Principles of Physics

David Halliday , Robert Resnick , Jearl Walker

Chapter 21

Coulomb's Law - all with Video Answers

Educators

Chapter Questions

Problem 1

Identify $\mathrm{X}$ in the following nuclear reactions: (a) ${ }^{1} \mathrm{H}+{ }^{13} \mathrm{Al} \rightarrow$ $\mathrm{X}+\mathrm{n} ;(\mathrm{b}){ }^{29} \mathrm{Cu}+{ }^{1} \mathrm{H} \rightarrow \mathrm{X} ;(\mathrm{c}){ }^{15} \mathrm{~N}+{ }^{1} \mathrm{H} \rightarrow{ }^{4} \mathrm{He}+\mathrm{X} .$ Appendix $\mathrm{F}$ will help.

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 2

In Fig. 21-11, four particles form a square. The charges are $q_{1}=q_{4}=Q$ and $q_{2}=q_{3}=q \cdot$ (a) What is $Q / q$ if the net electrostatic force on particles 2 and 3 is zero? (b) Is there any value of $q$ that makes the net electrostatic force on each of the four particles zero? Explain.

Brandy Heflin

Numerade Educator

Problem 3

In Fig. 21-11, the particles have charges $q_{1}=-q_{2}=300 \mathrm{nC}$ and $q_{3}=-q_{4}=200 \mathrm{nC}$, and distance $a=5.0$ $\mathrm{cm}$. What are the (a) magnitude and (b) angle (relative to the $+x$ direction) of the net force on particle 3 ?

Vishal Gupta

Numerade Educator

Problem 4

(a) What is the magnitude of the electrostatic force between a singly charged sodium ion $\left(\mathrm{Na}^{+}\right.$, of charge $\left.+e\right)$ and an adjacent singly charged chlorine ion $\left(\mathrm{Cl}^{-}\right.$, of charge $\left.-e\right)$ in a salt crystal if their separation is $2.82 \times 10^{-10} \mathrm{~m}$ ? (b) What is the force magnitude if radiation removes another electron from the sodium ion?

Vishal Gupta

Numerade Educator

Problem 5

In Fig. 21-12, particle 1 of charge $+6.0 \mu \mathrm{C}$ and particle 2 of charge $-2.0$ $\mu \mathrm{C}$ are held at separation $L=10.0 \mathrm{~cm}$ on an $x$ axis. If particle 3 of unknown charge $q_{3}$ is to be located such that the net electrostatic force on it from particles 1 and 2 is zero, what must be the (a) $x$ and (b) $y$ coordinates of particle 3 ?

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 6

Three particles are fixed on an $x$ axis. Particle 1 of charge $q_{1}$ is at $x=-a$, and particle 2 of charge $q_{2}$ is at $x=+a$. If their net electrostatic force on particle 3 of charge $+Q$ is to be zero, what must be the ratio $q_{1} / q_{2}$ when particle 3 is at (a) $x=+0.750 a$ and (b) $x=+1.50 a$ ?

Sheh Lit Chang

University of Washington

Problem 7

In Fig. 21-12, particle 1 of charge $+q$ and particle 2 of charge $+9.00 q$ are held at separation $L=8.00 \mathrm{~cm}$ on an $x$ axis. If particle 3 of charge $q_{3}$ is to be located such that the three particles remain in place when released, what must be the (a) $x$ and (b) $y$ coordinates of particle 3, and $(c)$ the ratio $q_{3} / q ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 8

In Fig. 21-12, particles 1 and 2 are fixed in place on an $x$ axis, at a separation of $L=6.00 \mathrm{~cm}$. Their charges are $q_{1}=+e$ and $q_{2}=-27 e$. Particle 3 with charge $q_{3}=+4 e$ is to be placed on the line between particles 1 and 2, so that they produce a net electrostatic force $\vec{F}_{3, \text { net }}$ on it. (a) At what coordinate should particle 3 be placed to minimize the magnitude of that force? (b) What is that minimum magnitude?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 9

As a cat rubs its back along a carpet, it acquires a charge of $+8.2 \times 10^{-7} \mathrm{C}$. How many electrons did it lose to the carpet?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 10

Figure 21-13 shows electrons 1 and 2 on an $x$ axis and charged ions 3 and 4 of identical charge $-q$ and at identical angles $\theta$. Electron 2 is free to move; the other three particles are fixed in place at horizontal distances $R$ from electron 2 and are intended to hold electron 2 in place. For physically possible values of $q \leq 5 e$, what are the (a) largest, (b) second largest, and (c) third largest values of $\theta$ for which electron 2 is held in place?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 11

The magnitude of the electrostatic force between point charges $q_{1}=26.0 \mu \mathrm{C}$ and $q_{2}=47.0 \mu \mathrm{C}$ is initially $5.70 \mathrm{~N}$. The separation is then changed such that the force magnitude is then $0.570$ N. (a) What is the ratio of the new separation to the initial separation? (b) What is the new separation?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 12

In the return stroke of a typical lightning bolt, a current of $2.8 \times 10^{4}$ A exists for $20 \mu$. How much charge is transferred in this event?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 13

A nonconducting spherical shell, with an inner radius of $4.0$ $\mathrm{cm}$ and an outer radius of $5.0 \mathrm{~cm}$, has charge spread nonuniformly through its volume between its inner and outer surfaces. The volume charge density $\rho$ is the charge per unit volume, with the unit coulomb per cubic meter. For this shell $\rho=b / r$, where $r$ is the distance in meters from the center of the shell and $b=3.0 \mu \mathrm{C} / \mathrm{m}^{2}$. What is the net charge in the shell?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 14

A current of $0.300 \mathrm{~A}$ through your chest can send your heart into fibrillation, ruining the normal rhythm of heartbeat and disrupting the flow of blood (and thus oxygen) to your brain. If that current persists for $1.50 \mathrm{~min}$, how many conduction electrons pass through your chest?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 15

In Fig. $21-14 a$, particles 1 and 2
have charge $20.0 \mu \mathrm{C}$ each and are held
at separation distance $d=0.75 \mathrm{~m}$. (a)
What is the magnitude of the electro-
static force on particle 1 due to parti-
cle $2 ?$ In Fig. $21-14 b$, particle 3 of
charge $20.0 \mu \mathrm{C}$ is positioned so as to
complete an equilateral triangle. (b)
What is the magnitude of the net elec-
trostatic force on particle 1 due to particles 2 and 3 ?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 16

In Fig. 21-15, three identical conducting spheres initially have the following charges: sphere $A, 4 Q ;$ sphere $B,-12 Q ;$ and sphere $C, 0 .$ Spheres $A$ and $B$ are fixed in place, with a center-to-center separation that is much larger than the spheres. Two experiments are conducted. In experiment 1 , sphere $C$ is touched to sphere $A$, then (separately) to sphere $B$, and then (separately) to sphere $A$ again, and then it is removed. In experiment 2, starting with the same initial states, the procedure is changed: Sphere $C$ is touched to sphere $B$ and then (separately) to
Figure 21-15

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 17

Earth's atmosphere is constantly bombarded by cosmic ray protons that originate somewhere in space. If the protons all passed through the atmosphere, each square meter of Earth's surface would intercept protons at the average rate of 1500 protons per second. (a) What would be the electric current intercepted by the total surface area of the planet? (b) How much charge would be collected per day?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 18

Two tiny, spherical water drops, with identical charges of $-1.00 \times 10^{-16} \mathrm{C}$, have a center-to-center separation of $1.20 \mathrm{~cm}$. (a) What is the magnitude of the electrostatic force acting between them? (b) How many excess electrons are on each drop, giving it its charge imbalance?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 19

The magnitude of the electrostatic force between two identical ions that are separated by a distance of $10.0 \times 10^{-10} \mathrm{~m}$ is $9.25 \times 10^{-10} \mathrm{~N}$. (a) What is the charge of each ion? (b) How many electrons are "missing" from each ion (thus giving the ion its charge imbalance)? (c) What is the force magnitude if the separation is halved?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 20

Figure 21-16a shows charged particles 1 and 2 that are fixed in place on an $x$ axis. Particle 1 has a charge with a magnitude of $\left|q_{1}\right|=$ $8.00 e$. Particle 3 of charge $q_{3}=+7.00 e$ is initially on the $x$ axis near particle 2 . Then particle 3 is gradually moved in the positive direction of the $x$ axis. As a result, the magnitude of the net electrostatic force $\vec{F}_{2 \text { net }}$ on particle 2 due to particles 1 and 3 changes. Figure $21-16 b$ gives the $x$ component of that net force as a function of the position $x$ of particle 3 . The scale of the $x$ axis is set by $x_{s}=0.80 \mathrm{~m}$. The plot has an asymptote of $F_{2, \text { net }}=1.5 \times 10^{-25} \mathrm{~N}$ as $x \rightarrow \infty$. As a multiple of $e$ and including the sign, what is the charge $q_{2}$ of particle 2 ?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 21

Two identical conducting spheres, fixed in place, attract each other with an electrostatic force of $0.108 \mathrm{~N}$ when their center-to-center separation is $50.0 \mathrm{~cm}$. The spheres are then connected by a thin conducting wire. When the wire is removed, the spheres repel each other with an electrostatic force of $0.144 \mathrm{~N}$. Of the initial charges on the spheres, with a positive net charge, what was (a) the negative charge on one of them and (b) the positive charge on the other?

Neelesh Sharma

Numerade Educator

Problem 22

Two equally charged particles are held $3.2 \times 10^{-3} \mathrm{~m}$ apart and then released from rest. The initial acceleration of the first particle is observed to be $6.0 \mathrm{~m} / \mathrm{s}^{2}$ and that of the second to be $9.0 \mathrm{~m} / \mathrm{s}^{2}$. If the mass of the first particle is $6.3 \times 10^{-7} \mathrm{~kg}$, what are (a) the mass of the second particle and (b) the magnitude of the charge of each particle?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 23

The charges and coordinates of two charged particles held fixed in an $x y$ plane are $q_{1}=+3.0 \mu \mathrm{C}, x_{1}=3.5 \mathrm{~cm}, y_{1}=0.50 \mathrm{~cm}$, and $q_{2}=-4.0 \mu \mathrm{C}, x_{2}=-2.0 \mathrm{~cm}, y_{2}=1.5 \mathrm{~cm}$. Find the (a) magnitude and (b) direction of the electrostatic force on particle 1 due to particle $2 .$ At what (c) $x$ and (d) $y$ coordinates should a third particle of charge $q_{3}=+6.0 \mu \mathrm{C}$ be placed such that the net electrostatic force on particle 1 due to particles 2 and 3 is zero?

Neelesh Sharma

Numerade Educator

Problem 24

Identical isolated conducting spheres 1 and 2 have equal charges and are separated by a distance that is large compared with their diameters (Fig. 21-17a). The electrostatic force acting on sphere 2 due to sphere 1 is $\vec{F}$. Suppose now that a third identical sphere 3 , having an insulating handle and initially neutral, is touched first to sphere 1

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 25

In Fig. 21-18, particles 2 and 4, of charge $-e$, are fixed in place on a $y$ axis, at $y_{2}=-10.0 \mathrm{~cm}$ and $y_{4}=5.00 \mathrm{~cm} .$ Particles 1 and 3, of charge $-e$, can be moved along charge $-e$, can be moved along the $x$ axis. Particle 5, of charge $+e$, is fixed at the origin. Initially particle 1 is at $x_{1}=-10.0 \mathrm{~cm}$ and

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 26

In Fig. 21-19a, three posi-
tively charged particles are fixed $\quad A \quad B C$

Neelesh Sharma

Numerade Educator

Problem 27

In crystals of the salt cesium chloride, cesium ions $\mathrm{Cs}^{+}$form the eight corners of a cube and a chlorine ion $\mathrm{Cl}^{-}$is at the cube's center (Fig. 21-20). The edge length of the cube is $0.40 \mathrm{~nm}$. The $\mathrm{Cs}^{+}$ions are each deficient by one electron (and thus each has a charge of $+e$ ), and the $\mathrm{Cl}^{-}$ion has one excess electron (and thus has a charge of $-e$ ). (a) What is the magnitude of the net electrostatic force exerted on the $\mathrm{Cl}^{-}$ion by the eight $\mathrm{Cs}^{+}$ions at the corners of the cube? (b) If one of the Cs $^{+}$ions is missing, the crystal is said to have a defect; what is the magnitude of the net electrostatic force exerted on the $\mathrm{Cl}^{-}$ion by the seven remaining $\mathrm{Cs}^{+}$ions?

Sri Datta Vikas Buchemmavari

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 28

Two particles are fixed on an $x$ axis. Particle 1 of charge $50 \mu \mathrm{C}$ is located at $x=-2.0 \mathrm{~cm}$; particle 2 of charge $Q$ is located at $x=3.0 \mathrm{~cm}$. Particle 3 of charge magnitude $20 \mu \mathrm{C}$ is released from rest on the $y$ axis at $y=2.0 \mathrm{~cm}$. What is the value of $Q$ if the initial acceleration of particle 3 is in the positive direction of (a) the $x$ axis and (b) the $y$ axis?

Neelesh Sharma

Numerade Educator

Problem 29

Calculate the number of coulombs of positive charge in 500 $\mathrm{cm}^{3}$ of (neutral) water. (Hint: A hydrogen atom contains one proton; an oxygen atom contains eight protons.)

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 30

Electrons and positrons are produced by the nuclear transformations of protons and neutrons known as beta decay. (a) If a proton transforms into a neutron, is an electron or a positron produced? (b) If a neutron transforms into a proton, is an electron or a positron produced?

Salamat Ali

Numerade Educator

Problem 31

In Fig. $21-21$, particles 1 and 2 of
charge $q_{1}=q_{2}=+4 e$ are on a $y$ axis at
distance $d=17.0 \mathrm{~cm}$ from the origin.
Particle 3 of charge $q_{3}=+8 e$ is moved
gradually along the $x$ axis from $x=0$ to
$x=+5.0 \mathrm{~m} .$ At what values of $x$ will the
magnitude of the electrostatic force on
the third particle from the other two
particles be (a) minimum and $(b)$ maxi-
mum? What are the $(c)$ minimum and $(d)$ maximum magnitudes?

Neelesh Sharma

Numerade Educator

Problem 32

In Fig. $21-22 a$, particle 1 (of charge $q_{1}$ ) and particle 2 (of charge $q_{2}$ ) are fixed in place on an $x$ axis, $8.00 \mathrm{~cm}$ apart. Particle 3 (of charge $q_{3}=+6.00 \times 10^{-19} \mathrm{C}$ ) is to be placed on the line between particles 1 and 2 so that they produce a net electrostatic force $\vec{F}_{3, \text { net }}$ on it. Figure $21-22 b$ gives the $x$ component of that force versus the coordinate $x$ at which particle 3 is placed. The scale of the $x$ axis is set by $x_{x}=8.0 \mathrm{~cm}$. What are (a) the sign of charge $q_{1}$ and (b) the ratio $q_{2} / q_{1}$ ?

Neelesh Sharma

Numerade Educator

Problem 33

A particle of charge $3.00 \mu \mathrm{C}$ is separated by $0.120 \mathrm{~m}$ from a particle of charge $-1.50 \mu \mathrm{C}$ (a) What is the magnitude of the electrostatic force between them? (b) What must their separation be to reduce that force by an order of magnitude?

Neelesh Sharma

Numerade Educator

Problem 34

Figure 21-23 shows an arrangement of four charged particles, with angle $\theta=35.0^{\circ}$ and distance $d=2.00$ $\mathrm{cm}$. Particle 2 has charge $q_{2}=$ $+8.00 \times 10^{-19}$ C;particles 3 and 4 have charges $q_{3}=q_{4}=-1.60 \times 10^{-19} \mathrm{C}$. (a) What is distance $D$ between the origin and particle 2 if the net electrostatic force on particle 1 due to the other particles is zero? (b) If par-
Figure 21-23 Problem $34 .$ ticles 3 and 4 were moved closer to the $x$ axis but maintained their symmetry about that axis, would the required value of $D$ be greater than, less than, or the same as in part (a)?

Neelesh Sharma

Numerade Educator

Problem 35

In Fig. 21-24, three charged particles lie on an $x$ axis. Particles 1 and 2 are fixed in place. Particle 3 is free to move, but the net elec-

Sheh Lit Chang

University of Washington

Problem 36

Figure 21-25a shows an arrangement of three charged particles separated by distance $d .$ Particles $A$ and $C$ are fixed on the $x$ axis, but particle $B$ can be moved along a circle centered on particle $A$. During the movement, a radial line between $A$ and $B$ makes an angle $\theta$ relative to the positive direction of the $x$ axis (Fig. 21-25b). The curves in Fig. 21-25c give, for two situations, the magnitude $F_{\text {net }}$ of the net electrostatic force on particle $A$ due to the other particles. That net force is given as a function of angle $\theta$ and as a multiple of a basic amount $F_{0}$. For example on curve 1, at $\theta=180^{\circ}$, we see that $F_{\text {net }}=2 F_{0}$. (a) For the situation corresponding to curve 1 , what is the ratio of the charge of particle $C$ to that of particle $B$ (including sign)? (b) For the situation corresponding to curve 2, what is that ratio?

Salamat Ali

Numerade Educator

Problem 37

Of the charge $Q$ initially on a tiny sphere, a portion $q$ is to be transferred to a second, nearby sphere. Both spheres can be treated as particles and are fixed with a certain separation. (a) For what value of $q / Q$ will the electrostatic force between the two spheres be maximized? What are the (b) smaller and (c) larger values of $q / Q$ that give a force magnitude that is $75 \%$ of that maximum?

Bettina Hanlon

Numerade Educator