electric potential energy of
the system as the particle
at the lower left corner in
Figure P25.27 is brought
to this position from infi-
nitely far away. Assume the
other three particles in Fig-
ure P25.27 remain fixed in
28. Three particles with equal posi-
tive charges q are at the corners
of an equilateral triangle of side a
as shown in Figure P25.28. (a) At
what point, if any, in the plane of
the particles is the electric poten-
tial zero? (b) What is the electric
potential at the position of one of
the particles due to the other two
particles in the triangle?
29. Five particles with equal negative charges 2q are
placed symmetrically around a circle of radius R. Cal-
culate the electric potential at the center of the circle.
30. Review. A light, unstressed spring has length d. Two
identical particles, each with charge q, are connected
to the opposite ends of the spring. The particles are
held stationary a distance d apart and then released at
the same moment. The system then oscillates on a fric-
tionless, horizontal table. The spring has a bit of inter-
nal kinetic friction, so the oscillation is damped. The
particles eventually stop vibrating when the distance
between them is 3d. Assume the system of the spring
and two charged particles is isolated. Find the increase
in internal energy that appears in the spring during
31. Review. Two insulating spheres have radii 0.300 cm
and 0.500 cm, masses 0.100 kg and 0.700 kg, and uni-
formly distributed charges 22.00 mC and 3.00 mC.
They are released from rest when their centers are
separated by 1.00 m. (a) How fast will each be moving
when they collide? (b) What If? If the spheres were
conductors, would the speeds be greater or less than
those calculated in part (a)? Explain.
32. Review. Two insulating spheres have radii r
, and uniformly distributed charges
. They are released from rest when their cen-
ters are separated by a distance d. (a) How fast is each
moving when they collide? (b) What If? If the spheres
were conductors, would their speeds be greater or less
than those calculated in part (a)? Explain.
33. How much work is required to assemble eight identical
charged particles, each of magnitude q, at the corners
of a cube of side s?
34. Four identical particles, each having charge q and mass
m, are released from rest at the vertices of a square of
side L. How fast is each particle moving when their dis-
tance from the center of the square doubles?
35. In 1911, Ernest Rutherford and his assistants Geiger
and Marsden conducted an experiment in which they
by the two 2.00-mC charges on the charge q? (b) What
is the electric field at the origin due to the two 2.00-mC
particles? (c)What is the electric potential at the ori-
gin due to the two 2.00-mC particles?
x = 0.800 m
x = -0.800 m
20. At a certain distance from a charged particle, the mag-
nitude of the electric field is 500 V/m and the electric
potential is 23.00 kV. (a) What is the distance to the
particle? (b) What is the magnitude of the charge?
21. Four point charges each having charge Q are located at
the corners of a square having sides of length a. Find
expressions for (a) the total electric potential at the
center of the square due to the four charges and
(b) the work required to bring a fifth charge q from
infinity to the center of the square.
22. The three charged particles in
Figure P25.22 are at the vertices
of an isosceles triangle (where d 5
2.00cm). Taking q5 7.00 mC,
calculate the electric potential at
point A, the midpoint of the base.
23. A particle with charge 1q is at
the origin. A particle with charge
22q is at x 5 2.00 m on the x axis.
(a) For what finite value(s) of x
is the electric field zero? (b) For
what finite value(s) of x is the electric potential zero?
24. Show that the amount of work required to assemble
four identical charged particles of magnitude Q at the
corners of a square of side s is 5.41k
25. Two particles each with charge 12.00mC are located
on the x axis. One is at x 5 1.00 m, and the other is at
x 5 21.00m. (a) Determine the electric potential on
the y axis at y5 0.500 m. (b) Calculate the change in
electric potential energy of the system as a third
charged particle of 23.00mC is brought from infinitely
far away to a position on the y axis at y 5 0.500 m.
26. Two charged particles of equal mag-
nitude are located along the y axis
equal distances above and below the
x axis as shown in Figure P25.26.
(a) Plot a graph of the electric
potential at points along the x axis
over the interval 23a , x, 3a. You
should plot the potential in units
Q/a. (b)Let the charge of the
particle located at y 5 2a be nega-
tive. Plot the potential along the y
axis over the interval 24a , y , 4a.
27. Four identical charged particles (q 5 110.0 mC) are
located on the corners of a rectangle as shown in Fig-
ure P25.27. The dimensions of the rectangle are L 5
60.0 cm and W 5 15.0 cm. Calculate the change in
chapter 25 electric potential
at B. (c)Represent what the electric field looks
like by drawing at least eight field lines.
41. The electric potential inside a charged spherical con-
ductor of radius R is given by V 5 k
Q/R, and the
potential outside is given by V 5 k
Q/r. Using E
2dV/dr, derive the electric field (a) inside and (b) out-
side this charge distribution.
42. It is shown in Example 25.7 that the potential at a point
P a distance a above one end of a uniformly charged
rod of length , lying along the x axis is
Use this result to derive an expression for the y compo-
nent of the electric field at P.
Section 25.5 Electric Potential Due
to Continuous Charge Distributions
43. Consider a ring of radius R with the total charge Q
spread uniformly over its perimeter. What is the poten-
tial difference between the point at the center of the ring
and a point on its axis a distance 2R from the center?
44. A uniformly charged insulating rod of
length 14.0 cm is bent into the shape
of a semicircle as shown in Figure
P25.44. The rod has a total charge of
27.50 mC. Find the electric potential
at O, the center of the semicircle.
45. A rod of length L (Fig. P25.45) lies
along the x axis with its left end at the
origin. It has a nonuniform charge
scattered alpha particles (nuclei of helium atoms) from
thin sheets of gold. An alpha particle, having charge
12e and mass 6.643 10227 kg, is a product of certain
radioactive decays. The results of the experiment led
Rutherford to the idea that most of an atom’s mass is
in a very small nucleus, with electrons in orbit around
it. (This is the planetary model of the atom, which we’ll
study in Chapter 42.) Assume an alpha particle, ini-
tially very far from a stationary gold nucleus, is fired
with a velocity of 2.00 3 107 m/s directly toward the
nucleus (charge 179e). What is the smallest distance
between the alpha particle and the nucleus before the
alpha particle reverses direction? Assume the gold
nucleus remains stationary.
Section 25.4 obtaining the Value of the Electric Field
from the Electric Potential
36. Figure P25.36 repre-
sents a graph of the
electric potential in a
region of space versus
position x, where the
electric field is paral-
lel to the x axis. Draw
a graph of the x compo-
nent of the electric field
versus x in this region.
37. The potential in a region between x 5 0 and x 5 6.00 m
is V 5 a 1 bx, where a 5 10.0 V and b 5 27.00 V/m.
Determine (a) the potential at x 5 0, 3.00 m, and 6.00 m
and (b)the magnitude and direction of the electric
field at x 5 0, 3.00 m, and 6.00 m.
38. An electric field in a region of space is parallel to the
x axis. The electric potential varies with position as
shown in Figure P25.38. Graph the x component of the
electric field versus position in this region of space.
39. Over a certain region of space, the electric potential is
V 5 5x 2 3x2y 1 2yz2. (a) Find the expressions for the
x, y, and z components of the electric field over this
region. (b) What is the magnitude of the field at the
point P that has coordinates (1.00, 0, 22.00) m?
40. Figure P25.40 shows several equipotential lines, each
labeled by its potential in volts. The distance between
the lines of the square grid represents 1.00 cm. (a) Is
the magnitude of the field larger at A or at B? Explain
how you can tell. (b) Explain what you can determine
Numerical values are in volts.
Problems 45 and 46.
dielectric strength of air. Any more charge leaks off in
sparks as shown in Figure P25.52. Assume the dome has
a diameter of 30.0 cm and is surrounded by dry air with
a “breakdown” electric field of 3.00 3 106 V/m. (a) What
is the maximum potential of the dome? (b) What is the
maximum charge on the dome?
53. Why is the following situation impossible? In the Bohr model
of the hydrogen atom, an electron moves in a circular
orbit about a proton. The model states that the electron
can exist only in certain allowed orbits around the pro-
ton: those whose radius r satisfies r 5 n2(0.052 9 nm),
where n5 1, 2, 3, . . . . For one of the possible allowed
states of the atom, the electric potential energy of the
system is 213.6eV.
54. Review. In fair weather, the electric field in the air at
a particular location immediately above the Earth’s
surface is 120 N/C directed downward. (a) What is the
surface charge density on the ground? Is it positive or
negative? (b) Imagine the surface charge density is
uniform over the planet. What then is the charge of
the whole surface of the Earth? (c) What is the Earth’s
electric potential due to this charge? (d) What is the
difference in potential between the head and the feet
of a person 1.75 m tall? (Ignore any charges in the
atmosphere.) (e) Imagine the Moon, with 27.3% of the
radius of the Earth, had a charge 27.3% as large, with
the same sign. Find the electric force the Earth would
then exert on the Moon. (f) State how the answer to
part (e) compares with the gravitational force the
Earth exerts on the Moon.
55. Review. From a large distance away, a particle of mass
2.00g and charge 15.0 mC is fired at 21.0i
toward a second particle, originally stationary but free
to move, with mass 5.00 g and charge 8.50 mC. Both
particles are constrained to move only along the x axis.
(a) At the instant of closest approach, both particles
will be moving at the same velocity. Find this velocity.
(b) Find the distance of closest approach. After the
interaction, the particles will move far apart again. At
this time, find the velocity of (c)the 2.00-g particle
and (d) the 5.00-g particle.
56. Review. From a large distance away, a particle of mass m
and positive charge q
is fired at speed v in the positive
x direction straight toward a second particle, originally
stationary but free to move, with mass m
. Both particles are constrained to move only
along the x axis. (a)At the instant of closest approach,
both particles will be moving at the same velocity. Find
this velocity. (b)Find the distance of closest approach.
After the interaction, the particles will move far apart
again. At this time, find the velocity of (c)the particle of
and (d) the particle of mass m
57. The liquid-drop model of the atomic nucleus suggests
high-energy oscillations of certain nuclei can split
the nucleus into two unequal fragments plus a few
density l 5 ax, where a is a positive constant. (a) What
are the units of a? (b) Calculate the electric potential
46. For the arrangement described in Problem 45, calcu-
late the electric potential at point B, which lies on the
perpendicular bisector of the rod a distance b above
the x axis.
47. A wire having a uniform linear charge density l is bent
into the shape shown in Figure P25.47. Find the elec-
tric potential at point O.
Section 25.6 Electric Potential Due to a Charged Conductor
48. The electric field magnitude on the surface of an
irregularly shaped conductor varies from 56.0 kN/C to
28.0kN/C. Can you evaluate the electric potential on the
conductor? If so, find its value. If not, explain why not.
49. How many electrons should be removed from an ini-
tially uncharged spherical conductor of radius 0.300 m
to produce a potential of 7.50 kV at the surface?
50. A spherical conductor has a radius of 14.0 cm and a
charge of 26.0 mC. Calculate the electric field and the
electric potential at (a) r 5 10.0 cm, (b) r 5 20.0 cm,
and (c) r 5 14.0 cm from the center.
51. Electric charge can accumulate on an airplane in flight.
You may have observed needle-shaped metal extensions
on the wing tips and tail of an airplane. Their purpose
is to allow charge to leak off before much of it accu-
mulates. The electric field around the needle is much
larger than the field around the body of the airplane
and can become large enough to produce dielectric
breakdown of the air, discharging the airplane. To
model this process, assume two charged spherical con-
ductors are connected by a long conducting wire and
a 1.20-mC charge is placed on the combination. One
sphere, representing the body of the airplane, has a
radius of 6.00 cm; the other, representing the tip of the
needle, has a radius of 2.00 cm. (a) What is the electric
potential of each sphere? (b) What is the electric field
at the surface of each sphere?
Section 25.8 applications of Electrostatics
52. Lightning can be studied
with a Van de Graaff gen-
erator, which consists of a
spherical dome on which
charge is continuously
deposited by a moving
belt. Charge can be added
until the electric field at
the surface of the dome
becomes equal to the
chapter 25 electric potential
neutrons. The fission products acquire kinetic energy
from their mutual Coulomb repulsion. Assume the
charge is distributed uniformly throughout the volume
of each spherical fragment and, immediately before sep-
arating, each fragment is at rest and their surfaces are
in contact. The electrons surrounding the nucleus can
be ignored. Calculate the electric potential energy (in
electron volts) of two spherical fragments from a ura-
nium nucleus having the following charges and radii:
38e and 5.50 3 10215 m, and 54e and 6.20 3 10215 m.
58. On a dry winter day, you scuff your leather-soled shoes
across a carpet and get a shock when you extend the
tip of one finger toward a metal doorknob. In a dark
room, you see a spark perhaps 5 mm long. Make order-
of-magnitude estimates of (a) your electric potential
and (b) the charge on your body before you touch the
doorknob. Explain your reasoning.
59. The electric potential immediately outside a charged
conducting sphere is 200 V, and 10.0 cm farther
from the center of the sphere the potential is 150 V.
Determine (a) the radius of the sphere and (b) the
charge on it. The electric potential immediately out-
side another charged conducting sphere is 210 V, and
10.0 cm farther from the center the magnitude of the
electric field is 400 V/m. Determine (c)the radius of
the sphere and (d) its charge on it. (e) Are the answers
to parts (c) and (d) unique?
60. (a) Use the exact result from Example 25.4 to find the
electric potential created by the dipole described in
the example at the point (3a, 0). (b) Explain how this
answer compares with the result of the approximate
expression that is valid when x is much greater than a.
61. Calculate the work that must be done on charges
brought from infinity to charge a spherical shell of
radius R 5 0.100m to a total charge Q 5 125 mC.
62. Calculate the work that must be done on charges
brought from infinity to charge a spherical shell of
radius R to a total charge Q.
63. The electric potential everywhere on the xy plane is
where V is in volts and x and y are in meters. Determine
the position and charge on each of the particles that
create this potential.
64. Why is the following situ-
ation impossible? You set
up an apparatus in your
laboratory as follows.
The x axis is the symme-
try axis of a stationary,
uniformly charged ring
of radius R 5 0.500 m
and charge Q 5 50.0 mC
(Fig. P25.64). You place
a particle with charge
Q5 50.0 mC and mass m 5 0.100kg at the center of the
ring and arrange for it to be constrained to move only
along the x axis. When it is displaced slightly, the par-
ticle is repelled by the ring and accelerates along the x
axis. The particle moves faster than you expected and
strikes the opposite wall of your laboratory at 40.0 m/s.
65. From Gauss’s law, the electric field set up by a uniform
line of charge is
where r^ is a unit vector pointing radially away from
the line and l is the linear charge density along the
line. Derive an expression for the potential difference
between r 5 r
and r5 r
66. A uniformly charged filament lies along the x axis
between x 5 a 5 1.00 m and x 5 a 1 , 5 3.00 m as
shown in Figure P25.66. The total charge on the fila-
ment is 1.60nC. Calculate successive approximations
for the electric potential at the origin by modeling the
filament as (a) a single charged particle at x 5 2.00 m,
(b) two 0.800-nC charged particles at x 5 1.5 m and
x 5 2.5 m, and (c) four 0.400-nC charged particles at
x 5 1.25 m, x 5 1.75 m, x 5 2.25 m, and x 5 2.75 m.
(d) Explain how the results compare with the potential
given by the exact expression
67. The thin, uniformly charged rod
shown in Figure P25.67 has a lin-
ear charge density l. Find an
expression for the electric poten-
tial at P.
68. A Geiger–Mueller tube is a radia-
tion detector that consists of a
closed, hollow, metal cylinder
(the cathode) of inner radius r
and a coaxial cylindrical wire (the
anode) of radius r
The charge per unit length on the anode is l, and the
charge per unit length on the cathode is 2l. A gas fills
the space between the electrodes. When the tube is in
use (Fig. P25.68b) and a high-energy elementary par-
ticle passes through this space, it can ionize an atom
of the gas. The strong electric field makes the result-
ing ion and electron accelerate in opposite directions.
They strike other molecules of the gas to ionize them,
producing an avalanche of electrical discharge. The
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