University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 23

Electric Potential - all with Video Answers

Educators

Chapter Questions

Problem 1

A positive charge is released and moves along an electric field line. This charge moves to a position of
a) lower potential and lower potential energy.
b) lower potential and higher potential energy.
c) higher potential and lower potential energy.
d) higher potential and higher potential energy.

Ajay Singhal

Ajay Singhal

Numerade Educator

Problem 2

A proton is placed midway between points $A$ and $B$. The potential at point $A$ is $-20 \mathrm{~V}$, and the potential at point $B$ $+20 \mathrm{~V}$. The potential at the midpoint is $0 \mathrm{~V}$. The proton will
a) remain at rest.
b) move toward point $B$ with constant velocity.
c) accelerate toward point $A$.
d) accelerate toward point $B$.
e) move toward point $A$ with constant velocity.

Ajay Singhal

Numerade Educator

Problem 3

What would be the consequence of setting the potential at $+100 \mathrm{~V}$ at infinity, rather than taking it to be zero there?
a) Nothing; the field and the potential would have the same values at every finite point.
b) The electric potential would become infinite at every finite point, and the electric field could not be defined.
c) The electric potential everywhere would be $100 \mathrm{~V}$ higher, and the electric field would be the same.
d) It would depend on the situation. For example, the potential due to a positive point charge would drop off more slowly with distance, so the magnitude of the electric field would be less.

Ajay Singhal

Numerade Educator

Problem 4

In which situation is the electric potential the highest?
a) at a point $1 \mathrm{~m}$ from a point charge of $1 \mathrm{C}$
b) at a point $1 \mathrm{~m}$ from the center of a uniformly charged spherical shell of radius $0.5 \mathrm{~m}$ with a total charge of $1 \mathrm{C}$
c) at a point $1 \mathrm{~m}$ from the center of a uniformly charged rod of length $1 \mathrm{~m}$ and with a total charge of $1 \mathrm{C}$
d) at a point $2 \mathrm{~m}$ from a point charge of $2 \mathrm{C}$
e) at a point $0.5 \mathrm{~m}$ from a point charge of $0.5 \mathrm{C}$

Austin Lewandowski

Austin Lewandowski

Numerade Educator

Problem 5

The amount of work done to move a positive point charge $q$ on an equipotential surface of $1000 \mathrm{~V}$ relative to that on an equipotential surface of $10 \mathrm{~V}$ is
a) the same.
d) dependent on the
b) less. distance the charge
c) more. moves.

Ajay Singhal

Numerade Educator

Problem 6

A solid conducting sphere of radius $R$ is centered about the origin of an $x y z$ -coordinate system. A total charge $Q$ is distributed uniformly on the surface of the sphere. Assuming, as usual, that the electric potential is zero at an infinite distance, what is the electric potential at the center of the conducting sphere?
a) zero
c) $Q / 2 \pi \epsilon_{0} R$
b) $Q / \epsilon_{0} R$
d) $Q / 4 \pi \epsilon_{0} R$

Austin Lewandowski

Austin Lewandowski

Numerade Educator

Problem 7

Which of the following angles between an electric dipole moment and an applied electric field will result in the most stable state?
a) $0 \mathrm{rad}$
d) The electric dipole moment is
b) $\pi / 2$ rad not stable under any condition in
c) $\pi$ rad an applied electric field.

Ajay Singhal

Numerade Educator

Problem 8

A positive point charge is to be moved from point $A$ to point $B$ in the vicinity of an electric dipole. Which of the three paths shown in the figure will result in the most work being done by the dipole's electric field on the point charge?
a) path 1
b) path 2
c) path 3
d) The work is the same on all three paths.

Ajay Singhal

Numerade Educator

Problem 9

Each of the following pairs of charges are separated by a distance $d$. Which pair has the highest potential energy?
a) $+5 \mathrm{C}$ and $+3 \mathrm{C}$
d) $-5 \mathrm{C}$ and $+3 \mathrm{C}$
b) $+5 \mathrm{C}$ and $-3 \mathrm{C}$
e) All pairs have the
$\begin{array}{ll}\text { c) }-5 \mathrm{C} \text { and }-3 \mathrm{C} & \text { same potential energy. }\end{array}$

Austin Lewandowski

Austin Lewandowski

Numerade Educator

Problem 10

A negatively charged particle revolves in a clockwise direction around a positively charged sphere. The work done on the negatively charged particle by the electric field of the sphere is
a) positive.
b) negative.
c) zero.

Ajay Singhal

Numerade Educator

Problem 11

High-voltage power lines are used to transport electricity cross country. These wires are favored resting places for birds. Why don't the birds die when they touch the wires?

Ajay Singhal

Numerade Educator

Problem 13

Can two equipotential lines cross? Why or why not?

Ajay Singhal

Numerade Educator

Problem 14

Why is it important, when soldering connectors onto a piece of electronic circuitry, to leave no pointy protrusions from the solder joints?

Ajay Singhal

Numerade Educator

Problem 15

Using Gauss's Law and the relation between electric potential and electric field, show that the potential outside a uniformly charged sphere is identical to the potential of a point charge placed at the center of the sphere and equal to the total charge of the sphere. What is the potential at the surface of the sphere? How does the potential change if the charge distribution is not uniform but has spherical (radial) symmetry?

Prem Bijarniya

Numerade Educator

Problem 16

A metal ring has a total charge $q$ and radius $R$, as shown in the figure. Without performing any calculations, predict the value of the electric potential and electric field at the center of the circle

Ajay Singhal

Numerade Educator

Problem 17

Find an integral expression for the electric potential at a point on the $z$ -axis a distance $H$ from a half-disk of radius $R$ (see the figure). The half-disk has uniformly distributed charge over its surface, with charge distribution $\sigma$.

Prem Bijarniya

Numerade Educator

Problem 18

An electron moves away from a proton. Describe how the potential it encounters changes. Describe how its potential energy is changing.

Rashmi Sinha

Numerade Educator

Problem 19

The electric potential energy of a continuous charge distribution can be found in a way similar to that used for systems of point charges in Section $23.6,$ by breaking the distribution up into suitable pieces. Find the electric potential energy of an arbitrary spherically symmetrical charge distribution, $\rho(r) .$ Do not assume that $\rho(r)$ represents a point charge, that it is constant, that it is piecewise-constant, or that it does or does not end at any finite radius, $r$. Your expression must cover all possibilities. Your expression may include an integral or integrals that cannot be evaluated without knowing the specific form of $\rho(r) .$ (Hint: A spherical pearl is built up of thin layers of nacre added one by one.)

Dominador Tan

Numerade Educator

Problem 20

In molecules of gaseous sodium chloride, the chloride ion has one more electron than proton, and the sodium ion has one more proton than electron. These ions are separated by about $0.24 \mathrm{nm}$. How much work would be required to increase the distance between these ions to $1.0 \mathrm{~cm} ?$

Eduard Sanchez

Numerade Educator

Problem 21

A metal ball with a mass of $3.00 \cdot 10^{-6} \mathrm{~kg}$ and a charge of $+5.00 \mathrm{mC}$ has a kinetic energy of $6.00 \cdot 10^{8} \mathrm{~J}$. It is traveling directly at an infinite plane of charge with a charge distribution of $+4.00 \mathrm{C} / \mathrm{m}^{2}$. If it is currently $1.00 \mathrm{~m}$ away from the plane of charge, how close will it come to the plane before stopping?

Ajay Singhal

Numerade Educator

Problem 22

An electron is accelerated from rest through a potential difference of $370 \mathrm{~V}$. What is its final speed?

Ajay Singhal

Numerade Educator

Problem 23

How much work would be done by an electric field in moving a proton from a point at a potential of $+180 . \mathrm{V}$ to a point at a potential of $-60.0 \mathrm{~V} ?$

Ajay Singhal

Numerade Educator

Problem 24

What potential difference is needed to give an alpha particle (composed of 2 protons and 2 neutrons) $200 \mathrm{keV}$ of kinetic energy?

Ajay Singhal

Numerade Educator

Problem 25

A proton, initially at rest, is accelerated through a potential difference of $500 .$ V. What is its final velocity?

Ajay Singhal

Numerade Educator

Problem 26

A $10.0-\mathrm{V}$ battery is connected to two parallel metal plates placed in a vacuum. An electron is accelerated from rest from the negative plate toward the positive plate.
a) What kinetic energy does the electron have just as it reaches the positive plate?
b) What is the speed of the electron just as it reaches the positive plate?

Ajay Singhal

Numerade Educator

Problem 27

A proton gun fires a proton from midway between two plates, $A$ and $B$, which are separated by a distance of $10.0 \mathrm{~cm} ;$ the proton initially moves at a speed of $150.0 \mathrm{~km} / \mathrm{s}$ toward plate $B$. Plate $A$ is kept at zero potential, and plate $B$ at a potential of $400.0 \mathrm{~V}$
a) Will the proton reach plate B?
b) If not, where will it turn around?
c) With what speed will it hit plate A?

Ajay Singhal

Numerade Educator

Problem 28

Fully stripped (all electrons removed) sulfur $\left({ }^{32} \mathrm{~S}\right)$ ions are accelerated in an accelerator from rest using a total voltage of $1.00 \cdot 10^{9} \mathrm{~V}$. ${ }^{32} \mathrm{~S}$ has 16 protons and 16 neutrons. The accelerator produces a beam consisting of $6.61 \cdot 10^{12}$ ions per second. This beam of ions is completely stopped in a beam dump. What is the total power the beam dump has to absorb?

Eduard Sanchez

Numerade Educator

Problem 29

Two point charges are located at two corners of a rectangle, as shown in the figure.
a) What is the electric potential at point $A ?$
b) What is the potential difference between points $A$ and $B ?$

Prem Bijarniya

Numerade Educator

Problem 30

Four identical point charges $(+1.61 \mathrm{nC})$ are placed at the corners of a rectangle, which measures $3.00 \mathrm{~m}$ by $5.00 \mathrm{~m}$. If the electric potential is taken to be zero at infinity, what is the potential at the geometric center of this rectangle?

Ajay Singhal

Numerade Educator

Problem 31

If a Van de Graff generator has an electric potential of $1.00 \cdot 10^{5} \mathrm{~V}$ and a diameter of $20.0 \mathrm{~cm},$ find how many more protons than electrons are on its surface.

Bettina Hanlon

Numerade Educator

Problem 32

One issue encountered during the exploration of Mars has been the accumulation of static charge on landroving vehicles, resulting in a potential of $100 . V$ or more. Calculate how much charge must be placed on the surface of a sphere of radius $1.00 \mathrm{~m}$ for the electric potential just above the surface to be $100 .$ V. Assume that the charge is uniformly distributed.

Eduard Sanchez

Numerade Educator

Problem 33

A charge $Q=$ $+5.60 \mu C$ is uniformly distributed on a thin cylindrical plastic shell. The radius, $R$, of the shell is $4.50 \mathrm{~cm}$. Calculate the electric potential at the origin of the $x y$ -coordinate system shown in the figure. Assume that the electric potential is zero at points infinitely far away from the origin.

Ajay Singhal

Numerade Educator

Problem 34

A hollow spherical conductor with a $5.0-\mathrm{cm}$ radius has a surface charge of $8.0 \mathrm{nC}$.
a) What is the potential $8.0 \mathrm{~cm}$ from the center of the sphere?
b) What is the potential $3.0 \mathrm{~cm}$ from the center of the sphere?
c) What is the potential at the center of the sphere?

Prem Bijarniya

Numerade Educator

Problem 35

Find the potential at the center of curvature of the (thin) wire shown in the figure. It has a (uniformly distributed) charge per unit length of $\lambda=$ $3.00 \cdot 10^{-8} \mathrm{C} / \mathrm{m}$ and a radius of
curvature of $R=8.00 \mathrm{~cm} .$

Ajay Singhal

Numerade Educator

Problem 36

Consider a dipole with charge $q$ and separation $d$. What is the potential a distance $x$ from the center of this dipole at an angle $\theta$ with respect to the dipole axis, as shown in the figure?

Prem Bijarniya

Numerade Educator

Problem 37

A spherical water drop $50.0 \mu \mathrm{m}$ in diameter has a uniformly distributed charge of $+20.0 \mathrm{pC}$. Find (a) the potential at its surface and (b) the potential at its center.

Prem Bijarniya

Numerade Educator

Problem 38

Consider an electron in the ground state of the hydrogen atom, separated from the proton by a distance of $0.0529 \mathrm{nm}$
a) Viewing the electron as a satellite orbiting the proton in the electrostatic potential, calculate the speed of the electron in its orbit.
b) Calculate an effective escape speed for the electron.
c) Calculate the energy of an electron having this speed, and from it determine the energy that must be given to the electron to ionize the hydrogen atom.

Eduard Sanchez

Numerade Educator

Problem 39

Four point charges are arranged in a square with side length $2 a$, where $a=2.7 \mathrm{~cm} .$ Three of the charges have magnitude $1.5 \mathrm{nC}$, and one of them has magnitude $-1.5 \mathrm{nC}$, as shown in the figure. What is the value of the electric potential generated by these \text { four point charges at point } P=(0,0, c), \text { where } c=4.1 \mathrm{~cm} ?

Eduard Sanchez

Numerade Educator

Problem 40

The plastic rod of length $L$ shown in the figure has the nonuniform linear charge distribution $\lambda=c x$, where $c$ is a positive constant. Find an expression for the electric potential at point $P$ on the $y$ -axis, a distance P $y$ from one end of the rod.

Ajay Singhal

Numerade Educator

Problem 41

An electric field varies in space according to this equation: $\vec{E}=E_{0} x e^{-x} \hat{x}$.
a) For what value of $x$ does the electric field have its largest value, $x_{\max } ?$
b) What is the potential difference between the points at $x=0$ and $x=x_{\max } ?$

Ajay Singhal

Numerade Educator

Problem 42

Derive an expression for electric potential along the axis (the $x$ -axis) of a disk with a hole in the center, as shown in the figure, where $R_{1}$ and $R_{2}$ are the inner and outer radii of the disk. What would the potential be if $R_{1}=0 ?$

Eduard Sanchez

Numerade Educator

Problem 43

An electric field is established in a nonuniform rod. A voltmeter is used to measure the potential difference between the left end of the rod and a point a distance $x$ from the left end. The process is repeated, and it is found that the data are described by the relationship $\Delta V=270 x^{2},$ where $\Delta V$ has the units $\mathrm{V} / \mathrm{m}^{2}$. What is the $x$ -component of the electric field at a point $13 \mathrm{~cm}$ from the left end?

Ajay Singhal

Numerade Educator

Problem 44

Two parallel plates are held at potentials of $+200.0 \mathrm{~V}$ and $-100.0 \mathrm{~V}$. The plates are separated by $1.00 \mathrm{~cm}$.
a) Find the electric field between the plates.
b) An electron is initially placed halfway between the plates. Find its kinetic energy when it hits the positive plate.

Ajay Singhal

Numerade Educator

Problem 45

A $2.50-\mathrm{mg}$ dust particle with a charge of $1.00 \mu \mathrm{C}$ falls at a point $x=2.00 \mathrm{~m}$ in a region where the electric potential varies according to $V(x)=\left(2.00 \mathrm{~V} / \mathrm{m}^{2}\right) x^{2}-\left(3.00 \mathrm{~V} / \mathrm{m}^{3}\right) x^{3}$
With what acceleration will the particle start moving after it touches down?

Ajay Singhal

Numerade Educator

Problem 46

The electric potential in a volume of space is given by $V(x, y, z)=x^{2}+x y^{2}+y z$. Determine the electric field in this region at the coordinate (3,4,5) .

Ajay Singhal

Numerade Educator

Problem 47

The electric potential inside a $10.0-\mathrm{m}$ -long linear particle accelerator is given by $V=\left(3000-5 x^{2} / \mathrm{m}^{2}\right) \mathrm{V},$ where $x$ is the distance from the left plate along the accelerator tube, as shown in the figure.
a) Determine an expression for the electric field along the accelerator tube.
b) A proton is released (from rest) at $x=4.00 \mathrm{~m}$. Calculate the acceleration of the proton just after it is released.
c) What is the impact speed of the proton when (and if) it collides with the plate?

Ajay Singhal

Numerade Educator

Problem 48

An infinite plane of charge has a uniform charge distribution of $+4.00 \mathrm{nC} / \mathrm{m}^{2}$ and is located in the $y z$ -plane at $x=0 . A+11.0 \mathrm{nC}$ fixed point charge is located at $x=+2.00 \mathrm{~m}$
a) Find the electric potential $V(x)$ on the $x$ -axis from $0<x<+2.00 \mathrm{~m}$
b) At what position(s) on the $x$ -axis between $x=0$ and $x=+2.00 \mathrm{~m}$ is the electric potential a minimum?
c) Where on the $x$ -axis between $x=0 \mathrm{~m}$ and $x=+2.00 \mathrm{~m}$ could a positive point charge be placed and not move?

Dominador Tan

Numerade Educator

Problem 49

Use $V=\frac{k q}{r}, E_{x}=-\frac{\partial V}{\partial x}, E_{y}=-\frac{\partial V}{\partial y},$ and $E_{z}=-\frac{\partial V}{\partial z}$ to
derive the expression for the electric field of a point charge, $q$

Eduard Sanchez

Numerade Educator

Problem 50

Show that an electron in a one-dimensional electri. cal potential $V(x)=A x^{2},$ where the constant $A$ is a positive real number, will execute simple harmonic motion about the origin. What is the period of that motion?

Eduard Sanchez

Numerade Educator

Problem 51

The electric field, $\vec{E}(\vec{r}),$ and the electric potential $V(\vec{r}),$ are calculated from the charge distribution, $\rho(\vec{r}),$ by integrating Coulomb's Law and then the electric field. In the other direction, the field and the charge distribution are determined from the potential by suitably differentiating. Suppose the electric potential in a large region of space is given by $V(r)=V_{0} \exp \left(-r^{2} / a^{2}\right),$ where $V_{0}$ and $a$ are constants and $r=\sqrt{x^{2}+y^{2}+z^{2}}$ is the distance from the origin.
a) Find the electric field $\vec{E}(\vec{r})$ in this region.
b) Determine the charge density $\rho(\vec{r})$ in this region, which gives rise to the potential and field.
c) Find the total charge in this region.
d) Roughly sketch the charge distribution that could give rise to such an electric field.

Dominador Tan

Numerade Educator

Problem 52

The electron beam emitted by an electron gun is controlled (steered) with two sets of parallel conducting plates: a horizontal set to control the vertical motion of the beam, and a vertical set to control the horizontal motion of the beam. The beam is emitted with an initial velocity of $2.00 \cdot 10^{7} \mathrm{~m} / \mathrm{s}$. The width of the plates is $d=5.00 \mathrm{~cm}$, the separation between the plates is $D=4.00 \mathrm{~cm},$ and the distance between the edge of the plates and a target screen is $L=40.0 \mathrm{~cm}$ In the absence of any applied voltage, the electron beam hits the origin of the $x y$ -coordinate system on the observation screen. What voltages need to be applied to the two sets of plates for the electron beam to hit a target placed on the observation screen at coordinates $(x, y)=(0 \mathrm{~cm}, 8.00 \mathrm{~cm}) ?$

Dominador Tan

Numerade Educator

Problem 53

Nuclear fusion reactions require that positively charged nuclei be brought into close proximity, against the electrostatic repulsion. As a simple example, suppose a proton is fired at a second, stationary proton from a large distance away. What kinetic energy must be given to the moving proton to get it to come within $1.00 \cdot 10^{-15} \mathrm{~m}$ of the target? Assume that there is a head-on collision and that the target is fixed in place.

Ajay Singhal

Numerade Educator

Problem 54

Fission of a uranium nucleus (containing 92 protons) produces a barium nucleus ( 56 protons) and a krypton nucleus ( 36 protons). The fragments fly apart as a result of electrostatic repulsion; they ultimately emerge with a total of $200 . \mathrm{MeV}$ of kinetic energy. Use this information to estimate the size of the uranium nucleus; that is, treat the barium and krypton nuclei as point charges and calculate the separation between them at the start of the process.

Ajay Singhal

Numerade Educator

Problem 55

A deuterium ion and a tritium ion each have charge $+e .$ What work is necessary to be done on the deuterium ion in order to bring it within $10^{-14} \mathrm{~m}$ of the tritium ion? This is the distance within which the two ions can fuse, as a result of strong nuclear interactions that overcome electrostatic repulsion, to produce a helium-5 nucleus. Express the work in electron-volts.

Eduard Sanchez

Numerade Educator

Problem 56

Three charges, $q_{1}, q_{2},$ and $q_{3},$ are located at the corners of an equilateral triangle with side length of $1.2 \mathrm{~m}$. Find the work done in each of the following cases:
a) to bring the first particle, $q_{1}=1.0 \mathrm{pC},$ to $P$ from infinity
b) to bring the second particle, $q_{2}=2.0 \mathrm{pC}$ to $Q$ from infinity
c) to bring the last particle, $q_{3}=3.0 \mathrm{pC}$ to $R$ from infinity
d) Find the total potential energy stored in the final configuration of $q_{1}, q_{2},$ and $q_{3}$

Dominador Tan

Numerade Educator

Problem 57

Two metal balls of mass $m_{1}=5.00 \mathrm{~g}$ (diameter = $5.00 \mathrm{~mm}$ ) and $m_{2}=8.00 \mathrm{~g}$ (diameter $=8.00 \mathrm{~mm}$ ) have positive charges of $q_{1}=5.00 \mathrm{nC}$ and $q_{2}=8.00 \mathrm{nC},$ respectively. A force holds them in place so that their centers are separated by $8.00 \mathrm{~mm}$. What will their velocities be after the force is removed and they are separated by a large distance?

Eduard Sanchez

Numerade Educator

Problem 58

Two protons at rest and separated by $1.00 \mathrm{~mm}$ are released simultaneously. What is the speed of either at the instant when the two are $10.0 \mathrm{~mm}$ apart?

Ajay Singhal

Numerade Educator

Problem 59

A $12-V$ battery is connected between a hollow metal sphere with a radius of $1 \mathrm{~m}$ and a ground, as shown in the figure. What are the electric field and the electric potential inside the hollow metal sphere?

Ajay Singhal

Numerade Educator

Problem 60

A solid metal ball with a radius of $3.00 \mathrm{~m}$ has a charge of $4.00 \mathrm{mC}$. If the electric potential is zero far away from the ball, what is the electric potential at each of the following positions?
a) at $r=0 \mathrm{~m},$ the center of the ball
b) at $r=3.00 \mathrm{~m},$ on the surface of the ball
c) at $r=5.00 \mathrm{~m}$

Ajay Singhal

Numerade Educator

Problem 61

An insulating sheet in the $x z$ -plane is uniformly charged with a charge distribution $\sigma=3.5 \cdot 10^{-6} \mathrm{C} / \mathrm{m}^{2}$. What is the change in potential when a charge of $Q=1.25 \mu C$ is moved from position $A$ to position $B$ in the figure?

Eduard Sanchez

Numerade Educator

Problem 62

Suppose that an electron inside a cathode ray tube starts from rest and is accelerated by the tube's voltage of $21.9 \mathrm{kV}$. What is the speed (in $\mathrm{km} / \mathrm{s}$ ) with which the electron (mass $=9.11 \cdot 10^{-31} \mathrm{~kg}$ ) hits the screen of the tube?

Ajay Singhal

Numerade Educator

Problem 63

A conducting solid sphere (radius of $R=18 \mathrm{~cm},$ charge of $q=6.1 \cdot 10^{-6} \mathrm{C}$ ) is shown in the figure. Calculate the electric potential at a point $24 \mathrm{~cm}$ from the center (point
$A$ ), a point on the surface (point $B$ ), and at the center of the sphere (point
C). Assume that the electric potential is zero at points infinitely far away from the origin of the coordinate system.

Eduard Sanchez

Numerade Educator

Problem 64

A classroom Van de Graaff generator accumulates a charge of $1.00 \cdot 10^{-6} \mathrm{C}$ on its spherical conductor, which has a radius of $10.0 \mathrm{~cm}$ and stands on an insulating column. Neglecting the effects of the generator base or any other objects or fields, find the potential at the surface of the sphere. Assume that the potential is zero at infinity.

Ajay Singhal

Numerade Educator

Problem 65

A Van de Graaff generator has a spherical conductor with a radius of $25.0 \mathrm{~cm}$. It can produce a maximum electric field of $2.00 \cdot 10^{6} \mathrm{~V} / \mathrm{m}$. What are the maximum voltage and charge that it can hold?

Ajay Singhal

Numerade Educator

Problem 66

A proton with a speed of $1.23 \cdot 10^{4} \mathrm{~m} / \mathrm{s}$ is moving from infinity directly toward a second proton. Assuming that the second proton is fixed in place, find the position where the moving proton stops momentarily before turning around.

Ajay Singhal

Numerade Educator

Problem 67

Two metal spheres of radii $r_{1}=10.0 \mathrm{~cm}$ and $r_{2}=$ $20.0 \mathrm{~cm},$ respectively, have been positively charged so that both have a total charge of $100, \mu C$
a) What is the ratio of their surface charge distributions?
b) If the two spheres are connected by a copper wire, how much charge flows through the wire before the system reaches equilibrium?

Ajay Singhal

Numerade Educator

Problem 68

The solid metal sphere of radius $a=0.200 \mathrm{~m}$ shown in the figure has a surface charge distribution of $\sigma$. The potential difference between the surface of the sphere and a point $P$ at a distance $r_{\mathrm{p}}=0.500 \mathrm{~m}$ from the center of the sphere is $\Delta V=V_{\text {surfhee }}-V_{\mathrm{p}}=+4 \pi \mathrm{V}=$
$+12.566 \mathrm{~V}$. Determine the value of $\sigma$.

Eduard Sanchez

Numerade Educator

Problem 69

A particle with a charge of $+5.0 \mu C$ is released from rest at a point on the $x$ -axis, where $x=0.10 \mathrm{~m}$. It begins to move as a result of the presence of a $+9.0-\mu C$ charge that remains fixed at the origin. What is the kinetic energy of the particle at the instant it passes the point $x=0.20 \mathrm{~m} ?$

Eduard Sanchez

Numerade Educator

Problem 70

The sphere in the figure has a radius of $2.00 \mathrm{~mm}$ and carries a $+2.00-\mu C$ charge uniformly distributed throughout its volume. What is the potential difference, $V_{\mathrm{B}}-V_{A},$ if the angle between the two radii to points $A$ and $B$ is $60.0^{\circ} ?$ Is the potential difference dependent on the angle? Would the answer be the same if the charge distribution had an angular dependence, $\rho=\rho(\theta)$

Dominador Tan

Numerade Educator

Problem 71

Two metallic spheres have radii of $10.0 \mathrm{~cm}$ and $5.00 \mathrm{~cm}$, respectively. The magnitude of the electric field on the surface of each sphere is $3600 . \mathrm{V} / \mathrm{m} .$ The two spheres are then connected by a long, thin metal wire. Determine the magnitude of the electric field on the surface of each sphere when they are connected.

Dominador Tan

Numerade Educator

Problem 72

A ring with charge $Q$ and radius $R$ is in the $y z$ -plane and centered on the origin. What is the electric potential a distance $x$ above the center of the ring? Derive the electric field from this relationship.

Ajay Singhal

Numerade Educator

Problem 73

A charge of $0.681 \mathrm{nC}$ is placed at $x=0 .$ Another charge of $0.167 \mathrm{nC}$ is placed at $x_{1}=10.9 \mathrm{~cm}$ on the $x$ -axis.
a) What is the combined electrostatic potential of these two charges at $x=20.1 \mathrm{~cm},$ also on the $x$ -axis?
b) At which point(s) on the $x$ -axis does this potential have a minimum?

Dominador Tan

Numerade Educator

Problem 74

A point charge of $+2.0 \mu C$ is located at $(2.5 \mathrm{~m}, 3.2 \mathrm{~m})$ A second point charge of $-3.1 \mu \mathrm{C}$ is located at $(-2.1 \mathrm{~m}, 1.0 \mathrm{~m})$.
a) What is the electric potential at the origin?
b) Along a line passing through both point charges, at what point(s) is (are) the electric potential(s) equal to zero?

Dominador Tan

Numerade Educator

Problem 74

A point charge of $+2.0 \mu C$ is located at $(2.5 \mathrm{~m}, 3.2 \mathrm{~m})$ A second point charge of $-3.1 \mu \mathrm{C}$ is located at $(-2.1 \mathrm{~m}, 1.0 \mathrm{~m})$
a) What is the electric potential at the origin?
b) Along a line passing through both point charges, at what point(s) is (are) the electric potential(s) equal to zero?

Dominador Tan

Numerade Educator

Problem 75

A total charge of $Q=4.2 \cdot 10^{-6} \mathrm{C}$ is placed on a conducting sphere (sphere 1 ) of radius $R=0.40 \mathrm{~m}$.
a) What is the electric potential, $V_{1},$ at the surface of sphere
1 assuming that the potential infinitely far away from it is zero? (Hint: What is the change in potential if a charge is brought from infinitely far away, where $V(\infty)=0,$ to the surface of the sphere?)
b) A second conducting sphere (sphere 2) of radius $r=0.10 \mathrm{~m}$ with an initial net charge of zero $(q=0)$ is connected to sphere 1 using a long thin metal wire. How much charge flows from sphere 1 to sphere 2 to bring them into equilibrium? What are the electric fields at the surfaces of the two spheres?

Dominador Tan

Numerade Educator

Problem 76

A thin line of charge is aligned along the positive $y$ -axis from $0 \leq y \leq L,$ with $L=4.0 \mathrm{~cm} .$
The charge is not uniformly distributed but has a charge per unit length of $\lambda=A y,$ with $A=$ $8.0 \cdot 10^{-7} \mathrm{C} / \mathrm{m}^{2}$. Assuming that
the electric potential is zero at infinite distance, find the electric potential at a point on the $x$ -axis as a function of $x$. Give the value of the electric potential at $x=3.0 \mathrm{~cm} .$

Ajay Singhal

Numerade Educator

Problem 77

Two fixed point charges are on the $x$ -axis. A charge of $-3.00 \mathrm{mC}$ is located at $x=+2.00 \mathrm{~m}$ and a charge of $+5.00 \mathrm{mC}$ is located at $x=-4.00 \mathrm{~m}$
a) Find the electric potential, $V(x),$ for an arbitrary point on the $x$ -axis.
b) At what position(s) on the $x$ -axis is $V(x)=0 ?$
c) Find $E(x)$ for an arbitrary point on the $x$ -axis.

Dominador Tan

Numerade Educator

Problem 78

One of the greatest physics experiments in history measured the charge-to-mass ratio of an electron, $q / m .$ If a uniform potential difference is created between two plates, atomized particles - each with an integral amount of charge-can be suspended in space. The assumption is that the particles of unknown mass, $M,$ contain a net number, $n$, of electrons of mass $m$ and charge $q .$ For a plate separation of $d,$ what is the potential difference necessary to suspend a particle of mass $M$ containing $n$ net electrons? What is the acceleration of the particle if the voltage is cut in half? What is the acceleration of the particle if the voltage is doubled?

Eduard Sanchez

Numerade Educator

Problem 79

A uniform linear charge distribution of total positive charge $Q$ has the shape of a half-circle of radius $R$, as shown in the figure.
a) Without performing any calculations, predict the electric potential produced by this linear charge distribution at point $O$.
b) Confirm, through direct calculations, your prediction of part (a).

Dominador Tan

Numerade Educator

Problem 80

A point charge $Q$ is placed a distance $R$ from the center of a conducting sphere of radius $a,$ with $R>a$ (the point charge is outside the sphere). The sphere is grounded, that is, connected to a distant, unlimited source and/or sink of charge at zero potential. (Neither the distant ground nor the connection directly affects the electric field in the vicinity of the charge and sphere.) As a result, the sphere acquires a charge opposite in sign to $Q$, and the point charge experiences an attractive force toward the sphere.
a) Remarkably, the electric field outside the sphere is the same as would be produced by the point charge $Q$ plus an imaginary mirror-image point charge $q$, with magnitude and location that make the set of points corresponding to the surface of the sphere an equipotential of potential zero. That is, the imaginary point charge produces the same field contribution outside the sphere as the actual surface charge on the sphere. Calculate the value and location of $q$. (Hint:
By symmetry, $q$ must lie somewhere on the axis that passes through the center of the sphere and the location of $Q .)$
b) Calculate the force exerted on point charge $Q$ and directed toward the sphere, in terms of the original quantities $Q, R,$ and $a$
c) Determine the actual nonuniform surface charge distribution on the conducting sphere.

Dominador Tan

Numerade Educator