University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 22

Electric Fields and Gauss's Law - all with Video Answers

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Chapter Questions

Problem 1

To be able to calculate the electric field created by a known distribution of charge using Gauss's Law, which of the following must be true?
a) The charge distribution must be in a nonconducting medium.
b) The charge distribution must be in a conducting medium.
c) The charge distribution must have spherical or cylindrical symmetry.
d) The charge distribution must be uniform.
e) The charge distribution must have a high degree of symmetry that allows assumptions about the symmetry of its electric field to be made.

Vishal Gupta

Vishal Gupta

Numerade Educator

Problem 2

An electric dipole consists of two equal and opposite charges situated a very small distance from each other. When the dipole is placed in a uniform electric field, which of the following statements is true?
a) The dipole will not experience any net force from the electric field; since the charges are equal and have opposite signs, the individual effects will cancel out.
b) There will be no net force and no net torque acting on the dipole.
c) There will be a net force but no net torque acting on the dipole.
d) There will be no net force, but there will (in general) be a net torque acting on dipole.

Ajay Singhal

Numerade Educator

Problem 3

A point charge, $+Q$, is located on the $x$ -axis at $x=a$, and a second point charge, $-Q$, is located on the $x$ -axis at $x=-a$. A Gaussian surface with radius $r=2 a$ is centered at the origin. The flux through this Gaussian surface is
a) zero.
b) greater than zero.
c) less than zero.
d) none of the above.

Ajay Singhal

Numerade Educator

Problem 4

A charge of $+2 q$ is placed at the center of an uncharged conducting shell. What will be the charges on the inner and outer surfaces of the shell, respectively?
a) $-2 q,+2 q$
b) $-q,+q$
c) $-2 q,-2 q$
d) $-2 q,+4 q$

Ajay Singhal

Numerade Educator

Problem 5

Two infinite nonconducting plates are parallel to each other, with a distance $d=10.0 \mathrm{~cm}$ between them, as shown in the figure. Each plate carries a uniform charge distribution of $\sigma=4.5 \mu \mathrm{C} / \mathrm{m}^{2} .$ What is the electric field, $\vec{E}$ at point $P\left(\right.$ with $\left.x_{P}=20.0 \mathrm{~cm}\right) ?$
a) $0 \mathrm{~N} / \mathrm{C}$
b) $2.54 \hat{x} \mathrm{~N} / \mathrm{C}$
c) $\left(-5.08 \cdot 10^{5}\right) \hat{x} \mathrm{~N} / \mathrm{C}$
d) $\left(5.08 \cdot 10^{5}\right) \hat{x} \mathrm{~N} / \mathrm{C}$
e) $\left(-1.02 \cdot 10^{6}\right) \hat{x} \mathrm{~N} / \mathrm{C}$
f) $\left(1.02 \cdot 10^{6}\right) \hat{x} \mathrm{~N} / \mathrm{C}$

Vishal Gupta

Numerade Educator

Problem 6

At which of the following locations is the electric field the strongest?
a) a point $1 \mathrm{~m}$ from a $1 \mathrm{C}$ point charge
b) a point $1 \mathrm{~m}$ (perpendicular distance) from the center of a $1-\mathrm{m}$ -long wire with $1 \mathrm{C}$ of charge distributed on it
c) a point $1 \mathrm{~m}$ (perpendicular distance) from the center of a $1-\mathrm{m}^{2}$ sheet of charge with $1 \mathrm{C}$ of charge distributed on it
d) a point $1 \mathrm{~m}$ from the surface of a charged spherical shell of charge $1 \mathrm{C}$ with a radius of $1 \mathrm{~m}$
e) a point $1 \mathrm{~m}$ from the surface of a charged spherical shell of charge $1 \mathrm{C}$ with a radius of $0.5 \mathrm{~m}$

Vishal Gupta

Numerade Educator

Problem 7

The electric flux through a spherical Gaussian surface of radius $R$ centered on a charge $Q$ is $1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right) .$ What is the electric flux through a cubic Gaussian surface of side $R$ centered on the same charge $Q ?$
a) less than $1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)$
b) more than $1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)$
c) equal to $1200 \mathrm{~N} /\left(\mathrm{C} \mathrm{m}^{2}\right)$
d) cannot be determined from the information given

Ajay Singhal

Numerade Educator

Problem 8

A single positive point charge, $q,$ is at one corner of a cube with sides of length $L$, as shown in the figure. The net electric flux through the three
net electric flux through the three adjacent sides is zero. The net electric flux through each of the other three sides is
a) $q / 3 \epsilon_{0}$.
b) $q / 6 \epsilon_{0}$.
c) $q / 24 \epsilon_{0}$.
d) $q / 8 \epsilon_{0}$.

Vishal Gupta

Numerade Educator

Problem 9

Three $-9-\mathrm{mC}$ point charges are located at (0,0) $(3 \mathrm{~m}, 3 \mathrm{~m})$, and $(3 \mathrm{~m},-3 \mathrm{~m})$. What is the magnitude of the electric field at $(3 \mathrm{~m}, 0) ?$
a) $0.9 \cdot 10^{7} \mathrm{~N} / \mathrm{C}$
b) $1.2 \cdot 10^{7} \mathrm{~N} / \mathrm{C}$
c) $1.8 \cdot 10^{7} \mathrm{~N} / \mathrm{C}$
d) $2.4 \cdot 10^{7} \mathrm{~N} / \mathrm{C}$
e) $3.6 \cdot 10^{7} \mathrm{~N} / \mathrm{C}$
f) $5.4 \cdot 10^{7} \mathrm{~N} / \mathrm{C}$
g) $10.8 \cdot 10^{7} \mathrm{~N} / \mathrm{C}$

Ajay Singhal

Numerade Educator

Problem 10

Which of the following statements is (are) true?
a) There will be no change in the charge on the inner surface of a hollow conducting sphere if additional charge is placed on the outer surface.
b) There will be some change in the charge on the inner surface of a hollow conducting sphere if additional charge is placed on the outer surface.
c) There will be no change in the charge on the inner surface of a hollow conducting sphere if additional charge is placed at the center of the sphere.
d) There will be some change in the charge on the inner surface of a hollow conducting sphere if additional charge is placed at the center of the sphere.

Ajay Singhal

Numerade Educator

Problem 11

Many people had been sitting in a car when it was struck by lightning. Why were they able to survive such an experience?

Ajay Singhal

Numerade Educator

Problem 12

Why is it a bad idea to stand under a tree in a thunderstorm? What should one do instead to avoid getting struck by lightning?

Ajay Singhal

Numerade Educator

Problem 13

Why do electric field lines never cross?

Ajay Singhal

Numerade Educator

Problem 14

How is it possible that the flux through a closed surface does not depend on where inside the surface the charge is located (that is, the charge can be moved around inside the surface with no effect whatsoever on the flux)? If the charge is moved from just inside to just outside the surface, the flux changes discontinuously to zero, according to Gauss's Law. Does this really happen? Explain.

Ajay Singhal

Numerade Educator

Problem 15

A solid conducting sphere of radius $r_{1}$ has a total charge of $+3 Q .$ It is placed inside (and concentric with) a conducting spherical shell of inner radius $r_{2}$ and outer radius $r_{3}$. Find the electric field in these regions: $r<r_{1}$, $r_{1}<r<r_{2}, r_{2}<r<r_{3}$, and $r>r_{3}$.

Pawan Yadav

Numerade Educator

Problem 16

A thin rod has end points at $x=\pm 100 \mathrm{~cm} .$ There is a charge of $Q$ uniformly distributed along the rod.
a) What is the electric field very close to the midpoint of the rod?
b) What is the electric field a few centimeters (perpendicularly) from the midpoint of the rod?
c) What is the electric field very far (perpendicularly) from the midpoint of the rod?

Juan Vazquez

Numerade Educator

Problem 17

A dipole is completely enclosed by a spherical surface. Describe how the total electric flux through this surface varies with the strength of the dipole.

Ajay Singhal

Numerade Educator

Problem 18

Repeat Example 22.3 , assuming that the charge distribution is $-\lambda$ for $-a<x<0$ and $+\lambda$ for $0<x<a$.

Keshav Singh

Numerade Educator

Problem 19

A negative charge is placed on a solid prolate spheroidal conductor (shown in cross section in the figure). Sketch the distribution of the charge on the conductor and the electric field lines due to the charge.

Ajay Singhal

Numerade Educator

Problem 20

Saint Elmos fire is an eerie glow that appears at the tips of masts and yardarms of sailing ships in stormy weather and at the tips and edges of the wings of aircraft in flight. St. Elmo's fire is an electrical phenomenon. Explain it, concisely.

Keshav Singh

Numerade Educator

Problem 21

A charge placed on a conductor of any shape forms a layer on the outer surface of the conductor. Mutual repulsion of the individual charge elements creates an outward pressure on this layer, called electrostatic stress. Treating the infinitesimal charge elements like tiles of a mosaic, calculate the magnitude of this electrostatic stress in terms of the surface charge density, $\sigma .$ Note that $\sigma$ need not be uniform over the surface.

Keshav Singh

Numerade Educator

Problem 22

An electric dipole is placed in a uniform electric field as shown in the figure. What motion will the dipole have in the electric field? Which way will it move? Which way will it rotate?

Ajay Singhal

Numerade Educator

Problem 23

A point charge, $q=4.00 \cdot 10^{-9} \mathrm{C},$ is placed on the $x$ -axis at the origin. What is the electric field produced at $x=25.0 \mathrm{~cm} ?$

Prashant Bana

Numerade Educator

Problem 24

$\mathrm{~A}+1.6-\mathrm{nC}$ point charge is placed at one corner of a square $(1.0 \mathrm{~m}$ on a side $),$ and $\mathrm{a}-2.4-\mathrm{n} \mathrm{C}$ charge is placed on the corner diagonally opposite. What is the magnitude of the electric field at either of the other two corners?

Vishal Gupta

Numerade Educator

Problem 25

$ \mathrm{~A}+48.00-\mathrm{nC}$ point charge is placed on the $x$ -axis at $x=4.000 \mathrm{~m},$ and $\mathrm{a}-24.00-\mathrm{n} \mathrm{C}$ point charge is placed on the $y$ -axis at $y=-6.000 \mathrm{~m} .$ What is the direction of the electric field at the origin?

Ajay Singhal

Numerade Educator

Problem 26

Two point charges are placed at two of the corners of a triangle as shown in the figure. Find the magnitude and the direction of the electric field at the third corner of the triangle.

Ajay Singhal

Numerade Educator

Problem 27

$\mathrm{~A}+5.0-\mathrm{C}$ charge is located at the origin. $\mathrm{A}-3.0-\mathrm{C}$
charge is placed at $x=1.0 \mathrm{~m}$. At what finite distance(s) along the $x$ -axis will the electric field be equal to zero?

Vishal Gupta

Numerade Educator

Problem 28

Three charges are on the $y$ -axis. Two of the charges, each $-q,$ are located $y=\pm d,$ and the third charge, $+2 q,$ is located at $y=0 .$ Derive an expression for the electric field at a point $P$ on the $x$ -axis.

Vishal Gupta

Numerade Educator

Problem 29

For the electric dipole shown in the figure, express the magnitude of the resulting electric field as a function of the perpendicular distance $x$ from the center of the dipole axis. Comment on what the magnitude is when $x \gg d$.

Ajay Singhal

Numerade Educator

Problem 30

Consider an electric dipole on the $x$ -axis and centered at the origin. At a distance $h$ along the positive $x$ -axis, the magnitude of electric field due to the electric dipole is given by $k(2 q d) / h^{3} .$ Find a distance perpendicular to the $x$ axis and measured from the origin at which the magnitude of the electric field stays the same.

Vishal Gupta

Numerade Educator

Problem 31

A small metal ball with a mass of $4.0 \mathrm{~g}$ and a charge of $5.0 \mathrm{mC}$ is located at a distance of $0.70 \mathrm{~m}$ above the ground in an electric field of $12 \mathrm{~N} / \mathrm{C}$ directed to the east. The ball is then released from rest. What is the velocity of the ball after it has moved downward a vertical distance of $0.30 \mathrm{~m} ?$

Keshav Singh

Numerade Educator

Problem 32

A charge per unit length $+\lambda$ is uniformly distributed along the positive $y$ -axis from $y=0$ to $y=+a$. A charge per unit length $-\lambda$ is uniformly distributed along the negative $y$ axis from $y=0$ to $y=-a$. Write an expression for the electric field (magnitude and direction) at a point on the $x$ -axis a distance $x$ from the origin.

Keshav Singh

Numerade Educator

Problem 33

A thin glass rod is bent into a semicircle of radius $R$. A charge $+Q$ is uniformly distributed along the upper half, and a charge $-Q$ is uniformly distributed along the lower half as shown in the figure. Find the magnitude and direction of the electric field $\vec{E}$ (in component form) at point $P$, the center of the semicircle.

Vishal Gupta

Numerade Educator

Problem 34

Two uniformly charged insulating rods are bent in a semicircular shape with radius $r=10.0 \mathrm{~cm} .$ If they are positioned so they form a circle but do not touch and have opposite charges of $+1.00 \mu \mathrm{C}$ and $-1.00 \mu \mathrm{C}$ find the magnitude and direction of the electric field at the center of the composite circular charge configuration.

Vishal Gupta

Numerade Educator

Problem 35

A uniformly charged rod of length $L$ with total charge $Q$ lies along the $y$ -axis, from $y=0$ to $y=L$. Find an expression for the electric field at the point $(d, 0)$ (that is, the point at $x=d$ on the $x$ -axis).

Keshav Singh

Numerade Educator

Problem 36

A charge $Q$ is distributed evenly on a wire bent into an arc of radius $R$, as shown in the figure. What is the electric field at the center of the
arc as a function of the angle $\theta$ ? Sketch a graph of the electric field as a function of $\theta$ for $0<\theta<180^{\circ}$.

Keshav Singh

Numerade Educator

Problem 37

A thin, flat washer is a disk with an outer diameter of $10.0 \mathrm{~cm}$ and a hole in the center with a diameter of $4.00 \mathrm{~cm} .$ The washer has a uniform charge distribution and a total charge of $7.00 \mathrm{nC}$. What is the electric field on the axis of the washer at a distance of $30.0 \mathrm{~cm}$ from the center of the washer?

Keshav Singh

Numerade Educator

Problem 38

Research suggests that the electric fields in some thunderstorm clouds can be on the order of $10.0 \mathrm{kN} / \mathrm{C}$. Calculate the magnitude of the electric force acting on a particle with two excess electrons in the presence of a $10.0-\mathrm{kN} / \mathrm{C}$ field.

Ajay Singhal

Numerade Educator

Problem 39

An electric dipole has opposite charges of $5.00 \cdot 10^{-15} \mathrm{C}$ separated by a distance of $0.400 \mathrm{~mm} .$ It is oriented at $60.0^{\circ}$ with respect to a uniform electric field of magnitude $2.00 \cdot 10^{3} \mathrm{~N} / \mathrm{C}$. Determine the magnitude of the torque exerted on the dipole by the electric field.

Ajay Singhal

Numerade Educator

Problem 40

Electric dipole moments of molecules are often measured in debyes $(\mathrm{D}),$ where $1 \mathrm{D}=3.34 \cdot 10^{-30} \mathrm{C} \mathrm{m} .$ For instance, the dipole moment of hydrogen chloride gas molecules is $1.05 \mathrm{D}$. Calculate the maximum torque such a molecule can experience in the presence of an electric field of magnitude $160.0 \mathrm{~N} / \mathrm{C}$.

Ajay Singhal

Numerade Educator

Problem 41

An electron is observed traveling at a speed of $27.5 \cdot 10^{6} \mathrm{~m} / \mathrm{s}$ parallel to an electric field of magnitude $11,400 \mathrm{~N} / \mathrm{C}$ How far will the electron travel before coming to a stop?

Vishal Gupta

Numerade Educator

Problem 42

Two charges, $+e$ and $-e,$ are a distance of $0.68 \mathrm{nm}$ apart in an electric field, $E,$ that has a magnitude of $4.4 \mathrm{kN} / \mathrm{C}$ and is directed at an angle of $45^{\circ}$ with the dipole axis. Calculate the dipole moment and thus the torque on the dipole in the electric field.

Vishal Gupta

Numerade Educator

Problem 43

A body of mass $M$, carrying charge $Q$, falls from rest from a height $h$ (above the ground) near the surface of the Earth, where the gravitational acceleration is $g$ and there is an electric field with a constant component $E$ in the vertical direction.
a) Find an expression for the speed, $v,$ of the body when it reaches the ground, in terms of $M, Q, h, g,$ and $E$.
b) The expression from part (a) is not meaningful for certain values of $M, g, Q,$ and $E$. Explain what happens in such cases.

Supratim Pal

Numerade Educator

Problem 44

A water molecule, which is electrically neutral but has a dipole moment of magnitude $p=6.20 \cdot 10^{-30} \mathrm{C} \mathrm{m},$ is $1.00 \mathrm{~cm}$ away from a point charge $q=+1.00 \mu \mathrm{C} .$ The dipole will align with the electric field due to the charge. It will also experience a net force, since the field is not uniform.
a) Calculate the magnitude of the net force. (Hint: You do not need to know the precise size of the molecule, only that it is much smaller than $1 \mathrm{~cm} .)$
b) Is the molecule attracted to or repelled by the point charge? Explain.

Vishal Gupta

Numerade Educator

Problem 45

A total of $3.05 \cdot 10^{6}$ electrons are placed on an initially uncharged wire of length $1.33 \mathrm{~m}$.
a) What is the magnitude of the electric field a perpendicular distance of $0.401 \mathrm{~m}$ away from the midpoint of the wire?
b) What is the magnitude of the acceleration of a proton placed at that point in space?
c) In which direction does the electric field force point in this case?

Vishal Gupta

Numerade Educator

Problem 46

Four charges are placed in a three-dimensional space. The charges have magnitudes $+3 q,-q,+2 q,$ and $-7 q .$ If a Gaussian surface encloses all the charges, what will be the electric flux through that surface?

Ajay Singhal

Numerade Educator

Problem 47

The six faces of a cubical box each measure $20.0 \mathrm{~cm}$ by $20 \mathrm{~cm},$ and the faces are numbered such that faces 1 and 6 are opposite to each other, as are faces 2 and $5,$ and faces 3 and $4 .$ The flux through each face is:

Vishal Gupta

Numerade Educator

Problem 48

A conducting solid sphere $\left(R=0.15 \mathrm{~m}, q=6.1 \cdot 10^{-6} \mathrm{C}\right)$ is
shown in the figure. Using Gauss's Law and two different Gaussian surfaces, determine the electric field (magnitude and direction) at point $A$, which is $0.000001 \mathrm{~m}$ outside the conducting sphere. (Hint: One Gaussian surface is a sphere, and the other is a small right cylinder.)

Keshav Singh

Numerade Educator

Problem 49

Electric fields of varying magnitudes are directed either inward or outward at right angles on the faces of a cube, as shown in the figure. What is the strength and direction of the field on the face $F ?$

Ajay Singhal

Numerade Educator

Problem 50

Consider a hollow spherical conductor with total charge $+5 e$. The outer and inner radii are $a$ and $b$, respectively. (a) Calculate the charge on the sphere's inner and outer surfaces if a charge of $-3 e$ is placed at the center of the sphere. (b) What is the total net charge of the sphere?

Ajay Singhal

Numerade Educator

Problem 51

A spherical aluminized Mylar balloon carries a charge $Q$ on its surface. You are measuring the electric field at a distance $R$ from the balloon's center. The balloon is slowly inflated, and its radius approaches but never reaches R. What happens to the electric field you measure as the balloon increases in radius. Explain.

Ajay Singhal

Numerade Educator

Problem 52

A hollow conducting spherical shell has an inner radius of $8.00 \mathrm{~cm}$ and an outer radius of $10.0 \mathrm{~cm} .$ The electric
field at the inner surface of the shell, $E_{\mathrm{i}}$, has a magnitude of $80.0 \mathrm{~N} / \mathrm{C}$ and points toward the center of the sphere, and the electric field at the outer surface, $E_{\infty}$ has a magnitude of $80.0 \mathrm{~N} / \mathrm{C}$ and points away from the center of the sphere (see the figure). Determine the magnitude of the charge on the inner surface and the outer surface of the spherical shell.

Keshav Singh

Numerade Educator

Problem 53

A $-6.00-n C$ point charge is located at the center of a conducting spherical shell. The shell has an inner radius of $2.00 \mathrm{~m},$ an outer radius of $4.00 \mathrm{~m},$ and a charge of $+7.00 \mathrm{nC}$
a) What is the electric field at $r=1.00 \mathrm{~m} ?$
b) What is the electric field at $r=3.00 \mathrm{~m} ?$
c) What is the electric field at $r=5.00 \mathrm{~m} ?$
d) What is the surface charge distribution, $\sigma,$ on the outside surface of the shell?

Keshav Singh

Numerade Educator

Problem 54

A solid, nonconducting sphere of radius $a$ has total charge $Q$ and a uniform charge distribution. Using Gauss's Law, determine the electric field (as a vector) in the regions $r<a$ and $r>a$ in terms of $Q$.

Keshav Singh

Numerade Educator

Problem 55

There is an electric field of magnitude $150 . \mathrm{N} / \mathrm{C}, \mathrm{di}-$ rected downward, near the surface of the Earth. What is the net electric charge on the Earth? You can treat the Earth as a spherical conductor of radius $6371 \mathrm{~km} .$

Vishal Gupta

Numerade Educator

Problem 56

A hollow metal sphere has inner and outer radii of $20.0 \mathrm{~cm}$ and $30.0 \mathrm{~cm},$ respectively. As shown in the figure, a solid metal sphere of radius $10.0 \mathrm{~cm}$ is located at the center of the hollow sphere. The electric field at a point $P,$ a distance of $15.0 \mathrm{~cm}$ from the center, is found to be $E_{1}=1.00 \cdot 10^{4} \mathrm{~N} / \mathrm{C},$ directed radially inward. At point $Q$,a distance of $35.0 \mathrm{~cm}$ from the center, the electric field is found to be $E_{2}=1.00 \cdot 10^{4} \mathrm{~N} / \mathrm{C}$
directed radially outward. Determine the total charge on
(a) the surface of the inner sphere, (b) the inner surface of the hollow sphere, and (c) the outer surface of the hollow sphere.

Keshav Singh

Numerade Educator

Problem 57

Two parallel, infinite, nonconducting plates are $10.0 \mathrm{~cm}$ apart and have charge distributions of $+1.00 \mu \mathrm{C} / \mathrm{m}^{2}$ and $-1.00 \mu \mathrm{C} / \mathrm{m}^{2} .$ What is the force on an electron in the space between the plates? What is the force on an electron located outside the two plates near the surface of one of the two plates?

Ajay Singhal

Numerade Educator

Problem 58

An infinitely long charged wire produces an electric field of magnitude $1.23 \cdot 10^{3} \mathrm{~N} / \mathrm{C}$ at a distance of $50.0 \mathrm{~cm}$ perpendicular to the wire. The direction of the electric field is toward the wire.
a) What is the charge distribution?
b) How many electrons per unit length are on the wire?

Ajay Singhal

Numerade Educator

Problem 59

A solid sphere of radius $R$ has a nonuniform charge distribution $\rho=A r^{2},$ where $A$ is a constant. Determine the total charge, $Q$, within the volume of the sphere.

Ajay Singhal

Numerade Educator

Problem 60

Two parallel, uniformly charged, infinitely long wires carry opposite charges with a linear charge density $\lambda=1.00 \mu \mathrm{C} / \mathrm{m}$ and are $6.00 \mathrm{~cm}$ apart. What is the magnitude and direction of the electric field at a point midway between them and $40.0 \mathrm{~cm}$ above the plane containing the two wires?

Keshav Singh

Numerade Educator

Problem 61

A sphere centered at the origin has a volume charge distribution of $120 \mathrm{nC} / \mathrm{cm}^{3}$ and a radius of $12 \mathrm{~cm}$. The sphere is centered inside a conducting spherical shell with an inner radius of $30.0 \mathrm{~cm}$ and an outer radius of $50.0 \mathrm{~cm}$. The charge on the spherical shell is $-2.0 \mathrm{mC}$. What is the magnitude and direction of the electric field at each of the following distances from the origin?
a) at $r=10.0 \mathrm{~cm}$
c) at $r=40.0 \mathrm{~cm}$
b) at $r=20.0 \mathrm{~cm}$
d) at $r=80.0 \mathrm{~cm}$

Meghan Miholics

Meghan Miholics

Numerade Educator

Problem 62

A thin, hollow, metal cylinder of radius $R$ has a surface charge distribution $\sigma$. A long, thin wire with a linear charge density $\lambda / 2$ runs through the center of the cylinder. Find an expression for the electric fields and the direction of the field at each of the following locations:
a) $r \leq R$
b) $r \geq R$

Keshav Singh

Numerade Educator

Problem 63

Two infinite sheets of charge are separated by $10.0 \mathrm{~cm}$ as shown in the figure. Sheet 1 has a surface charge distribution of $\sigma_{1}=3.00 \mu \mathrm{C} / \mathrm{m}^{2}$ and sheet 2 has a surface charge distribution of $\sigma_{2}=-5.00 \mu \mathrm{C} / \mathrm{m}^{2}$. Find the total electric field (magnitude and direction) at each of the following locations:
a) at point $P, 6.00 \mathrm{~cm}$ to the left of sheet 1
b) at point $P^{\prime} 6.00 \mathrm{~cm}$ to the right of sheet 1

Ajay Singhal

Numerade Educator

Problem 64

A conducting solid sphere of radius $20.0 \mathrm{~cm}$ is located with its center at the origin of a three-dimensional coordinate system. A charge of $0.271 \mathrm{nC}$ is placed on the sphere.
a) What is the magnitude of the electric field at point $(x, y, z)=$ $(23.1 \mathrm{~cm}, 1.1 \mathrm{~cm}, 0 \mathrm{~cm}) ?$
b) What is the angle of this electric field with the $x$ -axis at this point?
c) What is the magnitude of the electric field at point $(x, y, z)=$
$(4.1 \mathrm{~cm}, 1.1 \mathrm{~cm}, 0 \mathrm{~cm}) ?$

Keshav Singh

Numerade Educator

Problem 65

A solid nonconducting sphere of radius $a$ has a total charge $+Q$ uniformly distributed throughout its volume. The surface of the sphere is coated with a very thin (negligible thickness) conducting layer of gold. A total charge of $-2 Q$ is placed on this conducting layer. Use Gauss's Law to do the following.
a) Find the electric field $E(r)$ for $r<a$ (inside the sphere, up to and excluding the gold layer).
b) Find the electric field $E(r)$ for $r>a$ (outside the coated sphere, beyond the sphere and the gold layer).
c) Sketch the graph of $E(r)$ versus $r$. Comment on the continuity or discontinuity of the electric field, and relate this to the surface charge distribution on the gold layer.

Keshav Singh

Numerade Educator

Problem 66

A solid nonconducting sphere has a volume charge distribution given by $\rho(r)=(\beta / r) \sin (\pi r / 2 R) .$ Find the total charge contained in the spherical volume and the electric field in the regions $r<R$ and $r>R$. Show that the two expressions for the electric field equal each other at $r=R$.

LA

Lazarus Arnau

Numerade Educator

Problem 67

A very long cylindrical rod of nonconducting material with a $3.00-\mathrm{cm}$ radius is given a uniformly distributed positive charge of $6.00 \mathrm{nC}$ per centimeter of its length. Then a cylindrical cavity is drilled all the way through the rod, of radius $1 \mathrm{~cm},$ with its axis located $1.50 \mathrm{~cm}$ from the axis of the rod. That is, if, at some cross section of the rod, $x$ and $y$ -axes are placed so that the center of the rod is at $(x, y)$ $=(0,0) ;$ then the center of the cylindrical cavity is at $(x, y)$ $=(0,1.50) .$ The creation of the cavity does not disturb the charge on the remainder of the rod that has not been drilled away; it just removes the charge from the region in the cavity. Find the electric field at the point $(x, y)=(2.00,1.00)$.

Linda Winkler

Numerade Educator

Problem 68

What is the electric field at a point $P$, a distance $h=20.0 \mathrm{~cm}$ above an infinite sheet of charge, with a charge distribution of $1.3 \mathrm{C} / \mathrm{m}^{2}$ and hole of radius $5.0 \mathrm{~cm}$ with $P$ directly above the center of the hole, as shown in the figure? Plot the electric field as a function of $h$ in units of $\sigma /\left(2 \epsilon_{0}\right)$.

Narayan Hari

Numerade Educator

Problem 69

A cube has an edge length of $1.00 \mathrm{~m} .$ An electric field acting on the cube from outside has a constant magnitude of $150 \mathrm{~N} / \mathrm{C}$ and its direction is also constant but unspecified (not necessarily along any edges of the cube). What is the total charge within the cube?

Ajay Singhal

Numerade Educator

Problem 70

Consider a long horizontally oriented conducting wire with $\lambda=4.81 \cdot 10^{-12} \mathrm{C} / \mathrm{m} .$ A proton $\left(\mathrm{mass}=1.67 \cdot 10^{-27} \mathrm{~kg}\right)$
is placed $0.620 \mathrm{~m}$ above the wire and released. What is the magnitude of the initial acceleration of the proton?

Vishal Gupta

Numerade Educator

Problem 71

An infinitely long, solid cylinder of radius $R=9.00 \mathrm{~cm},$ with a uniform charge per unit of volume of $\rho=6.40 \cdot 10^{-8} \mathrm{C} / \mathrm{m}^{3},$ is centered about the $y$ -axis. Find the magnitude of the electric field at a radius $r=4.00 \mathrm{~cm}$ from the center of this cylinder.

Ajay Singhal

Numerade Educator

Problem 72

Carbon monoxide (CO) has a dipole moment of approximately $8.0 \cdot 10^{-30} \mathrm{C} \mathrm{m} .$ If the two atoms are separated by $1.2 \cdot 10^{-10} \mathrm{~m}$, find the net charge on each atom and the maximum amount of torque the molecule would experience in an electric field of $500.0 \mathrm{~N} / \mathrm{C}$.

Vishal Gupta

Numerade Educator

Problem 73

A solid metal sphere of radius $8.00 \mathrm{~cm},$ with a total charge of $10.0 \mu C$, is surrounded by a metallic shell with a radius of $15.0 \mathrm{~cm}$ carrying a $-5.00 \mu \mathrm{C}$ charge. The sphere and the shell are both inside a larger metallic shell of inner radius $20.0 \mathrm{~cm}$ and outer radius $24.0 \mathrm{~cm} .$ The sphere and the two shells are concentric.
a) What is the charge on the inner wall of the larger shell?
b) If the electric field outside the larger shell is zero, what is the charge on the outer wall of the shell?

Ajay Singhal

Numerade Educator

Problem 74

Find the vector electric fields needed to counteract the weight of (a) an electron and (b) a proton at the Earth's surface.

Vishal Gupta

Numerade Educator

Problem 75

There is an electric field of magnitude $150 .$ N/C, directed vertically downward, near the surface of the Earth. Find the acceleration (magnitude and direction) of an electron released near the Earth's surface.

Ajay Singhal

Numerade Educator

Problem 76

Two infinite, uniformly charged, flat nonconducting surfaces are mutually perpendicular. One of the surfaces has a charge distribution of $+30.0 \mathrm{pC} / \mathrm{m}^{2}$, and the other has a charge distribution of $-40.0 \mathrm{pC} / \mathrm{m}^{2}$. What is the magnitude of the electric field at any point not on either surface?

Ajay Singhal

Numerade Educator

Problem 77

A 30.0 -cm-long uniformly charged rod is sealed in a container. The total electric flux leaving the container is $1.46 \cdot 10^{6} \mathrm{~N} \mathrm{~m}^{2} / \mathrm{C}$. Determine the linear charge distribution on the rod.

Nishant Kumar

Numerade Educator

Problem 78

Suppose you have a large spherical balloon and you are able to measure the component $E_{n}$ of the electric field normal to its surface. If you sum $E_{n} d A$ over the whole surface area of the balloon and obtain a magnitude of $10 \mathrm{~N} \mathrm{~m}^{2} / \mathrm{C}$ what is the electric charge enclosed by the balloon?

Ajay Singhal

Numerade Educator

Problem 79

An object with mass $m=1.0 \mathrm{~g}$ and charge $q$ is placed at point $A$, which is $0.05 \mathrm{~m}$ above an infinitely large, uniformly charged, nonconducting sheet $\left(\sigma=-3.5 \cdot 10^{-5} \mathrm{C} / \mathrm{m}^{2}\right)$, as shown in the figure. Gravity is acting downward $\left(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\right)$.
Determine the number, $N$, of electrons that must be added to or removed from the object for the object to remain motionless above the charged plane.

Vishal Gupta

Numerade Educator

Problem 80

A long conducting wire with charge distribution $\lambda$ and radius $r$ produces an electric field of $2.73 \mathrm{~N} / \mathrm{C}$ just outside the surface of the wire. What is the magnitude of the electric field just outside the surface of another wire with charge distribution $0.81 \lambda$ and radius $6.5 r ?$

Vishal Gupta

Numerade Educator

Problem 81

There is a uniform charge distribution of $\lambda=$ $8.00 \cdot 10^{-8} \mathrm{C} / \mathrm{m}$ along a thin wire of length $L=6.00 \mathrm{~cm}$
The wire is then curved into a semicircle that is centered about the origin, so the radius of the semicircle is $R=L / \pi .$ Find the magnitude of the electric field at the center of the semicircle.

Vishal Gupta

Numerade Educator

Problem 82

A proton enters the gap between a pair of metal plates (an electrostatic separator) that produces a uniform, vertical electric field between them. Ignore the effect of gravity on the proton.
a) Assuming that the length of the plates is $15.0 \mathrm{~cm}$, and that the proton will approach the plates at a speed of $15.0 \mathrm{~km} / \mathrm{s}$ what electric field strength should the plates be designed to provide, if the proton must be deflected vertically by $1.50 \cdot 10^{-3} \mathrm{rad} ?$
b) What speed does the proton have after exiting the electric field?
c) Suppose the proton is one in a beam of protons that has been contaminated with positively charged kaons, particles whose mass is $494 \mathrm{MeV} / \mathrm{c}^{2}\left(8.81 \cdot 10^{-28} \mathrm{~kg}\right)$, compared to the
mass of the proton, which is $938 \mathrm{MeV} / \mathrm{c}^{2}\left(1.67 \cdot 10^{-27} \mathrm{~kg}\right)$
The kaons have $+1 e$ charge, just like the protons. If the electrostatic separator is designed to give the protons a deflection of $1.20 \cdot 10^{-3} \mathrm{rad}$, what deflection will kaons with the same momentum as the protons experience?

Janielle Madlansacay

Janielle Madlansacay

Numerade Educator

Problem 83

Consider a uniform nonconducting sphere with a charge $\rho=3.57 \cdot 10^{-6} \mathrm{C} / \mathrm{m}^{3}$ and a radius $R=1.72 \mathrm{~m}$. What is the magnitude of the electric field $0.530 \mathrm{~m}$ from the center of the sphere?

Vishal Gupta

Numerade Educator

Problem 84

A uniform sphere has a radius $R$ and a total charge $+Q,$ uniformly distributed throughout its volume. It is surrounded by a thick spherical shell carrying a total charge $-Q,$ also uniformly distributed, and having an outer radius of $2 R$. What is the electric field as a function of $R ?$

Rajesh Kumar

Numerade Educator

Problem 85

A uniform sphere has a radius $R$ and a total charge $+Q,$ uniformly distributed throughout its volume. It is surrounded by a thick spherical shell carrying a total charge $-Q,$ also uniformly distributed, and having an outer radius of $2 R$. What is the electric field as a function of $R ?$

Rajesh Kumar

Numerade Educator