Physics for Scientists and Engineers

Raymond A. Serway, John W. Jewett

Chapter 26

Capacitance and Dielectrics - all with Video Answers

Educators

Chapter Questions

Problem 1

(a) How much charge is on each plate of a $4.00-\mu \mathrm{F}$ capacitor when it is connected to a $12.0-\mathrm{V}$ battery? (b) If this same capacitor is connected to a $1.50-\mathrm{V}$ battery, what charge is stored?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 2

Two conductors having net charges of $+10.0 \mu \mathrm{C}$ and $-10.0 \mu \mathrm{C}$ have a potential difference of $10.0 \mathrm{V}$ between them. (a) Determine the capacitance of the system.
(b) What is the potential difference between the two conductors if the charges on each are increased to $+100 \mu \mathrm{C}$ and $-100 \mu \mathrm{C} ?$

Prashant Bana

Numerade Educator

Problem 3

An isolated charged conducting sphere of radius $12.0 \mathrm{cm}$ creates an electric field of $4.90 \times 10^{4} \mathrm{N} / \mathrm{C}$ at a distance $21.0 \mathrm{cm}$ from its center. (a) What is its surface charge density? (b) What is its capacitance?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 4

(a) If a drop of liquid has capacitance $1.00 \mathrm{pF},$ what is its radius? (b) If another drop has radius $2.00 \mathrm{mm},$ what is its capacitance? (c) What is the charge on the smaller drop if its potential is $100 \mathrm{V} ?$

Pawan Yadav

Numerade Educator

Problem 5

Two conducting spheres with diameters of $0.400 \mathrm{m}$ and $1.00 \mathrm{m}$ are separated by a distance that is large compared with the diameters. The spheres are connected by a thin wire and are charged to $7.00 \mu \mathrm{C}$. (a) How is this total charge shared between the spheres? (Ignore any charge on the wire.) (b) What is the potential of the system of spheres when the reference potential is taken to be $V=0$ at $r=\infty ?$

Justin Swantek

Numerade Educator

Problem 6

Regarding the Earth and a cloud layer $800 \mathrm{m}$ above the Earth as the "plates" of a capacitor, calculate the capacitance. Assume the cloud layer has an area of $1.00 \mathrm{km}^{2}$ and that the air between the cloud and the ground is pure and dry. Assume charge builds up on the cloud and on the ground until a uniform electric field of $3.00 \times 10^{6} \mathrm{N} / \mathrm{C}$ throughout the space between them makes the air break down and conduct electricity as a lightning bolt. What is the maximum charge the cloud can hold?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 7

An air-filled capacitor consists of two parallel plates, each with an area of $7.60 \mathrm{cm}^{2},$ separated by a distance of 1.80 mm. A 20.0-V potential difference is applied to these plates. Calculate (a) the electric field between the plates, (b) the surface charge density, (c) the capacitance, and (d) the charge on each plate.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 8

A 1 -megabit computer memory chip contains many $60.0-\mathrm{fF}$ capacitors. Fach capacitor has a plate area of $21.0 \times 10^{-12} \mathrm{m}^{2} .$ Determine the plate separation of such a capacitor (assume a parallel-plate configuration). The order of magnitude of the diameter of an atom is $10^{-10} \mathrm{m}=0.1 \mathrm{nm} .$ Express the plate separation in nanometers.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 9

When a potential difference of $150 \mathrm{V}$ is applied to the plates of a parallel-plate capacitor, the plates carry a surface charge density of $30.0 \mathrm{nC} / \mathrm{cm}^{2} .$ What is the spacing between the plates?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 10

A variable air capacitor used in a radio tuning circuit is made of $N$ semicircular plates each of radius $R$ and positioned a distance $d$ from its neighbors, to which it is electrically connected. As shown in Figure $\mathrm{P} 26.10,$ a second identical set of plates is enmeshed with its plates halfway between those of the first set. The second set can rotate as a unit. Determine the capacitance as a function of the angle of rotation $\theta,$ where $\theta=0$ corresponds to the maximum capacitance.
CAN'T COPY THE FIGURE

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 11

A 50.0 -m length of coaxial cable has an inner conductor that has a diameter of $2.58 \mathrm{mm}$ and carries a charge of $8.10 \mu \mathrm{C} .$ The surrounding conductor has an inner diameter of $7.27 \mathrm{mm}$ and a charge of $-8.10 \mu \mathrm{C} .$ (a) What is the capacitance of this cable? (b) What is the potential difference between the two conductors? Assume the region between the conductors is air.

Mayukh Banik

Numerade Educator

Problem 12

A $20.0-\mu \mathrm{F}$ spherical capacitor is composed of two concentric metal spheres, one having a radius twice as large as the other. The region between the spheres is a vacuum. Determine the volume of this region.

Mayukh Banik

Numerade Educator

Problem 13

An air-filled spherical capacitor is constructed with inner and outer shell radii of 7.00 and $14.0 \mathrm{cm},$ respectively. (a) Calculate the capacitance of the device. (b) What potential difference between the spheres results in a charge of $4.00 \mu \mathrm{C}$ on the capacitor?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 14

A small object of mass $m$ carries a charge $q$ and is suspended by a thread between the vertical plates of a parallel-plate capacitor. The plate separation is $d$. If the thread makes an angle $\theta$ with the vertical, what is the potential difference between the plates?

Rashmi Sinha

Numerade Educator

Problem 15

Find the capacitance of the Earth. (Suggestion: The outer conductor of the "spherical capacitor" may be considered as a conducting sphere at infinity where $V$ approaches zero.)

Mayukh Banik

Numerade Educator

Problem 16

Two capacitors, $C_{1}=5.00 \mu \mathrm{F}$ and $C_{2}=12.0 \mu \mathrm{F},$ are connected in parallel, and the resulting combination is connected to a $9.00-\mathrm{V}$ battery. (a) What is the equivalent capacitance of the combination? What are (b) the potential difference across each capacitor and (c) the charge stored on each capacitor?

Mayukh Banik

Numerade Educator

Problem 17

What If? The two capacitors of Problem 16 are now connected in series and to a $9.00-\mathrm{V}$ battery. Find (a) the equivalent capacitance of the combination, (b) the potential difference across each capacitor, and (c) the charge on each capacitor.

Mayukh Banik

Numerade Educator

Problem 18

Evaluate the equivalent capacitance of the configuration shown in Figure P26.18. All the capacitors are identical, and each has capacitance $C$.
CAN'T COPY THE FIGURE

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 19

Two capacitors when connected in parallel give an equivalent capacitance of $9.00 \mathrm{pF}$ and give an equivalent capacitance of $2.00 \mathrm{pF}$ when connected in series. What is the capacitance of each capacitor?

Ben Nicholson

Numerade Educator

Problem 20

Two capacitors when connected in parallel give an equivalent capacitance of $C_{p}$ and an equivalent capacitance of $C_{s}$ when connected in series. What is the capacitance of each capacitor?

Willis James

Numerade Educator

Problem 21

Four capacitors are connected as shown in Figure P26.21. (a) Find the equivalent capacitance between points $a$ and $b$. (b) Calculate the charge on each capacitor if $\Delta V_{a b}=15.0 \mathrm{V}$.
CAN'T COPY THE FIGURE

Mayukh Banik

Numerade Educator

Problem 22

Three capacitors are connected to a battery as shown in Figure P26.22. Their capacitances are $C_{1}=3 C, C_{2}=C$ and $C_{3}=5 C$. (a) What is the equivalent capacitance of this set of capacitors? (b) State the ranking of the capacitors according to the charge they store, from largest to smallest. (c) Rank the capacitors according to the potential differences across them, from largest to smallest.
(d) What If? If $C_{3}$ is increased, what happens to the charge stored by each of the capacitors?
CAN'T COPY THE FIGURE

Mayukh Banik

Numerade Educator

Problem 23

Consider the circuit shown in Figure $\mathrm{P} 26.23,$ where $C_{1}=6.00 \mu \mathrm{F}, C_{2}=3.00 \mu \mathrm{F},$ and $\Delta V=20.0 \mathrm{V} .$ Capacitor
$\bar{C}_{1}$ is first charged by the closing of switch $\mathrm{S}_{1} .$ Switch $\mathrm{S}_{1}$ is then opened, and the charged capacitor is connected to the uncharged capacitor by the closing of $S_{2} .$ Calculate the initial charge acquired by $C_{1}$ and the final charge on each capacitor.
CAN'T COPY THE FIGURE

Mayukh Banik

Numerade Educator

Problem 24

According to its design specification, the timer circuit delaying the closing of an elevator door is to have a capacitance of $32.0 \mu \mathrm{F}$ between two points $A$ and $B .$ (a) When one circuit is being constructed, the inexpensive but durable capacitor installed between these two points is found to have capacitance $34.8 \mu \mathrm{F}$. To meet the specification, one additional capacitor can be placed between the two points. Should it be in series or in parallel with the $34.8-\mu \mathrm{F}$ capacitor? What should be its capacitance?
(b) What If? The next circuit comes down the assembly line with capacitance $29.8 \mu \mathrm{F}$ between $A$ and $B .$ What additional capacitor should be installed in series or in parallel in that circuit, to meet the specification?

Mayukh Banik

Numerade Educator

Problem 25

A group of identical capacitors is connected first in series and then in parallel. The combined capacitance in parallel is 100 times larger than for the series connection. How many capacitors are in the group?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 26

Consider three capacitors $C_{1}, C_{2}, C_{3},$ and a battery. If $C_{1}$ is connected to the battery, the charge on $C_{1}$ is $30.8 \mu \mathrm{C}$. Now $C_{1}$ is disconnected, discharged, and connected in series with $C_{2} .$ When the series combination of $C_{2}$ and $C_{1}$ is connected across the battery, the charge on $C_{1}$ is $23.1 \mu \mathrm{C} .$ The circuit is disconnected and the capacitors discharged. Capacitor $C_{3},$ capacitor $C_{1},$ and the battery are connected in series, resulting in a charge on $C_{1}$ of $25.2 \mu \mathrm{C} .$ If, after being disconnected and discharged, $C_{1}, C_{2},$ and $C_{3}$ are connected in series with one another and with the battery, what is the charge on $C_{1} ?$

Mayukh Banik

Numerade Educator

Problem 27

Find the equivalent capacitance between points $a$ and $b$ for the group of capacitors connected as shown in Figure P26.27. Take $C_{1}=5.00 \mu \mathrm{F}, C_{2}=10.0 \mu \mathrm{F},$ and $C_{3}=2.00 \mu \mathrm{F}$.
CAN'T COPY THE FIGURE

Mayukh Banik

Numerade Educator

Problem 28

For the network described in the previous problem, if the potential difference between points $a$ and $b$ is $60.0 \mathrm{V},$ what charge is stored on $C_{3} ?$

Mayukh Banik

Numerade Educator

Problem 29

Find the equivalent capacitance between points $a$ and $b$ in the combination of capacitors shown in Figure $\mathrm{P} 26.29 .$
CAN'T COPY THE FIGURE

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 30

Some physical systems possessing capacitance continuously distributed over space can be modeled as an infinite array of discrete circuit elements. Examples are a microwave waveguide and the axon of a nerve cell. To practice analysis of an infinite array, determine the equivalent capacitance $C$ between terminals $X$ and $Y$ of the infinite set of capacitors represented in Figure $\mathrm{P} 26.30 .$ Each capacitor has capacitance $C_{0} .$ Suggestion: Imagine that the ladder is cut at the line $A B,$ and note that the equivalent capacitance of the infinite section to the right of $A B$ is also $C$.
CAN'T COPY THE FIGURE

Mayukh Banik

Numerade Educator

Problem 31

(a) A $3.00-\mu \mathrm{F}$ capacitor is connected to a $12.0-\mathrm{V}$ battery. How much energy is stored in the capacitor? (b) If the capacitor had been connected to a $6.00-\mathrm{V}$ battery, how much energy would have been stored?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 32

The immediate cause of many deaths is ventricular fibrillation, uncoordinated quivering of the heart as opposed to proper beating. An electric shock to the chest can cause momentary paralysis of the heart muscle, after which the heart will sometimes start organized beating again. A defibrillator (Fig. 26.14 ) is a device that applies a strong electric shock to the chest over a time interval of a few milliseconds. The device contains a capacitor of several microfarads, charged to several thousand volts. Electrodes called paddles, about $8 \mathrm{cm}$ across and coated with conducting paste, are held against the chest on both sides of the heart. Their handles are insulated to prevent injury to the operator, who calls, "Clear!" and pushes a button on one paddle to discharge the capacitor through the patient's chest. Assume that an energy of $300 \mathrm{J}$ is to be delivered from a $30.0-\mu \mathrm{F}$ capacitor. To what potential difference must it be charged?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 33

Two capacitors, $C_{1}=25.0 \mu \mathrm{F}$ and $C_{2}=5.00 \mu \mathrm{F},$ are connected in parallel and charged with a $100-\mathrm{V}$ power supply. (a) Draw a circuit diagram and calculate the total energy stored in the two capacitors. (b) What If? What potential difference would be required across the same two capacitors connected in series in order that the combination stores the same amount of energy as in (a)? Draw a circuit diagram of this circuit.

Mayukh Banik

Numerade Educator

Problem 34

A parallel-plate capacitor is charged and then disconnected from a battery. By what fraction does the stored energy change (increase or decrease) when the plate separation is doubled?

Mayukh Banik

Numerade Educator

Problem 35

As a person moves about in a dry environment, electric charge accumulates on his body. Once it is at high voltage, either positive or negative, the body can discharge via sometimes noticeable sparks and shocks. Consider a human body well separated from ground, with the typical capacitance $150 \mathrm{pF}$. (a) What charge on the body will produce a potential of $10.0 \mathrm{kV} ?$ (b) Sensitive electronic devices can be destroyed by electrostatic discharge from a person. A particular device can be destroyed by a discharge releasing an energy of $250 \mu \mathrm{J}$. To what voltage on the body does this correspond?

Mayukh Banik

Numerade Educator

Problem 36

A uniform electric field $E=3000 \mathrm{V} / \mathrm{m}$ exists within a certain region. What volume of space contains an energy equal to $1.00 \times 10^{-7} \mathrm{J}$ ? Express your answer in cubic meters and in liters.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 37

A parallel-plate capacitor has a charge $Q$ and plates of area $A$. What force acts on one plate to attract it toward the other plate? Because the electric field between the plates is $E=Q / A \epsilon_{0},$ you might think that the force is $F=Q E=Q^{2} / A \epsilon_{0} .$ This is wrong, because the field $E$ includes contributions from both plates, and the field created by the positive plate cannot exert any force on the positive plate. Show that the force exerted on each plate is actually $F=Q^{2} / 2 \epsilon_{0} A .$ (Suggestion: Let $C=\epsilon_{0} A / x$ for an arbitrary plate separation $x ;$ then require that the work done in separating the two charged plates be $W=\int F \, d x$.) The force exerted by one charged plate on another is sometimes used in a machine shop to hold a workpiece stationary.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 38

The circuit in Figure $P 26.38$ consists of two identical parallel metal plates connected by identical metal springs to a 100-V battery. With the switch open, the plates are uncharged, are separated by a distance $d=8.00 \mathrm{mm},$ and have a capacitance $C=2.00 \mu \mathrm{F}$. When the switch is closed, the distance between the plates decreases by a factor of 0.500. (a) How much charge collects on each plate and
(b) what is the spring constant for each spring? (Suggestion: Use the result of Problem $37 .$ )
CAN'T COPY THE FIGURE

Mayukh Banik

Numerade Educator

Problem 39

Review problem. A certain storm cloud has a potential of $1.00 \times 10^{8} \mathrm{V}$ relative to a tree. If, during a lightning storm, $50.0 \mathrm{C}$ of charge is transferred through this potential difference and $1.00 \%$ of the energy is absorbed by the tree, how much sap in the tree can be boiled away? Model the sap as water initially at $30.0^{\circ} \mathrm{C}$. Water has a specific heat of $4186 \mathrm{J} / \mathrm{kg}^{\circ} \mathrm{C},$ a boiling point of $100^{\circ} \mathrm{C},$ and a latent heat of vaporization of $2.26 \times 10^{6} \mathrm{J} / \mathrm{kg}$.

Mayukh Banik

Numerade Educator

Problem 40

Two identical parallel-plate capacitors, each with capacitance $C,$ are charged to potential difference $\Delta V$ and connected in parallel. Then the plate separation in one of the capacitors is doubled. (a) Find the total energy of the system of two capacitors before the plate separation is doubled. (b) Find the potential difference across each capacitor after the plate separation is doubled. (c) Find the total energy of the system after the plate separation is doubled. (d) Reconcile the difference in the answers to parts (a) and (c) with the law of conservation of energy.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 41

Show that the energy associated with a conducting sphere of radius $R$ and charge $Q$ surrounded by a vacuum is $U=k_{e} Q^{2} / 2 R$.

Mayukh Banik

Numerade Educator

Problem 42

Consider two conducting spheres with radii $R_{1}$ and $R_{2}$ They are separated by a distance much greater than either radius. A total charge $Q$ is shared between the spheres, subject to the condition that the electric potential energy of the system has the smallest possible value. The total charge $Q$ is equal to $q_{1}+q_{2},$ where $q_{1}$ represents the charge on the first sphere and $q_{2}$ the charge on the second. Because the spheres are very far apart, you can assume that the charge of each is uniformly distributed over its surface. You may use the result of Problem $41 .$ (a) Determine the values of $q_{1}$ and $q_{2}$ in terms of $Q, R_{1}$ and $R_{2} .$ (b) Show that the potential difference between the spheres is zero. (We saw in Chapter 25 that two conductors joined by a conducting wire will be at the same potential in a static situation. This problem illustrates the general principle that static charge on a conductor will distribute itself so that the electric potential energy of the system is a minimum.)

Mayukh Banik

Numerade Educator

Problem 43

Determine (a) the capacitance and (b) the maximum potential difference that can be applied to a Teflon-filled parallel-plate capacitor having a plate area of $1.75 \mathrm{cm}^{2}$ and plate separation of $0.0400 \mathrm{mm}$.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 44

(a) How much charge can be placed on a capacitor with air between the plates before it breaks down, if the area of each of the plates is $5.00 \mathrm{cm}^{2} ?$ (b) What If? Find the maximum charge if polystyrene is used between the plates instead of air.

Mayukh Banik

Numerade Educator

Problem 45

A commercial capacitor is to be constructed as shown in Figure $26.17 \mathrm{a}$. This particular capacitor is made from two strips of aluminum separated by a strip of paraffin-coated paper. Each strip of foil and paper is $7.00 \mathrm{cm}$ wide. The foil is $0.00400 \mathrm{mm}$ thick, and the paper is $0.0250 \mathrm{mm}$ thick and has a diclectric constant of $3.70 .$ What length should the strips have, if a capacitance of $9.50 \times 10^{-8} \mathrm{F}$ is desired before the capacitor is rolled up? (Adding a second strip of paper and rolling the capacitor effectively doubles its capacitance, by allowing charge storage on both sides of each strip of foil.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 46

The supermarket sells rolls of aluminum foil, of plastic wrap, and of waxed paper. Describe a capacitor made from supermarket materials. Compute order-of-magnitude estimates for its capacitance and its breakdown voltage.

Mayukh Banik

Numerade Educator

Problem 47

A parallel-plate capacitor in air has a plate separation of $1.50 \mathrm{cm}$ and a plate area of $25.0 \mathrm{cm}^{2} .$ The plates are charged to a potential difference of $250 \mathrm{V}$ and disconnected from the source. The capacitor is then immersed in distilled water. Determine (a) the charge on the plates before and after immersion, (b) the capacitance and potential difference after immersion, and (c) the change in energy of the capacitor. Assume the liquid is an insulator.

Mayukh Banik

Numerade Educator

Problem 48

A wafer of titanium dioxide $(\kappa=173)$ of area $1.00 \mathrm{cm}^{2}$ has a thickness of $0.100 \mathrm{mm}$. Aluminum is evaporated on the parallel faces to form a parallel-plate capacitor. (a) Calculate the capacitance. (b) When the capacitor is charged with a $12.0-\mathrm{V}$ battery, what is the magnitude of charge delivered to each plate? (c) For the situation in part (b), what are the free and induced surface charge densities? (d) What is the magnitude of the electric field?

Mayukh Banik

Numerade Educator

Problem 49

Each capacitor in the combination shown in Figure $\mathrm{P} 26.49$ has a breakdown voltage of $15.0 \mathrm{V}$. What is the breakdown voltage of the combination?
CAN'T COPY THE FIGURE

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 50

A small rigid object carries positive and negative $3.50-\mathrm{nC}$ charges. It is oriented so that the positive charge has coordinates $(-1.20 \mathrm{mm}, 1.10 \mathrm{mm})$ and the negative charge is at the point $(1.40 \mathrm{mm},-1.30 \mathrm{mm}) .$ (a) Find the electric dipole moment of the object. The object is placed in an electric field $\quad \mathbf{E}=(7800 \mathbf{i}-4900 \hat{\mathbf{j}}) \mathrm{N} / \mathrm{C} .$ (b) Find the
torque acting on the object. (c) Find the potential energy of the object-ficld system when the object is in this orientation. (d) If the oricntation of the object can change, find the difference between the maximum and minimum potential energies of the system.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 51

A small object with electric dipole moment $p$ is placed in a nonuniform electric field $\mathbf{E}=E(x)$ i. That is, the field is in the $x$ direction and its magnitude depends on the coordinate $x$. Let $\theta$ represent the angle between the dipole moment and the $x$ direction. (a) Prove that the dipole feels a net force
$$F=p\left(\frac{d E}{d x}\right) \cos \theta$$
in the direction toward which the field increases. (b) Consider a spherical balloon centered at the origin, with radius $15.0 \mathrm{cm}$ and carrying charge $2.00 \mu \mathrm{C}$. Evaluate $d E / d x$ at the point $(16 \mathrm{cm}, 0,0) .$ Assume a water droplet at this point has an induced dipole moment of $6.30 \hat{\mathrm{i}} \mathrm{nC} \cdot \mathrm{m}$ Find the force on it.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 52

A detector of radiation called a Geiger tube consists of a closed, hollow, conducting cylinder with a fine wire along its axis. Suppose that the internal diameter of the cylinder is $2.50 \mathrm{cm}$ and that the wire along the axis has a diameter of $0.200 \mathrm{mm} .$ The dielectric strength of the gas between the central wire and the cylinder is $1.20 \times 10^{6} \mathrm{V} / \mathrm{m} .$ Calculate the maximum potential difference that can be applied between the wire and the cylinder before breakdown occurs in the gas.

Mayukh Banik

Numerade Educator

Problem 53

The general form of Gauss's law describes how a charge creates an electric field in a material, as well as in vacuum. It is
$$\oint \mathbf{E} \cdot d \mathbf{A}=\frac{q}{\epsilon}$$
where $\epsilon=\kappa \epsilon_{0}$ is the permittivity of the material. (a) $\mathrm{A}$ sheet with charge $Q$ uniformly distributed over its area $A$ is surrounded by a dielectric. Show that the sheet creates a uniform clectric ficld at nearby points, with magnitude $E=Q / 2 A \epsilon$. (b) Two large sheets of area $A$, carrying opposite charges of equal magnitude $Q,$ are a small distance
$d$ apart. Show that they create uniform electric field in the space between them, with magnitude $E=Q / A \epsilon$
(c) Assume that the negative plate is at zero potential. Show that the positive plate is at potential Qd/Ae.
(d) Show that the capacitance of the pair of plates is $A \epsilon / d=\kappa A \epsilon_{0} / d$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 54

For the system of capacitors shown in Figure $\mathrm{P} 26.54,$ find
(a) the equivalent capacitance of the system, (b) the potential across each capacitor, (c) the charge on each capacitor, and (d) the total energy stored by the group.
CAN'T COPY THE FIGURE

Mayukh Banik

Numerade Educator

Problem 55

Four parallel metal plates $P_{1}, P_{2}, P_{3},$ and $P_{4},$ each of area $7.50 \mathrm{cm}^{2},$ are separated successively by a distance $d=$
$1.19 \mathrm{mm},$ as shown in Figure $\mathrm{P} 26.55 . \mathrm{P}_{1}$ is connected to the negative terminal of a battery, and $P_{2}$ to the positive terminal. The battery maintains a potential difference of $12.0 \mathrm{V}$
(a) If $P_{3}$ is connected to the negative terminal, what is the capacitance of the three-plate system $\mathrm{P}_{1} \mathrm{P}_{2} \mathrm{P}_{3} ?$ (b) What is the charge on $\mathrm{P}_{2} ?$ (c) If $\mathrm{P}_{4}$ is now connected to the positive terminal of the battery, what is the capacitance of the fourplate system $\mathrm{P}_{1} \mathrm{P}_{2} \mathrm{P}_{3} \mathrm{P}_{4} ?$ (d) What is the charge on $\mathrm{P}_{4} ?$
CAN'T COPY THE FIGURE

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 56

One conductor of an overhead electric transmission line is a long aluminum wire $2.40 \mathrm{cm}$ in radius. Suppose that at a particular moment it carries charge per length $1.40 \mu \mathrm{C} / \mathrm{m}$ and is at potential $345 \mathrm{kV}$. Find the potential $12.0 \mathrm{m}$ below the wire. Ignore the other conductors of the transmission line and assume the electric field is everywhere purely radial.

Mayukh Banik

Numerade Educator

Problem 57

Two large parallel metal plates are oriented horizontally and separated by a distance $3 d .$ A grounded conducting wire joins them, and initially each plate carries no charge. Now a third identical plate carrying charge $Q$ is inserted between the two plates, parallel to them and located a distance $d$ from the upper plate, as in Figure $\mathrm{P} 26.57$
(a) What induced charge appears on each of the two original plates? (b) What potential difference appears between the middle plate and each of the other plates? Each plate has area $A$.
CAN'T COPY THE FIGURE

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 58

A 2.00 -nF parallel-plate capacitor is charged to an initial potential difference $\Delta V_{i}=100 \mathrm{V}$ and then isolated. The dielectric material between the plates is mica, with a dielectric constant of $5.00 .$ (a) How much work is required to withdraw the mica sheet? (b) What is the potential difference of the capacitor after the mica is withdrawn?

Mayukh Banik

Numerade Educator

Problem 59

A parallel-plate capacitor is constructed using a dielectric material whose dielectric constant is 3.00 and whose dielectric strength is $2.00 \times 10^{8} \mathrm{V} / \mathrm{m} .$ The desired capacitance is $0.250 \mu \mathrm{F},$ and the capacitor must withstand a maximum potential difference of $4000 \mathrm{V}$. Find the minimum area of the capacitor plates.

Manish Kumar

Numerade Educator

Problem 60

A $10.0-\mu \mathrm{F}$ capacitor has plates with vacuum between them. Each plate carries a charge of magnitude $1000 \mu \mathrm{C}$. A particle with charge $-3.00 \mu \mathrm{C}$ and mass $2.00 \times 10^{-16} \mathrm{kg}$ is fired from the positive plate toward the negative plate with an initial speed of $2.00 \times 10^{6} \mathrm{m} / \mathrm{s} .$ Does it reach the negative plate? If so, find its impact speed. If not, what fraction of the way across the capacitor does it travel?

Mayukh Banik

Numerade Educator

Problem 61

A parallel-plate capacitor is constructed by filling the space between two square plates with blocks of three dielectric materials, as in Figure $\mathrm{P} 26.61 .$ You may assume that $\ell \gg d$. (a) Find an expression for the capacitance of the device in terms of the plate area $A$ and $d, \kappa_{1}, \kappa_{2},$ and \kappa $_{3}$. (b) Calculate the capacitance using the values $A=1.00 \mathrm{cm}^{2}, \quad d=2.00 \mathrm{mm}, \quad \kappa_{1}=4.90, \quad \kappa_{2}=5.60,$ and $\kappa_{3}=2.10$.
CAN'T COPY THE FIGURE

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 62

A 10.0 - $\mu$ F capacitor is charged to $15.0 \mathrm{V}$. It is next connected in series with an uncharged $5.00-\mu \mathrm{F}$ capacitor. The series combination is finally connected across a $50.0-\mathrm{V}$ battery, as diagrammed in Figure $\mathrm{P} 26.62 .$ Find the new potential differences across the 5 - $\mu \mathrm{F}$ and $10-\mu \mathrm{F}$ capacitors.
CAN'T COPY THE FIGURE

Mayukh Banik

Numerade Educator

Problem 63

(a) Two spheres have radii $a$ and $b$ and their centers are a distance $d$ apart. Show that the capacitance of this system is
$$C=\frac{4 \pi \epsilon_{0}}{\frac{1}{a}+\frac{1}{b}-\frac{2}{d}}$$
provided that $d$ is large compared with $a$ and $b$. (Suggestion:
Because the spheres are far apart, assume that the potential of each equals the sum of the potentials due to each sphere, and when calculating those potentials assume that $V=k_{e} Q / r$ applies.) $(\mathrm{b})$ Show that as $d$ approaches infinity the above result reduces to that of two spherical capacitors in series.

Mayukh Banik

Numerade Educator

Problem 64

A capacitor is constructed from two square plates of sides $\ell$ and separation $d .$ A material of diclectric constant $\kappa$ is inserted a distance $x$ into the capacitor, as shown in Figure $\mathrm{P} 26.64 .$ Assume that $d$ is much smaller than $x$
(a) Find the equivalent capacitance of the device. (b) Calculate the energy stored in the capacitor, letting $\Delta V$ represent the potential difference. (c) Find the direction and magnitude of the force exerted on the dielectric, assuming a constant potential difference $\Delta V$. Ignore friction.
(d) Obtain a numerical value for the force assuming that $\ell=5.00 \mathrm{cm}, \Delta V=2000 \mathrm{V}, d=2.00 \mathrm{mm},$ and the dielec-
tric is glass $(\kappa=4.50) .$ (Suggestion: The system can be considered as two capacitors connected in parallel.)
CAN'T COPY THE FIGURE

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 65

A capacitor is constructed from two square plates of sides $\ell$ and separation $d$, as suggested in Figure $\mathrm{P} 26.64 .$ You may assume that $d$ is much less than $\ell$. The plates carry charges $+Q_{0}$ and $-Q_{0} .$ A block of metal has a width $\ell$, a length $\ell$ and a thickness slightly less than $d$. It is inserted a distance $x$ into the capacitor. The charges on the plates are not disturbed as the block slides in. In a static situation, a metal prevents an electric field from penetrating inside it. The metal can be thought of as a perfect dielectric, with $\kappa \rightarrow \infty,$ (a) Calculate the stored energy as a function of $x$
(b) Find the direction and magnitude of the force that acts on the metallic block. (c) The area of the advancing front face of the block is essentially equal to $\ell d$. Considering the force on the block as acting on this face, find the stress (force per area) on it. (d) For comparison, express the energy density in the electric field between the capacitor plates in terms of $Q_{0}, \ell, d,$ and $\epsilon_{0}$.

Mayukh Banik

Numerade Educator

Problem 66

When considering the energy supply for an automobile, the energy per unit mass of the energy source is an important parameter. Using the following data, compare the energy per unit mass ( $J /$ kg) for gasoline, lead-acid batteries, and capacitors. (The ampere A will be introduced in the next chapter as the SI unit of electric current. $1 \mathrm{A}=1 \mathrm{C} / \mathrm{s} .)$
Gasoline: 126000 Btu/gal; density $=670 \mathrm{kg} / \mathrm{m}^{3}$ Lead-acid battery: $12.0 \mathrm{V} ; 100 \mathrm{A} \cdot \mathrm{h} ;$ mass $=16.0 \mathrm{kg}$
Capacitor: potential difference at full charge $=12.0 \mathrm{V}$ capacitance $=0.100 \mathrm{F} ;$ mass $=0.100 \mathrm{kg}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 67

An isolated capacitor of unknown capacitance has been charged to a potential difference of $100 \mathrm{V}$. When the charged capacitor is then connected in parallel to an uncharged $10.0-\mu \mathrm{F}$ capacitor, the potential difference across the combination is $30.0 \mathrm{V}$. Calculate the unknown capacitance.

Mayukh Banik

Numerade Educator

Problem 68

To repair a power supply for a stereo amplifier, an electronics technician needs a $100-\mu \mathrm{F}$ capacitor capable of withstanding a potential difference of $90 \mathrm{V}$ between the plates. The only available supply is a box of five $100-\mu \mathrm{F}$ capacitors, each having a maximum voltage capability of $50 \mathrm{V} .$ Can the technician substitute a combination of these capacitors that has the proper electrical characteristics? If so, what will be the maximum voltage across any of the capacitors used? (Suggestion: The technician may not have to use all the capacitors in the box.)

Mayukh Banik

Numerade Educator

Problem 69

A parallel-plate capacitor of plate separation $d$ is charged to a potential difference $\Delta V_{0}$. A dielectric slab of thickness $d$ and dielectric constant $\kappa$ is introduced between the plates while the battery remains connected to the plates. (a) Show that the ratio of energy stored after the diclectric is introduced to the energy stored in the empty capacitor is $U / U_{0}=\kappa .$ Give a physical explanation for this increase in stored energy. (b) What happens to the charge on the capacitor? (Note that this situation is not the same as in Example 26.7 , in which the battery was removed from the circuit before the dielectric was introduced.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 70

A vertical parallel-plate capacitor is half filled with a dielectric for which the dielectric constant is 2.00 (Fig. $\mathrm{P} 26.70 \mathrm{a}$ ). When this capacitor is positioned horizontally, what fraction of it should be filled with the same dielectric (Fig. P26.70b) in order for the two capacitors to have equal capacitance?
CAN'T COPY THE FIGURE

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 71

Capacitors $C_{1}=6.00 \mu \mathrm{F}$ and $C_{2}=2.00 \mu \mathrm{F}$ are charged as a parallel combination across a $250-\mathrm{V}$ battery. The capacitors are disconnected from the battery and from each other. They are then connected positive plate to negative plate and negative plate to positive plate. Calculate the resulting charge on each capacitor.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 72

Calculate the equivalent capacitance between the points $a$ and $b$ in Figure $P 26.72 .$ Note that this is not a simple series or parallel combination. (Suggestion: Assume a potential difference $\Delta V$ between points $a$ and $b .$ Write expressions for $\Delta V_{a b}$ in terms of the charges and capacitances for the various possible pathways from $a$ to $b,$ and require conservation of charge for those capacitor plates that are connected to each other.)
CAN'T COPY THE FIGURE

Mayukh Banik

Numerade Educator

Problem 73

The inner conductor of a coaxial cable has a radius of $0.800 \mathrm{mm},$ and the outer conductor's inside radius is $3.00 \mathrm{mm} .$ The space between the conductors is filled with polyethylene, which has a diclectric constant of 2.30 and a dielectric strength of $18.0 \times 10^{6} \mathrm{V} / \mathrm{m} .$ What is the maximum potential difference that this cable can withstand?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 74

You are optimizing coaxial cable design for a major manufacturer. Show that for a given outer conductor radius $b$ maximum potential difference capability is attained when the radius of the inner conductor is $a=b / e$ where $e$ is the base of natural logarithms.

Mayukh Banik

Numerade Educator

Problem 75

Determine the equivalent capacitance of the combination shown in Figure $\mathrm{P} 26.75 .$ (Suggestion: Consider the symmetry involved.)
CAN'T COPY THE FIGURE

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 76

- Consider two long, parallel, and oppositely charged wires of radius $d$ with their centers separated by a distance $D$ Assuming the charge is distributed uniformly on the surface of each wire, show that the capacitance per unit length of this pair of wires is
$$\frac{C}{\ell}=\frac{\pi \epsilon_{0}}{\ln [(D-d) / d]}$$

Mayukh Banik

Numerade Educator

Problem 77

Example 26.2 explored a cylindrical capacitor of length $\ell$ and radii $a$ and $b$ of the two conductors. In the What If? section, it was claimed that increasing $\ell$ by $10 \%$ is more effective in terms of increasing the capacitance than increasing $a$ by $10 \%$ if $b>2.85 a$. Verify this claim mathematically.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator