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Fundamentals of Physics

David Halliday, Robert Resnick, Jearl Walker

Chapter 23

Electric Fields - all with Video Answers

Educators

Chapter Questions

06:57

Problem 1

(a) Calculate the number of electrons in a small, electrically neutral silver pin that has a mass of $10.0 \mathrm{~g}$. Silver has 47 electrons per atom, and its molar mass is $107.87 \mathrm{~g} / \mathrm{mol}$. (b) Electrons are added to the pin until the net negative charge is $1.00 \mathrm{mC}$. How many electrons are added for every $10^{9}$ electrons already present?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
05:07

Problem 2

(a) Two protons in a molecule are separated by a distance of $3.80 \times 10^{-10} \mathrm{~m}$. Find the electric force exerted by one proton on the other. (b) How does the magnitude of this force compare with the magnitude of the gravitational force between the two protons? (c) What must be the charge-to-mass ratio of a particle if the magnitude of the gravitational force between two of these particles equals the magnitude of the electric force between them?

Keshav Singh
Keshav Singh
Numerade Educator
04:00

Problem 3

Richard Feynman once said that if two persons stood at arm's length from each other and each person had $1 \%$ more electrons than protons, the force of repulsion between them would be enough to lift a "weight" equal to that of the entire Earth. Carry out an order-ofmagnitude calculation to substantiate this assertion.

Keshav Singh
Keshav Singh
Numerade Educator
03:56

Problem 4

Two small silver spheres, each with a mass of $10.0 \mathrm{~g}$, are separated by $1.00 \mathrm{~m}$. Calculate the fraction of the electrons in one sphere that must be transferred to the other to produce an attractive force of $1.00 \times 10^{4} \mathrm{~N}$ (about 1 ton) between the spheres. (The number of electrons per atom of silver is 47 , and the number of atoms per gram is Avogadro's number divided by the molar mass of silver, $107.87 \mathrm{~g} / \mathrm{mol}$.)

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
04:26

Problem 5

Suppose that $1.00 \mathrm{~g}$ of hydrogen is separated into electrons and protons. Suppose also that the protons are placed at the Earth's north pole and the electrons are placed at the south pole. What is the resulting compressional force on the Earth?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
03:27

Problem 6

Two identical conducting small spheres are placed with their centers $0.300 \mathrm{~m}$ apart. One is given a charge of $12.0 \mathrm{nC}$, and the other is given a charge of $-18.0 \mathrm{nC}$. (a) Find the electric force exerted on one sphere by the other. (b) The spheres are connected by a conducting wire. Find the electric force between the two after equilibrium has occurred.

Keshav Singh
Keshav Singh
Numerade Educator
04:06

Problem 7

Three point charges are located at the corners of an equilateral triangle, as shown in Figure P23.7. Calculate the net electric force on the $7.00-\mu \mathrm{C}$ charge.

Keshav Singh
Keshav Singh
Numerade Educator
06:39

Problem 8

Two small beads having positive charges $3 q$ and $q$ are fixed at the opposite ends of a horizontal insulating rod extending from the origin to the point $x=d$. As shown in Figure P23.8, a third small charged bead is free to slide on the rod. At what position is the third bead in equilibrium? Can it be in stable equilibrium?

Vishal Gupta
Vishal Gupta
Numerade Educator
05:56

Problem 9

Review Problem. In the Bohr theory of the hydrogen atom, an electron moves in a circular orbit about a proton, where the radius of the orbit is $0.529 \times 10^{-10} \mathrm{~m}$.
(a) Find the electric force between the two. (b) If this force causes the centripetal acceleration of the electron, what is the speed of the electron?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
05:19

Problem 10

Review Problem. Two identical point charges each having charge $+q$ are fixed in space and separated by a distance $d$. A third point charge $-Q$ of mass $m$ is free to move and lies initially at rest on a perpendicular bisector of the two fixed charges a distance $x$ from the midpoint of the two fixed charges (Fig. P23.10). (a) Show that if $x$ is small compared with $d$, the motion of $-Q$ is simple harmonic along the perpendicular bisector. Determine the period of that motion. (b) How fast will the charge $-Q$ be moving when it is at the midpoint between the two fixed charges, if initially it is released at a distance $x=a \ll d$ from the midpoint?

Keshav Singh
Keshav Singh
Numerade Educator
04:12

Problem 11

What are the magnitude and direction of the electric field that will balance the weight of (a) an electron and
(b) a proton? (Use the data in Table 23.1.)

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:48

Problem 12

An object having a net charge of $24.0 \mu \mathrm{C}$ is placed in a uniform electric field of $610 \mathrm{~N} / \mathrm{C}$ that is directed vertically. What is the mass of this object if it "floats" in the field?

Vishal Gupta
Vishal Gupta
Numerade Educator
03:42

Problem 13

In Figure P23.13, determine the point (other than infinity) at which the electric field is zero.

Keshav Singh
Keshav Singh
Numerade Educator
01:49

Problem 14

An airplane is flying through a thundercloud at a height of $2000 \mathrm{~m}$. (This is a very dangerous thing to do because of updrafts, turbulence, and the possibility of electric discharge.) If there are charge concentrations of $+40.0 \mathrm{C}$ at a height of $3000 \mathrm{~m}$ within the cloud and of $-40.0 \mathrm{C}$ at a height of $1000 \mathrm{~m}$, what is the electric field $E$ at the aircraft?

Keshav Singh
Keshav Singh
Numerade Educator
08:09

Problem 15

Three charges are at the corners of an equilateral triangle, as shown in Figure P23.7. (a) Calculate the electric field at the position of the $2.00-\mu \mathrm{C}$ charge due to the $7.00-\mu \mathrm{C}$ and $-4.00-\mu \mathrm{C}$ charges. (b) Use your answer to part (a) to determine the force on the $2.00-\mu \mathrm{C}$ charge.

Vishal Gupta
Vishal Gupta
Numerade Educator
View

Problem 16

Three point charges are arranged as shown in Figure P23.16. (a) Find the vector electric field that the $6.00-\mathrm{n} \mathrm{C}$ and $-3.00-\mathrm{nC}$ charges together create at the origin. (b) Find the vector force on the $5.00$ -nC charge.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:37

Problem 17

Three equal positive charges $q$ are at the corners of an equilateral triangle of side $a$, as shown in Figure P23.17.
(a) Assume that the three charges together create an electric field. Find the location of a point (other than $\infty$ ) where the electric field is zero. (Hint: Sketch the field lines in the plane of the charges.) (b) What are the magnitude and direction of the electric field at $P$ due to the two charges at the base?

Keshav Singh
Keshav Singh
Numerade Educator
02:21

Problem 18

Two $2.00-\mu$ C point charges 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 field on the $y$ axis at $y=$ $0.500 \mathrm{~m} .$ (b) Calculate the electric force on a $-3.00-\mu \mathrm{C}$ charge placed on the $y$ axis at $y=0.500 \mathrm{~m}$.

Narayan Hari
Narayan Hari
Numerade Educator
12:00

Problem 19

Four point charges are at the corners of a square of side $a$, as shown in Figure P23.19. (a) Determine the magnitude and direction of the electric field at the location of charge $q$ (b) What is the resultant force on $q ?$

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
09:21

Problem 20

A point particle having charge $q$ is located at point $\left(x_{0}, y_{0}\right)$ in the $x y$ plane. Show that the $x$ and $y$ components of the electric field at point $(x, y)$ due to this charge $q$ are
$$
\begin{aligned}
E_{x} &=\frac{k_{e} q\left(x-x_{0}\right)}{\left[\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}\right]^{3 / 2}} \\
E_{y} &=\frac{k_{e} q\left(y-y_{0}\right)}{\left[\left(x-x_{0}\right)^{2}+\left(y-y_{0}\right)^{2}\right]^{3 / 2}}
\end{aligned}
$$

Susan Hallstrom
Susan Hallstrom
Numerade Educator
01:48

Problem 21

Consider the electric dipole shown in Figure P23.21. Show that the electric field at a distant point along the $x$ axis is $E_{x} \cong 4 k_{e} q a / x^{3}$.

Keshav Singh
Keshav Singh
Numerade Educator
02:44

Problem 22

Consider $n$ equal positive point charges each of magnitude $Q / n$ placed symmetrically around a circle of radius $R$. (a) Calculate the magnitude of the electric field $E$ at a point a distance $x$ on the line passing through the center of the circle and perpendicular to the plane of the circle. (b) Explain why this result is identical to the one obtained in Example $23.8$.

Keshav Singh
Keshav Singh
Numerade Educator
02:12

Problem 23

Consider an infinite number of identical charges (each of charge $q$ ) placed along the $x$ axis at distances $a, 2 a$, $3 a, 4 a, \ldots$ from the origin. What is the electric field at the origin due to this distribution? Hint: Use the fact that
$$
1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\cdots=\frac{\pi^{2}}{6}
$$

Keshav Singh
Keshav Singh
Numerade Educator
11:04

Problem 24

A rod $14.0 \mathrm{~cm}$ long is uniformly charged and has a total charge of $-22.0 \mu \mathrm{C}$. Determine the magnitude and direction of the electric field along the axis of the rod at a point $36.0 \mathrm{~cm}$ from its center.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
06:56

Problem 25

A continuous line of charge lies along the $x$ axis, extending from $x=+x_{0}$ to positive infinity. The line carries a uniform linear charge density $\lambda_{0}$. What are the magnitude and direction of the electric field at the origin?

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
04:46

Problem 26

A line of charge starts at $x=+x_{0}$ and extends to positive infinity. If the linear charge density is $\lambda=\lambda_{0} x_{0} / x$, determine the electric field at the origin.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
05:43

Problem 27

A uniformly charged ring of radius $10.0 \mathrm{~cm}$ has a total charge of $75.0 \mu \mathrm{C}$. Find the electric field on the axis of the ring at (a) $1.00 \mathrm{~cm}$, (b) $5.00 \mathrm{~cm}$, (c) $30.0 \mathrm{~cm}$, and
(d) $100 \mathrm{~cm}$ from the center of the ring.

Dading Chen
Dading Chen
Numerade Educator
03:08

Problem 28

Show that the maximum field strength $E_{\max }$ along the axis of a uniformly charged ring occurs at $x=a / \sqrt{2}$ (see Fig. 23.17) and has the value $Q /\left(6 \sqrt{3} \pi \epsilon_{0} a^{2}\right)$.

Keshav Singh
Keshav Singh
Numerade Educator
06:21

Problem 29

A uniformly charged disk of radius $35.0 \mathrm{~cm}$ carries a charge density of $7.90 \times 10^{-3} \mathrm{C} / \mathrm{m}^{2}$. Calculate the electric field on the axis of the disk at (a) $5.00 \mathrm{~cm}$,
(b) $10.0 \mathrm{~cm}$,
(c) $50.0 \mathrm{~cm}$, and
(d) $200 \mathrm{~cm}$ from the center of the disk.

Dading Chen
Dading Chen
Numerade Educator
05:19

Problem 30

Example $23.9$ derives the exact expression for the electric field at a point on the axis of a uniformly charged disk. Consider a disk of radius $R=3.00 \mathrm{~cm}$ having a uniformly distributed charge of $+5.20 \mu \mathrm{C}$. (a) Using the result of Example 23.9, compute the electric field at a point on the axis and $3.00 \mathrm{~mm}$ from the center. Compare this answer with the field computed from the nearfield approximation $E=\sigma / 2 \epsilon_{0} .$ (b) Using the result of Example 23.9, compute the electric field at a point on the axis and $30.0 \mathrm{~cm}$ from the center of the disk. Compare this result with the electric field obtained by treating the disk as a $+5.20-\mu$ C point charge at a distance of $30.0 \mathrm{~cm}$.

Keshav Singh
Keshav Singh
Numerade Educator
04:59

Problem 31

The electric field along the axis of a uniformly charged disk of radius $R$ and total charge $Q$ was calculated in $\mathrm{Ex}-$ ample 23.9. Show that the electric field at distances $x$ that are great compared with $R$ approaches that of a point charge $Q=\sigma \pi R^{2}$. (Hint: First show that $x /\left(x^{2}+R^{2}\right)^{1 / 2}=\left(1+R^{2} / x^{2}\right)^{-1 / 2}$, and use the binomial expansion $(1+\delta)^{n} \approx 1+n \delta$ when $\left.\delta \ll 1 .\right)$

Keshav Singh
Keshav Singh
Numerade Educator
01:33

Problem 32

A piece of Styrofoam having a mass $m$ carries a net charge of $-q$ and floats above the center of a very large horizontal sheet of plastic that has a uniform charge density on its surface. What is the charge per unit area on the plastic sheet?

Keshav Singh
Keshav Singh
Numerade Educator
10:11

Problem 33

A uniformly charged insulating rod of length $14.0 \mathrm{~cm}$ is bent into the shape of a semicircle, as shown in Figure P23.33. The rod has a total charge of $-7.50 \mu$ C. Find the magnitude and direction of the electric field at $O$, the center of the semicircle.

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
08:01

Problem 34

(a) Consider a uniformly charged right circular cylindrical shell having total charge $Q$, radius $R$, and height
h. Determine the electric field at a point a distance $d$ from the right side of the cylinder, as shown in Figure P23.34. (Hint: Use the result of Example $23.8$ and treat the cylinder as a collection of ring charges.) (b) Consider now a solid cylinder with the same dimensions and carrying the same charge, which is uniformly distributed through its volume. Use the result of Example $23.9$ to find the field it creates at the same point.

Keshav Singh
Keshav Singh
Numerade Educator
03:37

Problem 35

A thin rod of length $\ell$ and uniform charge per unit length $\lambda$ lies along the $x$ axis, as shown in Figure P23.35.
(a) Show that the electric field at $P$, a distance $y$ from the rod, along the perpendicular bisector has no $x$ component and is given by $E=2 k_{e} \lambda \sin \theta_{0} / y .$ (b) Using your result to part (a), show that the field of a rod of infinite length is $E=2 k_{e} \lambda / y$. (Hint: First calculate the field at $P$ due to an element of length $d x$, which has a charge $\lambda d x$. Then change variables from $x$ to $\theta$, using the facts that $x=y \tan \theta$ and $d x=y \sec ^{2} \theta d \theta$, and integrate over $\theta$.)
$y$

Keshav Singh
Keshav Singh
Numerade Educator
04:38

Problem 36

Three solid plastic cylinders all have a radius of $2.50 \mathrm{~cm}$ and a length of $6.00 \mathrm{~cm}$. One (a) carries charge with uniform density $15.0 \mathrm{nC} / \mathrm{m}^{2}$ everywhere on its surface. Another (b) carries charge with the same uniform density on its curved lateral surface only. The third (c) carries charge with uniform density $500 \mathrm{nC} / \mathrm{m}^{3}$ throughout the plastic. Find the charge of each cylinder.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
08:15

Problem 37

Eight solid plastic cubes, each $3.00 \mathrm{~cm}$ on each edge, are glued together to form each one of the objects (i, ii, iii, and iv) shown in Figure P23.37.
(a) If each object carries charge with a uniform density of $400 \mathrm{nC} / \mathrm{m}^{3}$ throughout its volume, what is the charge of each object? (b) If each object is given charge with a uniform density of $15.0 \mathrm{nC} / \mathrm{m}^{2}$ everywhere on its exposed surface, what is the charge on each object? (c) If charge is placed only on the edges where perpendicular surfaces meet, with a uniform density of $80.0 \mathrm{pC} / \mathrm{m}$, what is the charge of each object?

Keshav Singh
Keshav Singh
Numerade Educator
01:20

Problem 38

A positively charged disk has a uniform charge per unit area as described in Example 23.9. Sketch the electric field lines in a plane perpendicular to the plane of the disk passing through its center.

Banhishikha Sinha
Banhishikha Sinha
Numerade Educator
01:09

Problem 39

A negatively charged rod of finite length has a uniform charge per unit length. Sketch the electric field lines in a plane containing the rod.

Keshav Singh
Keshav Singh
Numerade Educator
04:11

Problem 40

Figure P23.40 shows the electric field lines for two point charges separated by a small distance. (a) Determine the ratio $q_{1} / q_{2}$. (b) What are the signs of $q_{1}$ and $q_{2}$ ?

Vishal Gupta
Vishal Gupta
Numerade Educator
07:41

Problem 41

An electron and a proton are each placed at rest in an electric field of $520 \mathrm{~N} / \mathrm{C}$. Calculate the speed of each particle $48.0 \mathrm{~ns}$ after being released.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
03:56

Problem 42

A proton is projected in the positive $x$ direction into a region of uniform electric field $\mathbf{E}=-6.00 \times 10^{5} \mathbf{i} \mathrm{N} / \mathrm{C}$. The proton travels $7.00 \mathrm{~cm}$ before coming to rest. Determine (a) the acceleration of the proton, (b) its initial speed, and (c) the time it takes the proton to come to rest.

Keshav Singh
Keshav Singh
Numerade Educator
04:30

Problem 43

A proton accelerates from rest in a uniform electric field of $640 \mathrm{~N} / \mathrm{C}$. At some later time, its speed has reached $1.20 \times 10^{6} \mathrm{~m} / \mathrm{s}$ (nonrelativistic, since $v$ is much less than the speed of light). (a) Find the acceleration of the proton. (b) How long does it take the proton to reach this speed? (c) How far has it moved in this time? (d) What is its kinetic energy at this time?

Keshav Singh
Keshav Singh
Numerade Educator
02:33

Problem 44

The electrons in a particle beam each have a kinetic energy of $1.60 \times 10^{-17} \mathrm{~J} .$ What are the magnitude and direction of the electric field that stops these electrons in a distance of $10.0 \mathrm{~cm}$ ?

Keshav Singh
Keshav Singh
Numerade Educator
01:42

Problem 45

The electrons in a particle beam each have a kinetic energy $K$. What are the magnitude and direction of the electric field that stops these electrons in a distance $d$ ?

Keshav Singh
Keshav Singh
Numerade Educator
08:22

Problem 46

A positively charged bead having a mass of $1.00 \mathrm{~g}$ falls from rest in a vacuum from a height of $5.00 \mathrm{~m}$ in a uniform vertical electric field with a magnitude of $1.00 \times 10^{4} \mathrm{~N} / \mathrm{C}$. The bead hits the ground at a speed of $21.0 \mathrm{~m} / \mathrm{s}$. Determine (a) the direction of the electric field (up or down) and (b) the charge on the bead.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
05:06

Problem 47

A proton moves at $4.50 \times 10^{5} \mathrm{~m} / \mathrm{s}$ in the horizontal direction. It enters a uniform vertical electric field with a magnitude of $9.60 \times 10^{3} \mathrm{~N} / \mathrm{C}$. Ignoring any gravitational effects, find (a) the time it takes the proton to travel $5.00 \mathrm{~cm}$ horizontally, (b) its vertical displacement after it has traveled $5.00 \mathrm{~cm}$ horizontally, and (c) the horizontal and vertical components of its velocity after it has traveled $5.00 \mathrm{~cm}$ horizontally.

Keshav Singh
Keshav Singh
Numerade Educator
09:51

Problem 48

An electron is projected at an angle of $30.0^{\circ}$ above the horizontal at a speed of $8.20 \times 10^{5} \mathrm{~m} / \mathrm{s}$ in a region where the electric field is $\mathbf{E}=390 \mathbf{j} \mathrm{N} / \mathrm{C}$. Neglecting the effects of gravity, find (a) the time it takes the electron to return to its initial height, (b) the maximum height it reaches, and (c) its horizontal displacement when it reaches its maximum height.

David Morabito
David Morabito
Numerade Educator
04:17

Problem 49

Protons are projected with an initial speed $v_{i}=9.55 \times 10^{3} \mathrm{~m} / \mathrm{s}$ into a region where a uniform electric field $\mathbf{E}=(-720 \mathbf{j}) \mathrm{N} / \mathrm{C}$ is present, as shown in Figure P23.49. The protons are to hit a target that lies at a horizontal distance of $1.27 \mathrm{~mm}$ from the point where the protons are launched. Find (a) the two projection angles $\theta$ that result in a hit and (b) the total time of flight for each trajectory.

Keshav Singh
Keshav Singh
Numerade Educator
07:50

Problem 50

Three point charges are aligned along the $x$ axis as shown in Figure $\mathrm{P} 23.50$. Find the electric field at (a) the position $(2.00,0)$ and (b) the position $(0,2.00)$.

Keshav Singh
Keshav Singh
Numerade Educator
06:38

Problem 51

A uniform electric field of magnitude $640 \mathrm{~N} / \mathrm{C}$ exists between two parallel plates that are $4.00 \mathrm{~cm}$ apart. A proton is released from the positive plate at the same instant that an electron is released from the negative plate. (a) Determine the distance from the positive plate at which the two pass each other. (Ignore the electrical attraction between the proton and electron.)
(b) Repeat part (a) for a sodium ion $\left(\mathrm{Na}^{+}\right)$ and a chlorine ion $\left(\mathrm{Cl}^{-}\right)$.

Keshav Singh
Keshav Singh
Numerade Educator
04:31

Problem 52

A small, $2.00$ -g plastic ball is suspended by a $20.0-\mathrm{cm}-$ long string in a uniform electric field, as shown in Figure $\mathrm{P} 23.52$. If the ball is in equilibrium when the string makes a $15.0^{\circ}$ angle with the vertical, what is the net charge on the ball?

Meghan Miholics
Meghan Miholics
Numerade Educator
03:40

Problem 53

A charged cork ball of mass $1.00 \mathrm{~g}$ is suspended on a light string in the presence of a uniform electric field, as shown in Figure $\mathrm{P} 23.53$. When $\mathbf{E}=(3.00 \mathbf{i}+$ $5.00 \mathbf{j}) \times 10^{5} \mathrm{~N} / \mathrm{C}$, the ball is in equilibrium at $\theta=37.0^{\circ} .$ Find (a) the charge on the ball and (b) the tension in the string.

Keshav Singh
Keshav Singh
Numerade Educator
02:50

Problem 54

A charged cork ball of mass $m$ is suspended on a light string in the presence of a uniform electric field, as shown in Figure P23.53. When $\mathbf{E}=(A \mathbf{i}+B \mathbf{j}) \mathrm{N} / \mathrm{C}$, where $A$ and $B$ are positive numbers, the ball is in equilibrium at the angle $\theta$. Find (a) the charge on the ball and (b) the tension in the string.

Keshav Singh
Keshav Singh
Numerade Educator
05:30

Problem 55

Four identical point charges $(q=+10.0 \mu \mathrm{C})$ are located on the corners of a rectangle, as shown in Figure P23.55. The dimensions of the rectangle are $L=60.0 \mathrm{~cm}$ and $W=15.0 \mathrm{~cm} .$ Calculate the magnitude and direction of the net electric force exerted on the charge at the lower left corner by the other three charges.

Keshav Singh
Keshav Singh
Numerade Educator
09:28

Problem 56

Three identical small Styrofoam balls $(m=2.00 \mathrm{~g})$ are suspended from a fixed point by three nonconducting threads, each with a length of $50.0 \mathrm{~cm}$ and with negligible mass. At equilibrium the three balls form an equilateral triangle with sides of $30.0 \mathrm{~cm} .$ What is the common charge $q$ carried by each ball?

David Morabito
David Morabito
Numerade Educator
02:43

Problem 57

Two identical metallic blocks resting on a frictionless horizontal surface are connected by a light metallic spring having the spring constant $k=100 \mathrm{~N} / \mathrm{m}$ and an unstretched length of $0.300 \mathrm{~m}$, as shown in Figure P23.57a. A total charge of $Q$ is slowly placed on the system, causing the spring to stretch to an equilibrium length of $0.400 \mathrm{~m}$, as shown in Figure P23.57b. Determine the value of $Q$, assuming that all the charge resides on the blocks and that the blocks are like point charges.

Keshav Singh
Keshav Singh
Numerade Educator
01:37

Problem 58

Two identical metallic blocks resting on a frictionless horizontal surface are connected by a light metallic spring having a spring constant $k$ and an unstretched length $L_{i}$, as shown in Figure $\mathrm{P} 23.57 \mathrm{a}$. A total charge of $Q$ is slowly placed on the system, causing the spring to stretch to an equilibrium length $L$, as shown in Figure P23.57b. Determine the value of $Q$, assuming that all the charge resides on the blocks and that the blocks are like point charges.

Keshav Singh
Keshav Singh
Numerade Educator
06:38

Problem 59

Identical thin rods of length $2 a$ carry equal charges, $+Q$, uniformly distributed along their lengths. The rods lie along the $x$ axis with their centers separated by a distance of $b>2 a$ (Fig. P23.59). Show that the magnitude of the force exerted by the left rod on the right one is given by
$$
F=\left(\frac{k_{e} Q^{2}}{4 a^{2}}\right) \ln \left(\frac{b^{2}}{b^{2}-4 a^{2}}\right)
$$

Dading Chen
Dading Chen
Numerade Educator
06:33

Problem 60

A particle is said to be nonrelativistic as long as its speed is less than one-tenth the speed of light, or less than $3.00 \times 10^{7} \mathrm{~m} / \mathrm{s}$. (a) How long will an electron remain nonrelativistic if it starts from rest in a region of an electric field of $1.00 \mathrm{~N} / \mathrm{C} ?(\mathrm{~b})$ How long will a proton remain nonrelativistic in the same electric field?
(c) Electric fields are commonly much larger than $1 \mathrm{~N} / \mathrm{C}$. Will the charged particle remain nonrelativistic for a shorter or a longer time in a much larger electric field?

David Morabito
David Morabito
Numerade Educator
13:28

Problem 61

A line of positive charge is formed into a semicircle of radius $R=60.0 \mathrm{~cm}$, as shown in Figure $\mathrm{P} 23.61 .$ The charge per unit length along the semicircle is described by the expression $\lambda=\lambda_{0} \cos \theta .$ The total charge on the semicircle is $12.0 \mu \mathrm{C}$. Calculate the total force on a charge of $3.00 \mu \mathrm{C}$ placed at the center of curvature.

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
07:55

Problem 62

Two small spheres, each of mass $2.00 \mathrm{~g}$, are suspended by light strings $10.0 \mathrm{~cm}$ in length (Fig. P23.62). A uniform electric field is applied in the $x$ direction. The spheres have charges equal to $-5.00 \times 10^{-8} \mathrm{C}$ and $+5.00 \times 10^{-8} \mathrm{C}$. Determine the electric field that enables the spheres to be in equilibrium at an angle of $\theta=10.0^{\circ}$.

Mihajlo Grcic
Mihajlo Grcic
Numerade Educator
05:10

Problem 63

Two small spheres of mass $m$ are suspended from strings of length $\ell$ that are connected at a common point. One sphere has charge $Q$; the other has charge $2 Q$. Assume that the angles $\theta_{1}$ and $\theta_{2}$ that the strings make with the vertical are small. (a) How are $\theta_{1}$ and $\theta_{2}$ related?
(b) Show that the distance $r$ between the spheres is
$$
r \cong\left(\frac{4 k_{e} Q^{2} \ell}{m g}\right)^{1 / 3}
$$

Keshav Singh
Keshav Singh
Numerade Educator
04:57

Problem 64

Three charges of equal magnitude $q$ are fixed in position at the vertices of an equilateral triangle (Fig. P23.64). A fourth charge $Q$ is free to move along the positive $x$ axis under the influence of the forces exerted by the three fixed charges. Find a value for $s$ for which $Q$ is in equilibrium. You will need to solve a transcendental equation.

Nadir Iqbal
Nadir Iqbal
Numerade Educator
09:13

Problem 65

Four identical point charges, each having charge $+q$, are fixed at the corners of a square of side $L$. A fifth point charge $-Q$ lies a distance $z$ along the line perpendicular to the plane of the square and passing through the center of the square (Fig. $\mathrm{P} 23.65$ ).
(a) Show that the force exerted on $-Q$ by the other four charges is
$$
\mathbf{F}=-\frac{4 k_{e} q Q z}{\left(z^{2}+\frac{L^{2}}{2}\right)^{3 / 2}} \mathbf{k}
$$
Note that this force is directed toward the center of the square whether $z$ is positive $(-Q$ above the square) or negative $(-Q$ below the square $)$. (b) If $z$ is small compared with $L$, the above expression reduces to $\mathbf{F} \approx-($ constant $) z \mathbf{k}$. Why does this imply that the motion of $-Q$ is simple harmonic, and what would be the period of this motion if the mass of $-Q$ were $m ?$

David Morabito
David Morabito
Numerade Educator
06:58

Problem 66

A 1.00-g cork ball with a charge of $2.00 \mu \mathrm{C}$ is suspended vertically on a $0.500$ -m-long light string in the presence of a uniform, downward-directed electric field of magnitude $E=1.00 \times 10^{5} \mathrm{~N} / \mathrm{C}$. If the ball is displaced slightly from the vertical, it oscillates like a simple pendulum. (a) Determine the period of this oscillation. (b) Should gravity be included in the calculation for part (a)? Explain.

Vishal Gupta
Vishal Gupta
Numerade Educator
08:46

Problem 67

Three charges of equal magnitude $q$ reside at the corners of an equilateral triangle of side length $a$ (Fig. P23.67). (a) Find the magnitude and direction of the electric field at point $P$, midway between the negative charges, in terms of $k_{e}, q$, and $a$. (b) Where must a $-4 q$ charge be placed so that any charge located at $P$ experiences no net electric force? In part (b), let $P$ be the origin and let the distance between the $+q$ charge and $P$ be $1.00 \mathrm{~m}$.

David Morabito
David Morabito
Numerade Educator
05:42

Problem 68

Two identical beads each have a mass $m$ and charge $q$. When placed in a hemispherical bowl of radius $R$ with frictionless, nonconducting walls, the beads move, and at equilibrium they are a distance $R$ apart (Fig. $\mathrm{P} 23.68$ ). Determine the charge on each bead.

Ren Jie Tuieng
Ren Jie Tuieng
Numerade Educator
03:09

Problem 69

Eight point charges, each of magnitude $q$, are located on the corners of a cube of side $s$, as shown in Figure P23.69. (a) Determine the $x, y$, and $z$ components of the resultant force exerted on the charge located at point $A$ by the other charges. (b) What are the magnitude and direction of this resultant force?

Keshav Singh
Keshav Singh
Numerade Educator
03:07

Problem 70

Consider the charge distribution shown in Figure P23.69. (a) Show that the magnitude of the electric field at the center of any face of the cube has a value of $2.18 k_{e} q / s^{2}$. (b) What is the direction of the electric field at the center of the top face of the cube?

Keshav Singh
Keshav Singh
Numerade Educator
09:17

Problem 71

A line of charge with a uniform density of $35.0 \mathrm{nC} / \mathrm{m}$ lies along the line $y=-15.0 \mathrm{~cm}$, between the points with coordinates $x=0$ and $x=40.0 \mathrm{~cm}$. Find the electric field it creates at the origin.

Dading Chen
Dading Chen
Numerade Educator
09:10

Problem 72

Three point charges $q,-2 q$, and $q$ are located along the $x$ axis, as shown in Figure $\mathrm{P} 23.72$. Show that the electric field at $P(y \gg a)$ along the $y$ axis is
$$
\mathbf{E}=-k_{e} \frac{3 q a^{2}}{y^{4}} \mathbf{j}
$$
This charge distribution, which is essentially that of two electric dipoles, is called an electric quadrupole. Note that $\mathbf{E}$ varies as $r^{-4}$ for the quadrupole, compared with variations of $r^{-3}$ for the dipole and $r^{-2}$ for the monopole (a single charge ).

David Morabito
David Morabito
Numerade Educator
03:34

Problem 73

Review Problem. A negatively charged particle $-q$ is placed at the center of a uniformly charged ring, where the ring has a total positive charge $Q$, as shown in Example $23.8 .$ The particle, confined to move along the $x$ axis, is displaced a small distance $x$ along the axis (where $x \ll a$ ) and released. Show that the particle oscillates with simple harmonic motion with a frequency
$$
f=\frac{1}{2 \pi}\left(\frac{k_{e} q Q}{m a^{3}}\right)^{1 / 2}
$$

Keshav Singh
Keshav Singh
Numerade Educator
03:47

Problem 74

An electric dipole in a uniform electric field is displaced slightly from its equilibrium position, as shown in Figure $\mathrm{P} 23.74$, where $\theta$ is small and the charges are separated by a distance $2 a$. The moment of inertia of the dipole is $I$. If the dipole is released from this position, show that its angular orientation exhibits simple harmonic motion with a frequency
$$
f=\frac{1}{2 \pi} \sqrt{\frac{2 q a E}{I}}
$$

Keshav Singh
Keshav Singh
Numerade Educator