Principles of Physics a Calculus Based Text

Raymond A. Serway, John W. Jewett, Jr.

Chapter 20

Electric Potential and Capacitance - all with Video Answers

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Chapter Questions

Problem 1

A uniform electric field of magnitude $325 \mathrm{V} / \mathrm{m}$ is directed in the negative $y$ direction in Figure P20.1. The coordinates of point ? $\operatorname{are}(-0.200,-0.300) \mathrm{m}$ and those of point ? $\operatorname{are}(0.400$ 0.500) $\mathrm{m} .$ Calculate the electric potential difference $V_{?}-V_{?}$ using the dashed-line path.

Manish Kumar

Manish Kumar

Numerade Educator

Problem 2

How much work is done (by a battery, generator, or some other source of potential difference in moving Avogadro's number of electrons from an initial point where the electric potential is $9.00 \mathrm{V}$ to a point where the electric potential is $-5.00 \mathrm{V} ?($ The potential in each case is measured relative to a common reference point.)

Manish Kumar

Numerade Educator

Problem 3

Calculate the speed of a proton that is accelerated from rest through an electric potential difference of $120 \mathrm{V}$. Calculate the speed of an electron that is accelerated through the same electric potential difference.

Manish Kumar

Numerade Educator

Problem 4

A uniform electric field of magnitude $250 \mathrm{V} / \mathrm{m}$ is directed in the positive $x$ direction. $A+12.0-\mu C$ charge moves from the origin to the point $(x, y)=(20.0 \mathrm{cm}, 50.0 \mathrm{cm}) .$ (a) What is the change in the potential energy of the charge-field system? (b) Through what potential difference does the charge move?

Manish Kumar

Numerade Educator

Problem 5

An electron moving parallel to the $x$ axis has an initial speed of $3.70 \times 10^{6} \mathrm{m} / \mathrm{s}$ at the origin. Its speed is reduced to $1.40 \times 10^{5} \mathrm{m} / \mathrm{s}$ at the point $x=2.00 \mathrm{cm} .$ (a) Calculate the electric potential difference between the origin and that point. (b) Which point is at the higher potential?

Mayukh Banik

Numerade Educator

Problem 6

A block having mass $m$ and charge $+Q$ is connected to an insulating spring having a force constant $k .$ The block lies on a frictionless, insulating, horizontal track, and the system is immersed in a uniform electric field of magnitude $E$ directed as shown in Figure $\mathrm{P} 20.6 .$ The block is released from rest when the spring is unstretched (at $x=0$ ). We wish to show that the ensuing motion of the block is simple harmonic. (a) Consider the system of the block, the spring, and the electric field. Is this system isolated or nonisolated? (b) What kinds of potential energy exist within this system? (c) Call the initial configuration of the system that existing just as the block is released from rest. The final configuration is when the block momentarily comes to rest again. What is the value of $x$ when the block comes to rest momentarily? (d) At some value of $x$ we will call $x=x_{0},$ the block has zero net force on it. What analysis model describes the particle in this situation? (e) What is the value of $x_{0} ?$ (f) Define a new coordinate system $x^{\prime}$ such that $x^{\prime}=x-x_{0} .$ Show that $x^{\prime}$ satisfies a differential equation for simple harmonic motion. (g) Find the period of the simple harmonic motion. (h) How does the period depend on the electric field magnitude?

Dominador Tan

Numerade Educator

Problem 7

(a) Find the potential at a distance of $1.00 \mathrm{cm}$ from a proton. (b) What is the potential difference between two points that are $1.00 \mathrm{cm}$ and $2.00 \mathrm{cm}$ from a proton? (c) Repeat parts (a) and (b) for an electron.

Manish Kumar

Numerade Educator

Problem 8

Show that the amount of work required to assemble four identical charged particles of magnitude $Q$ at the corners of a square of side $s$ is $5.41 k_{e} Q^{2} / s$.

Mayukh Banik

Numerade Educator

Problem 9

Given two particles with $2.00-\mu \mathrm{C}$ charges as shown in Figure $P 20.9$ and a particle with charge $q=1.28 \times 10^{-18}$ Cat the origin, (a) what is the net force exerted by the two $2.00-\mu \mathrm{C}$ charges on the test charge $q$ ? (b) What is the electric field at the origin due to the two $2.00-\mu \mathrm{C}$ particles? (c) What is the electric potential at the origin due to the two $2.00-\mu \mathrm{C}$ particles?

Manish Kumar

Numerade Educator

Problem 10

Three particles with equal positive charges $q$ are at the corners of an equilateral triangle of side $a$ as shown in Figure $\mathrm{P} 20.10 .$ (a) At what point, if any, in the plane of the particles is the electric potential zero?
(b) What is the electric potential at the position of one of the particles due to the other two particles in the triangle?

Mayukh Banik

Numerade Educator

Problem 11

The three charged particles in Figure $P 20.11$ are at the vertices of an isosceles triangle (where $d=2.00 \mathrm{cm})$ Taking $q=7.00 \mu \mathrm{C},$ calculate the electric potential at point $A,$ the midpoint of the base.

Mayukh Banik

Numerade Educator

Problem 12

In $1911,$ Ernest Rutherford and his assistants Geiger and Marsden conducted an experiment in which they scattered alpha particles (nuclei of helium atoms from thin sheets of gold. An alpha particle, having charge $+2 e$ and mass $6.64 \times 10^{-27} \mathrm{kg},$ is a product of certain radioactive decays. The results of the experiment led Rutherford to the idea that most of an atom's mass is in a very small nucleus, with electrons in orbit around it. (This is the planetary model of the atom, which we'll study in Chapter $29 .$ ) Assume an alpha particle, initially very far from a stationary gold nucleus, is fired with a velocity of $2.00 \times 10^{7} \mathrm{m} / \mathrm{s}$ directly toward the nucleus (charge $+79 e$ ). What is the smallest distance between the alpha particle and the nucleus before the alpha particle reverses direction? Assume the gold nucleus remains stationary.

John Boyer

Numerade Educator

Problem 13

Four identical charged particles $(q=+10.0 \mu \mathrm{C})$ are located on the corners of a rectangle as shown in Figure $\mathrm{P} 20.13 .$ The dimensions of the rectangle are $L=60.0 \mathrm{cm}$ and $W=15.0 \mathrm{cm}$ Calculate the change in electric potential energy of the system as the particle at the lower left corner in Figure $\mathrm{P} 20.13$ is brought to this position from infinitely far away. Assume the other three particles in Figure $\mathrm{P} 20.13$ remain fixed in position.

Mayukh Banik

Numerade Educator

Problem 14

A light, unstressed spring has length $d$. Two identical particles, each with charge $q,$ are connected to the opposite ends of the spring. The particles are held stationary a distance $d$ apart and then released at the same moment. The system then oscillates on a frictionless, horizontal table. The spring has a bit of internal kinetic friction, so the oscillation is damped. The particles eventually stop vibrating when the distance between them is $3 d$. Assume the system of the spring and two charged particles is isolated. Find the increase in internal energy that appears in the spring during the oscillations.

Mayukh Banik

Numerade Educator

Problem 15

Two insulating spheres have radii $0.300 \mathrm{cm}$ and $0.500 \mathrm{cm},$ masses $0.100 \mathrm{kg}$ and $0.700 \mathrm{kg},$ and uniformly distributed charges $-2.00 \mu \mathrm{C}$ and $3.00 \mu \mathrm{C}$. They are released from rest when their centers are separated by $1.00 \mathrm{m}$
(b) What If? If the spheres were conductors, would the speeds be greater or less than those calculated in part (a)? Explain.

Manish Kumar

Numerade Educator

Problem 16

Two insulating spheres have radii $r_{1}$ and $r_{2}$ masses $m_{1}$ and $m_{2},$ and uniformly distributed charges $-q_{1}$ and $q_{2} .$ They are released from rest when their centers are separated by a distance $d$. (a) How fast is each moving when they collide? (b) What If? If the spheres were conductors, would their speeds be greater or less than those calculated in part (a)? Explain.

Mayukh Banik

Numerade Educator

Problem 17

Two particles each with charge $+2.00 \mu \mathrm{C}$ are located on the $x$ axis. One is at $x=1.00 \mathrm{m},$ and the other is at $x=-1.00 \mathrm{m}$
(a) Determine the electric potential on the $y$ axis at $y=0.500 \mathrm{m} .$ (b) Calculate the change in electric potential energy of the system as a third charged particle of $-3.00 \mu \mathrm{C}$ is brought from infinitely far away to a position on the $y$ axis at $y=0.500 \mathrm{m}$.

Mayukh Banik

Numerade Educator

Problem 18

Two charged particles create influences at the origin, described by the expressions $$8.99 \times 10^{9} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}\left[-\frac{7.00 \times 10^{-9} \mathrm{C}}{(0.0700 \mathrm{m})^{2}} \cos 70.0^{\circ} \hat{\mathrm{i}}\right.$$ $$\left.-\frac{7.00 \times 10^{-9} \mathrm{C}}{(0.0700 \mathrm{m})^{2}} \sin 70.0^{\circ} \hat{\mathrm{j}}+\frac{8.00 \times 10^{-9} \mathrm{C}}{(0.0300 \mathrm{m})^{2}} \hat{\mathbf{j}}\right]$$ and $$8.99 \times 10^{9} \mathrm{N} \cdot \mathrm{m}^{2} / \mathrm{C}^{2}\left[\frac{7.00 \times 10^{-9} \mathrm{C}}{0.0700 \mathrm{m}}-\frac{8.00 \times 10^{-9} \mathrm{C}}{0.0300 \mathrm{m}}\right]$$ (a) Identify the locations of the particles and the charges on them. (b) Find the force on a -16.0 nC charge placed at the origin and (c) the work required to move this third charge to the origin from a very distant point.

Dominador Tan

Numerade Educator

Problem 19

Two particles, with charges of $20.0 \mathrm{nC}$ and $-20.0 \mathrm{nC}$
are placed at the points with coordinates $(0,4.00 \mathrm{cm})$ and $(0,-4.00 \mathrm{cm})$ as shown in Figure P20.19. A particle with charge $10.0 \mathrm{nC}$ is located at the origin. (a) Find the electric potential energy of the configuration of the three fixed charges.
(b) A fourth particle, with a mass of $2.00 \times 10^{-13} \mathrm{kg}$
and a charge of $40.0 \mathrm{nC}$ is released from rest at the point $(3.00 \mathrm{cm}, 0) .$ Find its speed after it has moved freely to a very large distance away.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 20

At a certain distance from a charged particle, the magnitude of the electric field is $500 \mathrm{V} / \mathrm{m}$ and the electric potential is $-3.00 \mathrm{kV} .$ (a) What is the distance to the particle?
(b) What is the magnitude of the charge?

Mayukh Banik

Numerade Educator

Problem 21

A particle with charge $+q$ is at the origin. A particle with charge $-2 q$ is at $x=2.00 \mathrm{m}$ on the $x$ axis. (a) For what finite value(s) of $x$ is the electric field zero? (b) For what finite value(s) of $x$ is the electric potential zero?

Rashmi Sinha

Numerade Educator

Problem 22

The electric potential inside a charged spherical conductor of radius $R$ is given by $V=k_{e} Q / R,$ and the potential outside is given by $V=k_{e} Q / r$. Using $E_{r}=-d V / d r$, derive the electric field (a) inside and (b) outside this charge distribution.

Mayukh Banik

Numerade Educator

Problem 23

Over a certain region of space, the electric potential is $V=5 x-3 x^{2} y+2 y z^{2} .$ (a) Find the expressions for the $x, y$ and $z$ components of the electric field over this region.
(b) What is the magnitude of the field at the point $P$ that has coordinates (1.00,0,-2.00) $\mathrm{m} ?$

Mayukh Banik

Numerade Educator

Problem 24

The potential in a region between $x=0$ and $x=6.00 \mathrm{m}$ is $V=a+b x,$ where $a=10.0 \mathrm{V}$ and $b=-7.00 \mathrm{V} / \mathrm{m} .$ Determine (a) the potential at $x=0,3.00 \mathrm{m},$ and $6.00 \mathrm{m}$ and (b) the magnitude and direction of the electric field at $x=0,3.00 \mathrm{m},$ and $6.00 \mathrm{m}$.

Mayukh Banik

Numerade Educator

Problem 25

Consider a ring of radius $R$ with the total charge $Q$ spread uniformly over its perimeter. What is the potential difference between the point at the center of the ring and a point on its axis a distance $2 R$ from the center?

Mayukh Banik

Numerade Educator

Problem 26

A rod of length $L$ (Fig. P20.26) lies along the $x$ axis with its left end at the origin. It has a nonuniform charge density $\lambda=\alpha x,$ where $\alpha$ is a positive constant. (a) What are the units of $\alpha$ ? (b) Calculate the electric potential at $A$

Sheh Lit Chang

University of Washington

Problem 27

For the arrangement described in Problem $26,$ calculate the electric potential at point $B,$ which lies on the perpendicular bisector of the rod a distance $b$ above the $x$ axis.

Dominador Tan

Numerade Educator

Problem 28

A wire having a uniform linear charge density $\lambda$ is bent into the shape shown in Figure $\mathrm{P} 20.28$. Find the electric potential at point $O$.

Sheh Lit Chang

University of Washington

Problem 29

A uniformly charged insulating rod of length $14.0 \mathrm{cm}$ is bent into the shape of a semicircle as shown in Figure $\mathrm{P} 20.29 .$ The rod has a total charge of $-7.50 \mu \mathrm{C}$. Find the electric potential at $O,$ the center of the semicircle.

Mayukh Banik

Numerade Educator

Problem 30

How many electrons should be removed from an initially uncharged spherical conductor of radius $0.300 \mathrm{m}$ to produce a potential of $7.50 \mathrm{kV}$ at the surface?

Mayukh Banik

Numerade Educator

Problem 31

Electric charge can accumulate on an airplane in flight. You may have observed needle-shaped metal extensions on the wing tips and tail of an airplane. Their purpose is to allow charge to leak off before much of it accumulates. The electric field around the needle is much larger than the field around the body of the airplane and can become large enough to produce dielectric breakdown of the air, discharging the airplane. To model this process, assume two charged spherical conductors are connected by a long conducting wire and a $1.20-\mu \mathrm{C}$ charge is placed on the combination. One sphere, representing the body of the airplane, has a radius of $6.00 \mathrm{cm} ;$ the other, representing the tip of the needle, has a radius of $2.00 \mathrm{cm} .$ (a) What is the electric potential of each sphere? (b) What is the electric field at the surface of each sphere?

Mayukh Banik

Numerade Educator

Problem 32

A spherical conductor has a radius of $14.0 \mathrm{cm}$ and charge of $26.0 \mu \mathrm{C}$. Calculate the electric field and the electric potential (a) $r=10.0 \mathrm{cm},$ (b) $r=20.0 \mathrm{cm},$ and (c) $r=14.0 \mathrm{cm}$ from the center.

Penny Riley

Numerade Educator

Problem 33

(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?

Manish Kumar

Numerade Educator

Problem 34

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 35

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 36

A spherical capacitor consists of a spherical conducting shell of radius $b$ and charge $-Q$ that is concentric with a smaller conducting sphere of radius $a$ and charge $+Q \text { (Fig. } P 20.36)$
(a) Show that its capacitance is $$C=\frac{a b}{k_{e}(b-a)}$$ (b) Show that as $b$ approaches infinity, the capacitance approaches the value $a / k_{e}=4 \pi \epsilon_{0} a$.

Dominador Tan

Numerade Educator

Problem 37

An air-filled capacitor consists of two parallel plates, each with an area of $7.60 \mathrm{cm}^{2}$, separated by a distance of $1.80 \mathrm{mm} .$ 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 38

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 $\mathrm{P} 20.38,$ a second identical set of plates is enmeshed with the first set. Each plate in the second set is halfway between two plates 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.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 39

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 $-8.10 \mu \mathrm{C}$. Assume the region between the conductors is air. (a) What is the capacitance of this cable? (b) What is the potential difference between the two conductors?

Mayukh Banik

Numerade Educator

Problem 40

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 41

(a) Regarding the Earth and a cloud layer $800 \mathrm{m}$ above the Earth as the "plates" of a capacitor, calculate the capacitance of the Earth-cloud layer system. Assume the cloud layer has an area of $1.00 \mathrm{km}^{2}$ and 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. (b) What is the maximum charge the cloud can hold?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 42

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 -V battery. Find (a) the equivalent capacitance of the combination, (b) the potential difference across each capacitor, and (c) the charge stored on each capacitor.

Ben Nicholson

Numerade Educator

Problem 43

The two capacitors of Problem $42\left(C_{1}=5.00 \mu \mathrm{F}\right.$ and $C_{2}=12.0 \mu \mathrm{F}$ ) 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.

Manish Kumar

Numerade Educator

Problem 44

(a) Find the equivalent capacitance between points $a$ and $b$ for the group of capacitors connected as shown in Figure $\mathrm{P} 20.44 .$ Take $C_{1}=5.00 \mu \mathrm{F}$ $C_{2}=10.0 \quad \mu \mathrm{F},$ and $C_{3}=2.00 \mu \mathrm{F}$
(b) What charge is stored on $C_{\mathrm{g}}$ if the potential difference between points $a$ and $b$ is $60.0 \mathrm{V} ?$

Vishal Gupta

Numerade Educator

Problem 45

Four capacitors are connected as shown in Figure $\mathrm{P} 20.45$
(a) Find the equivalent capacitance between points $a$ and $b$
(b) Calculate the charge on each capacitor, taking $\Delta V_{a b}=15.0 \mathrm{V}$.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 46

Why is the following situation impossible? A technician is testing a circuit that contains a capacitance $C .$ He realizes that a better design for the circuit would include a capacitance $\frac{7}{3} C$ rather than $C .$ He has three additional capacitors, each with capacitance $C$. By combining these additional capacitors in a certain combination that is then placed in parallel with the original capacitor, he achieves the desired capacitance.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 47

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 $B .$ 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. (a) Should it be in series or in parallel with the $34.8-\mu \mathrm{F}$ capacitor?
(b) What should be its capacitance? (c) What If? The next circuit comes down the assembly line with capacitance $29.8 \mu \mathrm{F}$ between $A$ and $B .$ To meet the specification, what additional capacitor should be installed in series or in parallel in that circuit?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 48

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

Ben Nicholson

Numerade Educator

Problem 49

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 50

Three capacitors are connected to a battery as shown in Figure P20.50. 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? Assume $C_{3}$ is increased. Explain what happens to the charge stored by each capacitor.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 51

Find the equivalent capacitance between points $a$ and $b$ in the combination of capacitors shown in Figure $\mathrm{P} 20.51$

Vishal Gupta

Numerade Educator

Problem 52

Consider the circuit shown in Figure $\mathrm{P} 20.52$ where $C_{1}=6.00 \mu \mathrm{F}, C_{2}=$
$3.00 \mu \mathrm{F},$ and $\Delta V=20.0 \mathrm{V}$
Capacitor $C_{1}$ is first charged by closing switch $\mathrm{S}_{1} .$ Switch $\mathrm{S}_{1}$ is then opened, and the charged capacitor is connected to the uncharged capacitor by closing $\mathrm{S}_{2}$. Calculate (a) the initial charge acquired by $C_{1}$ and (b) the final charge on each capacitor.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 53

As a person moves about in a dry environment, electric charge accumulates on the person's body. Once it is at high voltage, either positive or negative, the body can discharge via sparks and shocks. Consider a human body isolated from ground, with the typical capacitance 150 pF. (a) What charge on the body will produce a potential of $10.0 \mathrm{kVP}$
(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 situation correspond?

Dominador Tan

Numerade Educator

Problem 54

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 the force is $F=Q E=Q^{2} / A \epsilon_{0}$ This conclusion 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 $F=Q^{2} / 2 A \epsilon_{0}$ Suggestion: Let $C=\epsilon_{0} A / x$ for an arbitrary plate separation $x$ and note that the work done in separating the two charged plates is $W=\int F d x$.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 55

Two identical parallel-plate capacitors, each with capacitance $10.0 \mu \mathrm{F}$, are charged to potential difference $50.0 \mathrm{V}$ and then disconnected from the battery. They are then connected to each other in parallel with plates of like sign connected. Finally, 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.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 56

Two identical parallel-plate capacitors, each with capacitance $C$, are charged to potential difference $\Delta V$ and then disconnected from the battery. They are then connected to each other in parallel with plates of like sign connected. Finally, 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.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 57

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 -V power supply.
(a) Draw a circuit diagram and (b) calculate the total energy stored in the two capacitors. (c) What If? What potential difference would be required across the same two capacitors connected in series for the combination to store the same amount of energy as in part (b)? (d) Draw a circuit diagram of the circuit described in part (c).

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 58

Consider two conducting spheres with radii $R_{1}$ and $R_{2}$ separated by a distance much greater than either radius. A total charge $Q$ is shared between the spheres. We wish to show that when the electric potential energy of the system has a minimum value, the potential difference between the spheres is zero. 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 the charge of each is uniformly distributed over its surface. (a) Show that the energy associated with a single conducting sphere of radius $R$ and charge $q$ surrounded by a vacuum is $U=k_{e} q^{2} / 2 R$
(b) Find the total energy of the system of two spheres in terms of $q_{1},$ the total charge $Q,$ and the radii $R_{1}$ and $R_{2}$
(c) To minimize the energy, differentiate the result to part
(b) with respect to $q_{1}$ and set the derivative equal to zero. Solve for $q_{1}$ in terms of $Q$ and the radii. (d) From the result to part $(\mathrm{c}),$ find the charge $q_{2} .$ (e) Find the potential of each sphere. (f) What is the potential difference between the spheres?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 59

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

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 60

The immediate cause of many deaths is ventricular fibrillation, which is an uncoordinated quivering of the heart. An electric shock to the chest can cause momentary paralysis of the heart muscle, after which the heart sometimes resumes its proper beating. One type of defibrillator (Fig. $\mathrm{P} 20.60$ ) applies a strong electric shock to the chest over a time interval of a few milliseconds.
This device contains a capacitor of several microfarads, charged to several thousand volts. Electrodes called paddles are held against the chest on both sides of the heart, and the capacitor is discharged through the patient's chest. Assume 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?

Dominador Tan

Numerade Educator

Problem 61

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 62

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

Supratim Pal

Numerade Educator

Problem 63

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 64

A commercial capacitor is to be constructed as shown in Figure $\mathrm{P} 20.64$ This particular capacitor is made from two strips of aluminum foil 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 dielectric 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 would effectively double its capacitance by allowing charge storage on both sides of each strip of foil.)

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 65

(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 66

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. Assume the liquid is an insulator. 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.

Mayukh Banik

Numerade Educator

Problem 67

Lightning can be studied with a Van de Graaff generator, which consists of a spherical dome on which charge is continuously deposited by a moving belt. Charge can be added until the electric field at the surface of the dome becomes equal to the dielectric strength of air. Any more charge leaks off in sparks as shown in Figure $\mathrm{P} 20.67$. Assume the dome has a diameter of $30.0 \mathrm{cm}$ and is surrounded by dry air with a "breakdown" electric field of $3.00 \times 10^{6} \mathrm{V} / \mathrm{m}$. (a) What is the maximum potential of the dome? (b) What is the maximum charge on the dome?

Mayukh Banik

Numerade Educator

Problem 68

Review. A storm cloud and the ground represent the plates of a capacitor. During a storm, the capacitor has a potential difference of $1.00 \times 10^{8} \mathrm{V}$ between its plates and a charge of 50.0 C. A lightning strike delivers $1.00 \%$ of the energy of the capacitor to a tree on the ground. 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 $4186 \mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C}$, a boiling point of $100^{\circ} \mathrm{C}$, and a latent heat of vaporization of $2.26 \times 10^{6} \mathrm{J} / \mathrm{kg}$.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 69

Review. From a large distance away, a particle of mass $2.00 \mathrm{g}$ and charge $15.0 \mu \mathrm{C}$ is fired at $21.0 \hat{\mathrm{i}} \mathrm{m} / \mathrm{s}$ straight toward a second particle, originally stationary but free to move, with mass $5.00 \mathrm{g}$ and charge $8.50 \mu \mathrm{C}$. Both particles are constrained to move only along the $x$ axis. (a) At the instant of closest approach, both particles will be moving at the same velocity. Find this velocity. (b) Find the distance of closest approach. After the interaction, the particles will move far apart again. At this time, find the velocity of (c) the $2.00-\mathrm{g}$ particle and (d) the 5.00 -g particle.

Dominador Tan

Numerade Educator

Problem 70

From a large distance away, a particle of mass $m_{1}$ and positive charge $q_{1}$ is fired at speed $v$ in the positive $x \mathrm{di}-$ rection straight toward a second particle, originally stationary but free to move, with mass $m_{2}$ and positive charge $q_{2} .$ Both particles are constrained to move only along the $x$ axis. (a) At the instant of closest approach, both particles will be moving at the same velocity. Find this velocity. (b) Find the distance of closest approach. After the interaction, the particles will move far apart again. At this time, find the velocity of (c) the particle of mass $m_{1}$ and (d) the particle of mass $m_{2}$.

Dominador Tan

Numerade Educator

Problem 71

A model of a red blood cell portrays the cell as a capacitor with two spherical plates. It is a positively charged conducting liquid sphere of area $A$, separated by an insulating membrane of thickness $t$ from the surrounding negatively charged conducting fluid. Tiny electrodes introduced into the cell show a potential difference of $100 \mathrm{mV}$ across the membrane. Take the membrane's thickness as $100 \mathrm{nm}$ and its dielectric constant as $5.00 .$ (a) Assume that a typical red blood cell has a mass of $1.00 \times 10^{-12} \mathrm{kg}$ and density $1100 \mathrm{kg} / \mathrm{m}^{3} .$ Calculate its volume and its surface area.
(b) Find the capacitance of the cell. (c) Calculate the charge on the surfaces of the membrane. How many electronic charges does this charge represent? (Suggestion: The chapter text models the Earth's atmosphere as a capacitor with two spherical plates.)

Dominador Tan

Numerade Educator

Problem 72

Why is the following situation impossible? In the Bohr model of the hydrogen atom, an electron moves in a circular orbit about a proton. The model states that the electron can exist only in certain allowed orbits around the proton: those whose radius $r$ satisfies $r=n^{2}(0.0529 \mathrm{nm}),$ where $n=1,2$ $3, \ldots .$ For one of the possible allowed states of the atom, the electric potential energy of the system is $-13.6 \mathrm{eV}$.

Khaled Yasein

Numerade Educator

Problem 73

The liquid-drop model of the atomic nucleus suggests high-energy oscillations of certain nuclei can split the nucleus into two unequal fragments plus a few neutrons. The fission products acquire kinetic energy from their mutual Coulomb repulsion. Assume the charge is distributed uniformly throughout the volume of each spherical fragment and, immediately before separating, each fragment is at rest and their surfaces are in contact. The electrons surrounding the nucleus can be ignored. Calculate the electric potential energy (in electron volts) of two spherical fragments from a uranium nucleus having the following charges and radii: $38 e$ and $5.50 \times 10^{-15} \mathrm{m},$ and $54 e$ and $6.20 \times 10^{-15} \mathrm{m}$

Dominador Tan

Numerade Educator

Problem 74

A Geiger-Mueller tube is a radiation detector that consists of a closed, hollow, metal cylinder (the cathode) of inner radius $r_{a}$ and a coaxial cylindrical wire (the anode) of radius $r_{b}$ (Fig. $\mathrm{P} 20.74 \mathrm{a}$ ). The charge per unit length on the anode is $\lambda,$ and the charge per unit length on the cathode is $-\lambda .$ A gas fills the space between the electrodes. When the tube is in use (Fig. $\mathrm{P} 20.74 \mathrm{b}$ ) and a high-energy elementary particle passes through this space, it can ionize an atom of the gas. The strong electric field makes the resulting ion and electron accelerate in opposite directions. They strike other molecules of the gas to ionize them, producing an avalanche of electrical discharge. The pulse of electric current between the wire and the cylinder is counted by an external circuit. (a) Show that the magnitude of the electric potential difference between the wire and the cylinder is $$\Delta V=2 k_{e} \lambda \ln \left(\frac{r_{a}}{r_{b}}\right)$$ (b) Show that the magnitude of the electric field in the space between cathode and anode is
$$E=\frac{\Delta V}{\ln \left(r_{a} / r_{b}\right)}\left(\frac{1}{r}\right)$$ where $r$ is the distance from the axis of the anode to the
point where the field is to be calculated.

Dominador Tan

Numerade Educator

Problem 75

Assume that the internal diameter of the Geiger-Mueller tube described in Problem 20.74 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 $1.20 \times 10^{6} \mathrm{V} / \mathrm{m}$. Use the result of Problem 20.74 to calculate the maximum potential difference that can be applied between the wire and the cylinder before breakdown occurs in the gas.

Dominador Tan

Numerade Educator

Problem 76

Four balls, each with mass $m$,are connected by four nonconducting strings to form a square with side $a$ as shown in Figure $\mathrm{P} 20.76 .$ The assembly is placed on a nonconducting, frictionless, horizontal surface. Balls 1 and 2 each have charge $q,$ and balls 3 and 4 are uncharged. After the string connecting balls 1 and 2 is cut, what is the maximum speed of balls 3 and 4 ?

Dominador Tan

Numerade Educator

Problem 77

Calculate the work that must be done on charges brought from infinity to charge a spherical shell of radius $R=0.100 \mathrm{m}$ to a total charge $Q=125 \mu \mathrm{C}$.

Mayukh Banik

Numerade Educator

Problem 78

Calculate the work that must be done on charges brought from infinity to charge a spherical shell of radius $R$ to a total charge $Q$.

Mayukh Banik

Numerade Educator

Problem 79

A 2.00 -nF parallel-plate capacitor is charged to an initial potential difference $\Delta V_{i}=100 \mathrm{V}$ and is 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 across the capacitor after the mica is withdrawn?

Manish Kumar

Numerade Educator

Problem 80

Why is the following situation impossible? You set up an apparatus in your laboratory as follows. The $x$ axis is the symmetry axis of a stationary, uniformly charged ring of radius $R=0.500 \mathrm{m}$ and charge $Q=50.0 \mu \mathrm{C} \text { (Fig. } \mathrm{P} 20.80)$ You place a particle with charge $Q=50.0 \mu \mathrm{C}$ and mass $m=0.100 \mathrm{kg}$ at the center of the ring and arrange for it to be constrained to move only along the $x$ axis. When it is displaced slightly, the particle is repelled by the ring and accelerates along the $x$ axis. The particle moves faster than you expected and strikes the opposite wall of your laboratory at $40.0 \mathrm{m} / \mathrm{s}$.

Manish Kumar

Numerade Educator

Problem 81

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 82

An electric dipole is located along the $y$ axis as shown in Figure $\mathrm{P} 20.82$ (page 696 ). The magnitude of its electric dipole moment is defined as $p=2 a q$. (a) At a point $P$, which is far from the dipole $(r>>a)$, show that the electric potential is $$V=\frac{k_{e} p \cos \theta}{r^{2}}$$ (b) Calculate the radial component $E_{r}$ and the perpendicular component $E_{\theta}$ of the associated electric field. Note that $E_{\theta}=-(1 / r)(\partial V / \partial \theta)$
Do these results seem reasonable for (c) $\theta=90^{\circ}$ and $0^{\circ}$ ? (d) For $r=0 ?$
(e) For the dipole arrangement shown in Figure $\mathrm{P} 20.82,$ express $V$ in terms of Cartesian coordinates using $r=\left(x^{2}+y^{2}\right)^{1 / 2}$ and $$\cos \theta=\frac{y}{\left(x^{2}+y^{2}\right)^{1 / 2}}$$ (f) Using these results and again $\operatorname{taking} r \gg a,$ calculate the field components $E_{x}$ and $E_{y}$.

Dominador Tan

Numerade Educator

Problem 83

A $10.0-\mu \mathrm{F}$ 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 -V battery as diagrammed in Figure $\mathrm{P} 20.83 .$ Find the new potential differences across the $5.00-\mu \mathrm{F}$ and $10.0-\mu \mathrm{F}$ capacitors after the switch is thrown closed.

Manish Kumar

Numerade Educator

Problem 84

Two large, parallel metal plates, each of area $A$, 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 shown in Figure $\mathrm{P} 20.84$. (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?

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 85

A capacitor is constructed from two square, metal lic plates of sides $\ell$ and separation $d .$ Charges $+Q$ and $-Q$ are placed on the plates, and the power supply is then removed. A material of dielectric constant $\kappa$ is inserted a distance $x$ into the capacitor as shown in Figure $\mathrm{P} 20.85 .$ Assume $d$ is much smaller than $x$
(a) Find the equivalent capacitance of the device. (b) Calculate the energy stored in the capacitor. (c) Find the direction and magnitude of the force exerted by the plates on the dielectric. (d) Obtain a numerical value for the force when $x=\ell / 2,$ assuming $\ell=5.00 \mathrm{cm}, d=2.00 \mathrm{mm}$
the dielectric is glass $(\kappa=4.50),$ and the capacitor was charged to $2.00 \times 10^{3} \mathrm{V}$ before the dielectric was inserted. Suggestion: The system can be considered as two capacitors connected in parallel.

Manish Kumar

Numerade Educator

Problem 86

Two square plates of sides $\ell$ are placed parallel to each other with separation $d$ as suggested in Figure $\mathrm{P} 20.86$ You may assume $d$ is much less than $\ell .$ The plates carry uniformly distributed static charges $+Q_{0}$ and $-Q_{0} .$ A block of metal has width $\ell$, length $\ell$, and thickness slightly less than $d$. It is inserted a distance $x$ into the space between the plates. The charges on the plates remain uniformly distributed 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 in the system 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) Express the energy density in the electric field between the charged plates in terms of $Q_{0}$ $\ell, d,$ and $\epsilon_{0} .$ (e) Explain how the answers to parts (c) and
(d) compare with each other.

Josh Broderick Phillips

Josh Broderick Phillips

Numerade Educator

Problem 87

Determine the equivalent capacitance of the combination shown in Figure $\mathrm{P} 20.87$ Suggestion: Consider the symmetry involved.

Manish Kumar

Numerade Educator