

kj

Karl Jacob

Numerade Educator

Problem 26

(II) Three conducting plates, each of area $A,$ are connected as shown in Fig. $22 .(a)$ Are the two capacitors thus formed connected in series or in parallel? $(b)$ Determine $C$ as a function of $d_{1}, d_{2},$ and $A$ . Assume $d_{1}+d_{2}$ is much less than the dimensions of the plates. (c) The middle plate can be moved (changing the values of $d_{1}$ and $d_{2} ),$ so as to vary the capacitance. What are the minimum and maximum values of the net capacitance?

Kai Chen

Princeton University

Problem 27

(II) Consider three capacitors, of capacitance 3600 $\mathrm{pF}$ , $5800 \mathrm{pF},$ and 0.0100$\mu \mathrm{F}$ . What maximum and minimum capacitance can you form from these? How do you make the connection in each case?

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Karl Jacob

Numerade Educator

Problem 28

(II) A $0.50-\mu \mathrm{F}$ and a $0.80-\mu \mathrm{F}$ capacitor are connected in series to a $9.0-\mathrm{V}$ battery. Calculate $(a)$ the potential difference across each capacitor and $(b)$ the charge on each. (c) Repeat parts $(a)$ and $(b)$ assuming the two capacitors are in parallel.

Kai Chen

Princeton University

Problem 29

$\begin{array}{ccc}{\text { (II) } \operatorname{In}} & {\text { Fig. }} & {23,} & {\text { suppose }} \\ {\text { (a) Determine }} & {\text { the equivalent }}\end{array}$ capacitance between points a and b. (b) Determine the charge on each capacitor and the potential difference across each in terms of $V .$

kj

Karl Jacob

Numerade Educator

Problem 30

(II) Suppose in Fig. 23 that $C_{1}=C_{2}=C_{3}=16.0 \mu \mathrm{F}$ and
$C_{4}=28.5 \mu \mathrm{F} .$ If the charge on $C_{2}$ is $Q_{2}=12.4 \mu \mathrm{C}$ , deter-
mine the charge on each of the other capacitors, the voltage across each capacitor, and the voltage $V_{\text { ab across the entire }}$ combination.

Supratim Pal

Numerade Educator

Problem 31

(11) The switch $S$ in Fig. 24 is connected downward so that capacitor $C_{2}$ becomes fully charged
by the battery of voltage $V_{0}$ . If the switch is then connected upward, determine the charge on each capacitor after the switching.

kj

Karl Jacob

Numerade Educator

Problem 32

(II) $(a)$ Determine the equivalent capacitance between points a and b for the combination of capacitors shown in Fig. $25,(b)$ Determine the charge on each capacitor and the voltage across each if $V_{\text { ba }}=V$ .

Kai Chen

Princeton University

Problem 33

(II) Suppose in Problem $32,$ Fig, $25,$ that $C_{1}=C_{3}=8.0 \mu \mathrm{F}$ $C_{2}=C_{4}=16 \mu \mathrm{F}, \quad$ and $\quad Q_{3}=23 \mu \mathrm{C} .$ Determine $(a)$ the charge on each of the other capacitors, $(b)$ the voltage across each capacitor, and (c) the voltage $V$ ba across the
combination.

kj

Karl Jacob

Numerade Educator

Problem 34

(II) Two capacitors connected in parallel produce an equivalent capacitance of 35.0$\mu \mathrm{F}$ but when connected in series the equivalent capacitance is only 5.5$\mu \mathrm{F}$ . What is the individual capacitance of each capacitor?

Kai Chen

Princeton University

Problem 35

(II) In the capacitance bridge shown in Fig. $26,$ a voltage $V_{0}$ . is applied and the variable capacitor $C_{1}$ is adjusted until there is zero voltage between points a and b as measured on the voltmeter $(\cdot(\overline{\mathrm{V}}) \cdot$ ). Determine the unknown capacitance $C_{x}$ if $C_{1}=8.9 \mu \mathrm{F}$ and the fixed capacitors have $C_{2}=18.0 \mu \mathrm{F} \quad$ and $\quad C_{3}=4.8 \mu \mathrm{F}$ Assume no charge flows through the voltmeter.

kj

Karl Jacob

Numerade Educator

Problem 36

(II) Two capacitors, $C_{1}=3200 \mathrm{pF}$ and $C_{2}=1800 \mathrm{pF},$ are connected in series to a $12.0-\mathrm{V}$ battery. The capacitors are later disconnected from the battery and connected directly to each other, positive plate to positive plate, and negative plate to negative plate. What then will be the charge on each capacitor?

Vishal Gupta

Numerade Educator

Problem 37

(II) $(a)$ Determine the equivalent capacitance of the circuit shown in Fig. $27 . \quad$ (b) If $C_{1}=C_{2}=2 C_{3}=24.0 \mu \mathrm{F}$ how much charge is stored on each capacitor when $V=35.0 \mathrm{V} ?$

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Karl Jacob

Numerade Educator

Problem 38

(1I) In Fig. $27,$ let $C_{1}=2.00 \mu \mathrm{F}, \quad C_{2}=3.00 \mu \mathrm{F}$
$C_{3}=4.00 \mu \mathrm{F}, \quad$ and $\quad V=24.0 \mathrm{V} .$ What is the potential
difference across each capacitor?

Kai Chen

Princeton University

Problem 39

(III) Suppose one plate of a parallel-plate capacitor is tilted so it makes a small angle $\theta$ with the other plate, as shown in Fig. $28 .$ Determine a formula for the capacitance $C$ in terms of $A, d,$ and $\theta$ where $A$ is the area of each plate and $\theta$ is small. Assume the plates are square.
[Hint: Imagine the capacitor as many infinitesimal capacitors in parallel.]

kj

Karl Jacob

Numerade Educator

Problem 40

(III) A voltage $V$ is applied to the capacitor network shown in Fig. $29 .$ (a) What is the equivalent capacitance? [Hint: Assume a potential difference $V_{\text { ab exists across across the }}$ network as shown; write potential differences for various pathways through the network from a to b in terms
of the charges on the capacitors and the capacitances. $]$ (b) Determine the equivalent capacitance
if $C_{2}=C_{4}=8.0 \mu \mathrm{F}$ and $C_{1}=C_{3}=C_{5}=4.5 \mu \mathrm{F}$

Kai Chen

Princeton University

Problem 41

(1) 2200 $\mathrm{V}$ is applied to a $2800-\mathrm{pF}$ capacitor. How much electric energy is stored?

kj

Karl Jacob

Numerade Educator

Problem 42

(I) There is an electric field near the Earth's surface whose intensity is about 150 $\mathrm{V} / \mathrm{m}$ . How much energy is stored per cubic meter in this field?

Kai Chen

Princeton University

Problem 43

(I) How much energy is stored by the electric field between two square plates, 8.0 $\mathrm{cm}$ on a side, separated by a $1.3-\mathrm{mm}$ air gap? The charges on the plates are equal and opposite and of magnitude 420$\mu \mathrm{C}$ .

kj

Karl Jacob

Numerade Educator

Problem 44

(II) A parallel-plate capacitor has fixed charges $+Q$ and $-Q .$ The separation of the plates is then tripled. (a) By what factor does the energy stored in the electric field change?
(b) How much work must be done to increase the separation of the plates from $d$ to 3.0$d ?$ The area of each plate is A.

Kai Chen

Princeton University

Problem 45

(1I) In Fig. $27,$ let $V=10.0 \mathrm{V}$ and $C_{1}=C_{2}=C_{3}=22.6 \mu \mathrm{F}$ . How much energy is stored in the capacitor network?

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Karl Jacob

Numerade Educator

Problem 46

(II) How much energy must a $28-\mathrm{V}$ battery expend to charge a $0.45-\mu \mathrm{F}$ and a $0.20-\mu \mathrm{F}$ capacitor fully when they are placed $(a)$ parallel, (b) in series? (c) How much
charge flowed from the battery in each case?

Kai Chen

Princeton University

Problem 47

(II) $(a)$ Suppose the outer radius $R_{\mathrm{a}}$ of a cylindrical capacitor was tripled, but the charge was kept constant. By what factor would the stored energy change? Where would the energy come from? (b) Repeat part $(a),$ assuming the voltage remains constant.

kj

Karl Jacob

Numerade Educator

Problem 48

(II) A $2.20-\mu \mathrm{F}$ capacitor is charged by a $12.0 . \mathrm{V}$ battery. It is disconnected from the battery and then connected to an uncharged $3.50-\mu \mathrm{F}$ capacitor (Fig. $20 ) .$ Determine the total stored energy $(a)$ before the two capacitors are connected, and (b) after they are connected. (c) What is the change in energy?

Kai Chen

Princeton University

Problem 49

(II) How much work would be required to remove a metal sheet from between the plates of a capacitor (as in Problem 18$a$ ), assuming: (a) the battery remains connected so the voltage remains constant; $(b)$ the battery is disconnected so the charge remains constant?

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Karl Jacob

Numerade Educator

Problem 50

(II) $(a)$ Show that each plate of a parallel-plate capacitor exerts a force
$$F=\frac{1}{2} \frac{Q^{2}}{\epsilon_{0} A}$$
on the other, by calculating dW/dx where $d W$ is the work needed to increase the separation by $d x$ . (b) Why does using $F=Q E,$ with $E$ being the electric field between the plates, give the wrong answer?

Kai Chen

Princeton University

Problem 51

(1I) Show that the electrostatic energy stored in the electric field outside an isolated spherical conductor of radius $r_{0}$ carrying a net charge $Q$ is
$$U=\frac{1}{8 \pi \epsilon_{0}} \frac{Q^{2}}{r_{0}}.$$
Do this in three ways: (a) Use Eq. 6 for the energy density in an electric field [Hint: Consider spherical shells of thickness $d r ] ;(b)$ use Eq. 5 together with the capacitance of an isolated sphere (Section 2 of "Capacitance, Dielectrics, Electric Energy Storage");(c) by calculating the work needed to bring all the charge $Q$ up from infinity in infinitesimal bits $d q$ .

kj

Karl Jacob

Numerade Educator

Problem 52

(II) When two capacitors are connected in parallel and then connected to a battery, the total stored energy is 5.0 times greater than when they are connected in series and then connected to the same battery. What is the ratio of the two capacitances? (Before the battery is connected in each case,
the capacitors are fully discharged.)

Kai Chen

Princeton University

Problem 53

(II) For commonly used CMOS (complementary metal oxide semiconductor) digital circuits, the charging of the component capacitors $C$ to their working potential difference $V$ accounts for the major contribution of its energy input requirements. Thus, if a given logical operation requires such circuitry to charge its capacitors $N$ times, we can assume that the operation requires an energy of
$N\left(\frac{1}{2} C V^{2}\right) .$ In the past 20 years, the capacitance in digital
circuits has been reduced by a factor of about 20 and the
voltage to which these capacitors are charged has been
reduced from 5.0 $\mathrm{V}$ to 1.5 $\mathrm{V}$ . Also, present-day alkaline
batteries hold about five times the energy of older batteries
Two present-day AA alkaline cells, each of which measures
1 cm diameter by 4 $\mathrm{cm}$ long, can power the logic circuitry of
a hand-held personal digital assistant (PDA) with its display
turned off for about two months. If an attempt was made to
construct a similar PDA (i.e., same digital capabilities so $N$
remains constant) 20 years ago, how many (older) AA
batteries would have been required to power its digital
circuitry for two months? Would this PDA fit in a pocket or
purse?

kj

Karl Jacob

Numerade Educator

Problem 54

(I) What is the capacitance of two square parallel plates
4.2 $\mathrm{cm}$ on a side that are separated by 1.8 $\mathrm{mm}$ of paraffin?

Kai Chen

Princeton University

Problem 55

(II) Suppose the capacitor in Example 11 of "Capacitance,
Dielectrics, Electric Energy Storage" remains connected to
the battery as the dielectric is removed. What will be the work
required to remove the dielectric in this case?

kj

Karl Jacob

Numerade Educator

Problem 56

(II) How much energy would be stored in the capacitor of
Problem 43 if a mica dielectric is placed between the plates?
Assume the mica is 1.3 $\mathrm{mm}$ thick (and therefore fills the
space between the plates.

Kai Chen

Princeton University

Problem 57

(II) In the DRAM computer chip of Problem $10,$ the cell
capacitor's two conducting parallel plates are separated by a
2.0 -nm thick insulating material with dielectric constant
$K=25 .$ (a) Determine the area $A\left(\operatorname{in} \mu \mathrm{m}^{2}\right)$ of the cell
capacitor's plates. $(b)$ In (older) "planar" designs, the capac-
itor was mounted on a silicon-wafer surface with its plates
parallel to the plane of the wafer. Assuming the plate area
$A$ accounts for half of the area of each cell, estimate how
many megabytes of memory can be placed on a 3.0 $\mathrm{cm}^{2}$
silicon wafer with the planar design? (1 byte $=8$ bits.

kj

Karl Jacob

Numerade Educator

Problem 58

(II) $A 3500$ -pF air-gap capacitor is connected to a $32-\mathrm{V}$
battery. If a piece of mica fills the space between the plates,
how much charge will flow from the battery?

Kai Chen

Princeton University

Problem 59

(II) Two different dielectrics each fill half the space between
the plates of a parallel-plate capacitor as shown in Fig, 30 .
Determine a formula for the capacitance in terms of $K_{1}, K_{2},$
the area $A$ of the plates, and the separation $d .[$Hint . Can you
consider this capacitor as two capacitors in series or in
parallel?]

kj

Karl Jacob

Numerade Educator

Problem 60

(II) Two different dielectrics fill the space between the plates
of a parallel-plate capacitor as shown in Fig. $31 .$ Determine a
formula for the capacitance in terms of $K_{1}, K_{2},$ the area $A,$ of
the plates, and the separation $d_{1}=d_{2}=d / 2 .$ [Hint: Can
you consider this capacitor as two capacitors in series or in
parallel?]

Kai Chen

Princeton University

Problem 61

(II) Repeat Problem 60 (Fig. 31) but assume the separation
$d_{1} \neq d_{2} .$

kj

Karl Jacob

Numerade Educator

Problem 62

(II) Two identical capacitors are connected in parallel and
each acquires a charge $Q_{0}$ when connected to a source of
voltage $V_{0} .$ The voltage source is disconnected and then a
dielectric $(K=3.2)$ is inserted to fill the space between
the plates of one of the capacitors. Determine $(a)$ the charge
now on each capacitor, and $(b)$ the voltage now across each
capacitor.

Kai Chen

Princeton University

Problem 63

(III) A slab of width $d$ and dielectric constant $K$ is inserted
a distance $x$ into the space between the square parallel
plates (of side $\ell$ ) of a capacitor as shown in Fig. 32. Deter-
mine, as a function of $x,(a)$ the capacitance, $(b)$ the energy
stored if the potential difference is $V_{0},$ and $(c)$ the magni-
tude and direction of the force exerted on the slab (assume
$V_{0}$ is constant)

kj

Karl Jacob

Numerade Educator

Problem 64

(III) The quantity of liquid (such as cryogenic liquid
nitrogen) available in its storage tank is often monitored by
a capacitive level sensor. This sensor is a vertically aligned
cylindrical capacitor with outer and inner conductor radii $R_{\mathrm{a}}$
and $R_{\mathrm{b}},$ whose length $\ell$ spans the height of the tank. When a
and $R_{b},$ whose length $\ell$ spans the height of the tank. When a
nonconducting liquid fills the tank to a height $h(5 \ell)$ from
the tank's bottom, the dielectric in the lower and upper
region between the cylindrical conductors is the liquid $\left(K_{\text { liq }}\right)$
and its vapor $\left(K_{\mathrm{v}}\right),$ respectively (Fig, $33 ) .$ (a) Determine a
formula for the fraction $F$ of the tank filled by liquid in
terms of the level-sensor capacitance $C .[$ Hint: Consider
the sensor as a combination of two capacitors. $.$ (b) By
connecting a capacitance-measuring instrument to the level
sensor, $F$ can be monitored. Assume the sensor dimensions
are $\ell=2.0 \mathrm{m}, \quad R_{\mathrm{n}}=5.0 \mathrm{mm}, \quad$ and $\quad R_{\mathrm{b}}=4.5 \mathrm{mm} .$ For
liquid nitrogen $\left(K_{\mathrm{liq}}=1.4, \quad K_{\mathrm{V}}=1.0\right),$ what values of $C$
(in pF) will correspond to the tank being completely full
and completely empty?

Kai Chen

Princeton University

Problem 65

(II) Show that the capacitor in Example 12 of "Capacitance,
Dielectrics, Electric Energy Storage" with dielectric inserted
can be considered as equivalent to three capacitors in series,
and using this assumption show that the same value for the
capacitance is obtained as was obtained in part $(g)$ of the
Example.

kj

Karl Jacob

Numerade Educator

Problem 66

(1I) Repeat Example 12 assuming the battery remains
connected when the dielectric is inserted. Also, what is the
free charge on the plates after the dielectric is added (let
this be part $(h)$ of this Problem $)$ ?

Kai Chen

Princeton University

Problem 67

(II) Using Example 12 as a model, derive a formula
for the capacitance of a parallel-plate capacitor whose
plates have area $A,$ separation $d,$ with a dielectric of dielec-
tric constant $K$ and thickness $\ell(\ell < d)$ placed between
the plates.

kj

Karl Jacob

Numerade Educator

Problem 68

(II) In Example 12 what percent of the stored energy is
stored in the electric field in the dielectric?

Kai Chen

Princeton University

Problem 69

(III) The capacitor shown in Fig. 34 is connected to a $90.0-\mathrm{V}$
battery. Calculate (and sketch) the electric field everywhere
between the capacitor plates. Find both the free charge on
the capacitor plate and the induced charge on the faces of
the glass dielectric plate.

kj

Karl Jacob

Numerade Educator

Problem 70

(a) A general rule for estimating the capacitance $C$ of an isolated conducting sphere with radius $r$ is $C($ in $\mathrm{pF}) \approx r($ in $\mathrm{cm})$ . That is, the numerical value of $C$ in $\mathrm{pF}$ is about the same as the numerical value of the sphere's radius in $\mathrm{cm} .$ Justify this rule. (b) Modeling the human body as a 1 - 1-m-radius conducting sphere, use the given rule to estimate your body's capacitance. (c) While walking across a carpet, you acquire an excess "static electricity" charge $Q$ and produce a 0.5 -cm spark when reaching out to touch a metallic door-knob. The dielectric strength of air is 30 $\mathrm{kV} / \mathrm{cm} .$ Use this information to estimate $Q($ in $\mu \mathrm{C}) .$

Kai Chen

Princeton University

Problem 71

A cardiac defibrillator is used to shock a heart that is beating erratically. A capacitor in this device is charged to 7.5 $\mathrm{kV}$ and stores 1200 $\mathrm{J}$ of energy. What is its capacitance?

kj

Karl Jacob

Numerade Educator

Problem 72

A homemade capacitor is assembled by placing two 9 -in. pie pans 5.0 $\mathrm{cm}$ apart and connecting them to the opposite terminals of a $9-\mathrm{V}$ battery. Estimate $(a)$ the capacitance,
$(b)$ the charge on each plate, $(c)$ the electric field halfway between the plates, and $(d)$ the work done by the battery to charge the plates, and (d) the work done by the battery to dielectric is inserted?

Kai Chen

Princeton University

Problem 73

An uncharged capacitor is connected to a 34.0 . V battery until it is fully charged, after which it is disconnected from the battery. A slab of paraffin is then inserted between the plates. What will now be the voltage between the plates?

kj

Karl Jacob

Numerade Educator

Problem 74

It takes 18.5 $\mathrm{J}$ of energy to move a $13.0-\mathrm{mC}$ charge from one plate of a $17.0-\mu \mathrm{F}$ capacitor to the other. How much charge is on each plate?

Kai Chen

Princeton University

Problem 75

A huge $3.0-\mathrm{F}$ capacitor has enough stored energy to heat 3.5 $\mathrm{kg}$ of water from $22^{\circ} \mathrm{C}$ to $95^{\circ} \mathrm{C}$ . What is the potential difference across the plates?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 76

A coaxial cable, Fig. $35,$ consists of an inner cylindrical conducting wire of radius $R_{b}$ surrounded by a dielectric insulator. Surrounding the dielectric insulator is an outer conducting sheath of radius $R_{a},$ which is usually "grounded." (a) Determine an expression for the capacitance per unit length of a cable whose insulator has dielectric constant $K .$ (b) For a given cable, $R_{b}=2.5 \mathrm{mm}$ The dielectric constant of the dielectric insulator is $K=2.6$ . Suppose that there is a potential of 1.0 $\mathrm{kV}$ between the inner conducting wire and the outer conducting sheath. Find the capacitance per meter of the cable.

Kai Chen

Princeton University

Problem 77

The electric field between the plates of a paper-separated $(K=3.75)$ capacitor is $9.21 \times 10^{4} \mathrm{V} / \mathrm{m} .$ The plates are 1.95 $\mathrm{mm}$ apart and the charge on each plate is 0.675$\mu \mathrm{C}$ . Determine the capacitance of this capacitor and the area of each plate.

kj

Karl Jacob

Numerade Educator

Problem 78

Capacitors can be used as "electric charge counters." Consider an initially uncharged capacitor of capacitance $C$ with its bottom plate grounded and its top plate connected to a source of electrons, (a) If $N$ electrons flow onto the capacitor's top plate, show that the resulting potential difference $V$ across the capacitor is directly proportional to $N .(b)$ Assume the voltage-measuring device can accurately resolve voltage changes of about 1 $\mathrm{mV}$ . What value of $C$ would be necessary to detect each new collected electron? (c) Using modern semiconductor technology, a micron-size capacitor can be constructed with parallel conducting plates separated by an insulating oxide of dielectric constant $K=3$ and thickness $d=100 \mathrm{nm}$ . To resolve the arrival of an individual electron on the plate of such a capacitor, determine the required value of $\ell($ in $\mu \mathrm{m})$ assuming square plates of side length $\ell .$

Kai Chen

Princeton University

Problem 79

A parallel-plate capacitor is isolated with a charge $\pm Q$ on each plate. If the separation of the plates is halved and a dielectric (constant $K )$ is inserted in place of air, by what factor does the energy storage change? To what do you attribute the change in stored potential energy? How does the new value of the electric field between the plates compare with the original value?

Vishal Gupta

Numerade Educator

Problem 80

In lightning storms, the potential difference between the Earth and the bottom of the thunderclouds can be as high as $35,000,000 \mathrm{V}$ . The bottoms of thunderclouds are typically 1500 $\mathrm{m}$ above the Earth, and may have an area of 120 $\mathrm{km}^{2} .$ Modeling the Earth-cloud system as a huge capacitor, calculate $(a)$ the capacitance of the Earth-cloud system, $(b)$ the charge stored in the "capacitor," and (c) the energy stored in the "capacitor."

Kai Chen

Princeton University

Problem 81

A multilayer film capacitor has a maximum voltage rating of 100 $\mathrm{V}$ and a capacitance of 1.0$\mu \mathrm{F}$ . It is made from alternating sheets of metal foil connected together, separated by films of polyester dielectric. The sheets are 12.0 $\mathrm{mm}$ by 14.0 $\mathrm{mm}$ and the total thickness of the capacitor is 6.0 $\mathrm{mm}$ (not counting the thickness of the insulator on the outside). The metal foil is actually a very thin layer of metal deposited directly on the dielectric, so most of the thickness of the capacitor is due to the dielectric. The dielectric strength of the polyester is about $30 \times 10^{6} \mathrm{V} / \mathrm{m} .$ Estimate the dielectric constant of the polyester material in the capacitor.

kj

Karl Jacob

Numerade Educator

Problem 82

A $3.5-\mu$ F capacitor is charged by a $12.4-\mathrm{V}$ battery and then is disconnected from the battery. When this capacitor $\left(C_{1}\right)$ is then connected to a second (initially uncharged) capacitor, $C_{2},$ the voltage on the first drops to 5.9 $\mathrm{V}$ . What is the value of $C_{2} ?$

Kai Chen

Princeton University

Problem 83

The power supply for a pulsed nitrogen laser has a $0.080-\mu \mathrm{F}$ capacitor with a maximum voltage rating of 25 $\mathrm{kV}$ . (a) Estimate how much energy could be stored in this capacitor.
(b) If 15$\%$ of this stored electrical energy is converted to light energy in a pulse that is $4.0-\mu \mathrm{s}$ long, what is the power of the laser pulse?

kj

Karl Jacob

Numerade Educator

Problem 84

A parallel-plate capacitor has square plates 12 $\mathrm{cm}$ on a side separated by 0.10 $\mathrm{mm}$ of plastic with a dielectric constant of $K=3.1 .$ The plates are connected to a battery, causing them to become oppositely charged. Since the oppositely charged plates attract each other, they exert a pressure on the dielectric. If this pressure is 40.0 $\mathrm{Pa}$ , what is the battery voltage?

Kai Chen

Princeton University

Problem 85

The variable capacitance of an old radio tuner consists of four plates connected together placed alternately between four other plates, also connected together (Fig. 36). Each plate is separated
from its neighbor by 1.6 $\mathrm{mm}$ of air. One set of plates can move so that the area of overlap of each plate varies from 2.0 $\mathrm{cm}^{2}$ to 9.0 $\mathrm{cm}^{2}$ . (a) Are these seven capacitors connected in series or in parallel? (b) Determine the range of capacitance values.

kj

Karl Jacob

Numerade Educator

Problem 86

A high-voltage supply can be constructed from a variable capacitor with interleaving plates which can be rotated as in Fig. 36 . A version of this type of capacitor with more plates has a capacitance which can be varied from 10 pF to 1 pF. (a) Initially, this capacitor is charged by a 7500 -V power supply when the capacitance is 8.0 $\mathrm{pF}$ . It is then disconnected from the power supply and the capacitance reduced to 1.0 $\mathrm{pF}$ by rotating the plates. What is the voltage across the capacitor now? (b) What is a major disadvantage of this as a high-voltage power supply?

Kai Chen

Princeton University

Problem 87

A 175 -pF capacitor is connected in series with an unknown capacitor, and as a series combination they are connected to a 25.0 -V battery. If the $175-$ pF capacitor stores 125 $\mathrm{pC}$ of charge on its plates, what is the unknown capacitance?

kj

Karl Jacob

Numerade Educator

Problem 88

A parallel-plate capacitor with plate area 2.0 $\mathrm{cm}^{2}$ and air- gap separation 0.50 $\mathrm{mm}$ is connected to a $12-\mathrm{V}$ battery, and fully charged. The battery is then disconnected. (a) What is the charge on the capacitor? (b) The plates are now pulled to a separation of 0.75 $\mathrm{mm}$ . What is the charge on the capacitor now? (c) What is the potential difference across the plates now? (d) How much work was required to pull the plates to their new separation?

Kai Chen

Princeton University

Problem 89

In the circuit shown in Fig. $37, C_{1}=1.0 \mu \mathrm{F}, \quad C_{2}=2.0 \mu \mathrm{F}$ $C_{3}=2.4 \mu \mathrm{F},$ and a voltage $V_{\mathrm{ab}}=24 \mathrm{V}$ is applied across points a and b. After $C_{1}$ is fully charged the switch is thrown to the right. What is the final charge and potential difference on each capacitor?

kj

Karl Jacob

Numerade Educator

Problem 90

The long cylindrical capacitor shown in Fig. 38 consists of four concentric cylinders, with respective radii $R_{\mathrm{a}}, R_{\mathrm{b}}, R_{c}$ and $R_{\mathrm{d}} .$ The cylinders $\mathrm{b}$ and $\mathrm{c}$ are joined by metal strips. Determine the capacitance per unit length of this arrangement. (Assume equal and opposite charges are placed on the innermost and outermost cylinders)

Kai Chen

Princeton University

Problem 91

A parallel-plate capacitor has plate area $A,$ plate separation $x,$ and has a charge $Q$ stored on its plates (Fig. $39 ) .$ Find the amount of work required to double the plate separation to $2 x,$ assuming the charge remains constant at $Q .$ Show that your answer is consistent with the change in energy stored by the capacitor. (Hint: See Example 10 of "Capacitance, Dielectrics, Electric Energy Storage.")

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 92

Consider the use of capacitors as memory cells. A charged capacitor would represent a one and an uncharged capacitor a zero. Suppose these capacitors were fabricated on a silicon chip and each has a capacitance of 30 femto-farads $\left(1 \mathrm{fF}=10^{-15} \mathrm{~F} .\right)$ The dielectric filling the space between the parallel plates has dielectric constant $K=25$ and a dielectric strength of $1.0 \times 10^{9} \mathrm{~V} / \mathrm{m} .(a)$ If the operating voltage is $1.5 \mathrm{~V}$, how many electrons would be stored on one of these capacitors when charged? $(b)$ If no safety factor is allowed, how thin a dielectric layer could we use for operation at $1.5 \mathrm{~V} ?(c)$ Using the layer thickness from your answer to part ( $b$ ), what would be the area of the capacitor plates?

Kai Chen

Princeton University

Problem 93

To get an idea how big a farad is, suppose you want to make a $1-\mathrm{F}$ air-filled parallel-plate capacitor for a circuit you are building. To make it a reasonable size, suppose you limit the plate area to 1.0 $\mathrm{cm}^{2} .$ What would the gap have to be between the plates? Is this practically achievable?

kj

Karl Jacob

Numerade Educator

Problem 94

A student wearing shoes with thin insulating soles is standing on a grounded metal floor when he puts his hand flat against the screen of a CRT computer monitor. The voltage inside the monitor screen, 6.3 $\mathrm{mm}$ from his hand, is $25,000 \mathrm{V}$ . The student's hand and the monitor form a capacitor; the student's a conductor, and there is another capacitor between the floor and his feet. Using reasonable numbers for hand and foot areas, estimate the student's voltage relative to the floor. Assume vinyl-soled shoes 1 $\mathrm{cm}$ thick.

Kai Chen

Princeton University

Problem 95

A parallel-plate capacitor with plate area $A=2.0 \mathrm{m}^{2}$ and plate separation $d=3.0 \mathrm{mm}$ is connected to a $45-\mathrm{V}$ battery (Fig. 40 $\mathrm{a} )$ . (a) Determine the charge on the capacitor, the electric field, the capacitance, and the energy stored in the capacitor. (b) With the capacitor still connected to the battery, a slab of plastic with dielectric strength $K=3.2$ is placed between the plates of the capacitor, so that the gap is completely filled with the dielectric. What are the new values of charge, electric field, capacitance, and the energy $U$ stored in the capacitor?

kj

Karl Jacob

Numerade Educator

Problem 96

Let us try to estimate the maximum "static electricity" charge that might result during each walking step across an insulating floor. Assume the sole of a person's shoe has area $A \approx 150 \mathrm{cm}^{2},$ and when the foot is lifted from the ground during each step, the sole acquires an excess charge $Q$ from rubbing contact with the floor. (a) Model the sole as a plane conducting surface with $Q$ uniformly distributed across it as the foot is lifted from the ground. If the dielectric strength of the air between the sole and floor as the foot is lifted is $E_{\mathrm{S}}=3 \times 10^{6} \mathrm{N} / \mathrm{C},$ determine $Q_{\mathrm{max}},$ the maximum possible excess charge that can be transferred to the sole during each step. (b) Modeling a person as an isolated conducting sphere of radius $r \approx 1 \mathrm{m},$ estimate a person's capacitance. (c) After lifting the foot from the floor, assume the excess charge $Q$ quickly redistributes itself over the entire surface area of the person. Estimate the maximum potential difference that the person can develop with respect to the floor.

Kai Chen

Princeton University

Problem 97

Paper has a dielectric constant $K=3.7$ and a dielectric strength of $15 \times 10^{6} \mathrm{V} / \mathrm{m}$ . Suppose that a typical sheet of paper has a thickness of 0.030 $\mathrm{mm}$ . You make a "homemade" capacitor by placing a sheet of $21 \times 14 \mathrm{cm}$ paper between two aluminum foil sheets (Fig. 41). The thickness of the aluminum foil is 0.040 $\mathrm{mm}$ . (a) What is the capacitance $C_{0}$ of your device? (b) About how much charge could you store on your capacitor before it would break down? (c) Show in a sketch how you could overlay sheets of paper and aluminum for a parallel combination. If you made 100 such capacitors, and connected the edges of the sheets in parallel so that you have a single large capacitor of capacitance $100 C_{0},$ how thick would your new large capacitor be? (d) What is the maximum voltage you can apply to this 100$C_{0}$ capacitor without break-down?

kj

Karl Jacob

Numerade Educator

Problem 98

(II) Six physics students were each given an air filled capacitor. Although the areas were different, the spacing between the plates, $d,$ was the same for all six capacitors, but was unknown. Each student made a measurement of the area $A$ and capacitance $C$ of their capacitor. Below is a Table for their data. Using the combined data and a graphing program or spreadsheet, determine the spacing $d$ between the plates.

Kai Chen

Princeton University