658
CHAPTE R 22 ELECTRIC FIELDS
1
d
d
2
d
2
d
2
A
B
C
–
+
x
Figure 22-69 Problem 87.
71 A charge of 20 nC is uniformly distributed along a straight
rod of length 4.0 m that is bent into a circular arc with a radius of
2.0 m. What is the magnitude of the electric field at the center of
curvature of the arc?
72 An electron is constrained to the central axis of the ring of
charge of radius R in Fig. 22-11, with z Ӷ R. Show that the electrostatic force on the electron can cause it to oscillate through the ring
center with an angular frequency
,
where q is the ring’s charge and m is the electron’s mass.
73
The electric field in an xy plane produced by a posiSSM
v ϭ A
eq
4p´ 0 mR
3
what time does the hour hand point in the same direction as the
electric field vector at the center of the dial? (Hint: Use symmetry.)
80 Calculate the electric dipole moment of an electron and a
proton 4.30 nm apart.
81 An electric field with an average magnitude of about 150 N/C
points downward in the atmosphere near Earth’s surface. We wish
to “float” a sulfur sphere weighing 4.4 N in this field by charging
the sphere. (a) What charge (both sign and magnitude) must be
used? (b) Why is the experiment impractical?
82 A circular rod has a radius of curvature R ϭ 9.00 cm and a
uniformly distributed positive charge Q ϭ 6.25 pC and subtends
an angle u ϭ 2.40 rad. What is the magnitude of the electric field
that Q produces at the center of curvature?
83
An electric dipole with dipole moment
is in an electric field
(a) What is the potential
energy of the electric dipole? (b) What is the torque acting on it?
(c) If an external agent turns the dipole until its electric dipole
moment is
how much work is done by the
agent?
84 In Fig. 22-68, a uniform, upward
electric field
of magnitude 2.00 ϫ
10
3
N/C has been set up between two
horizontal plates by charging the
lower plate positively and the upper
plate negatively. The plates have
length L ϭ 10.0 cm and separation d
ϭ 2.00 cm. An electron is then shot
between the plates from the left edge of the lower plate. The initial
velocity of the electron makes an angle u ϭ 45.0Њ with the lower
v
:
0
E
:
p
: ϭ (Ϫ4.00i ˆ ϩ 3.00j ˆ )(1.24 ϫ 10
Ϫ30
C иm),
E
: ϭ (4000 N/C)i ˆ .
p
: ϭ (3.00i ˆ ϩ 4.00j ˆ )(1.24 ϫ 10
Ϫ30
Cиm)
SSM
E
:
tively charged particle is
at the point
(3.0, 3.0) cm and
at the point (2.0, 0) cm. What are the (a)
x and (b) y coordinates of the particle? (c) What is the charge of
the particle?
74 (a) What total (excess) charge q must the disk in Fig. 22-15
have for the electric field on the surface of the disk at its center to
have magnitude 3.0 ϫ 10
6
N/C, the E value at which air breaks
down electrically, producing sparks? Take the disk radius as 2.5 cm.
(b) Suppose each surface atom has an effective cross-sectional
area of 0.015 nm
2
. How many atoms are needed to make up the
disk surface? (c) The charge calculated in (a) results from some of
the surface atoms having one excess electron. What fraction of
these atoms must be so charged?
75 In Fig. 22-66, particle 1 (of
charge ϩ1.00 mC), particle 2 (of
charge ϩ1.00 mC), and particle 3
(of charge Q) form an equilateral
triangle of edge length a. For what
value of Q (both sign and magnitude) does the net electric field produced by the particles at the center
of the triangle vanish?
76 In Fig. 22-67, an electric dipole swings
from an initial orientation i (u i ϭ 20.0Њ) to a
final orientation f (u f ϭ 20.0Њ) in a uniform
external electric field . The electric dipole
moment is 1.60 ϫ 10
Ϫ27
C иm; the field magnitude is 3.00 ϫ 10
6
N/C. What is the change in
the dipole’s potential energy?
77 A particle of charge Ϫq 1 is at the origin of
an x axis. (a) At what location on the axis should
a particle of charge Ϫ4q 1 be placed so that the
net electric field is zero at x ϭ 2.0 mm on the
axis? (b) If, instead, a particle of charge ϩ4q 1 is
placed at that location, what is the direction (relative to the positive
direction of the x axis) of the net electric field at x ϭ 2.0 mm?
78 Two particles, each of positive charge q, are fixed in place on a
y axis, one at y ϭ d and the other at y ϭ Ϫd. (a) Write an expression that gives the magnitude E of the net electric field at points on
the x axis given by x ϭ ad. (b) Graph E versus a for the range 0 Ͻ
a Ͻ 4. From the graph, determine the values of a that give (c) the
maximum value of E and (d) half the maximum value of E.
79 A clock face has negative point charges Ϫq, Ϫ2q, Ϫ3q, . . . ,
Ϫ12q fixed at the positions of the corresponding numerals. The
clock hands do not perturb the net field due to the point charges. At
E
:
100i ˆ N/C
7.2(4.0i ˆ ϩ 3.0j ˆ ) N/C
a
a
a
1
2
3
y
x
Figure 22-66 Problems 75
and 86.
f
p f
p i
E
θ
i
θ
Figure 22-67
Problem 76.
L
θ
d
v 0
E
Figure 22-68 Problem 84.
plate and has a magnitude of 6.00 ϫ 10
6
m/s. (a) Will the electron
strike one of the plates? (b) If so, which plate and how far horizontally from the left edge will the electron strike?
85 For the data of Problem 70, assume that the charge q on the
drop is given by q ϭ ne, where n is an integer and e is the elementary charge. (a) Find n for each given value of q. (b) Do a linear regression fit of the values of q versus the values of n and then use
that fit to find e.
86 In Fig. 22-66, particle 1 (of charge ϩ2.00 pC), particle 2 (of
charge Ϫ2.00 pC), and particle 3 (of charge ϩ5.00 pC) form an
equilateral triangle of edge length a ϭ 9.50 cm. (a) Relative to
the positive direction of the x axis, determine the direction of
the force
on particle 3 due to the other particles by sketching
electric field lines of the other particles. (b) Calculate the magnitude of .
87 In Fig. 22-69, particle 1 of charge q 1 ϭ 1.00 pC and particle 2
of charge q 2 ϭ Ϫ2.00 pC are fixed at a distance d ϭ 5.00 cm apart.
In unit-vector notation, what is the net electric field at points (a) A,
(b) B, and (c) C? (d) Sketch the electric field lines.
F
:
3
F
:
3
CHAPTE R 22 ELECTRIC FIELDS
1
d
d
2
d
2
d
2
A
B
C
–
+
x
Figure 22-69 Problem 87.
71 A charge of 20 nC is uniformly distributed along a straight
rod of length 4.0 m that is bent into a circular arc with a radius of
2.0 m. What is the magnitude of the electric field at the center of
curvature of the arc?
72 An electron is constrained to the central axis of the ring of
charge of radius R in Fig. 22-11, with z Ӷ R. Show that the electrostatic force on the electron can cause it to oscillate through the ring
center with an angular frequency
,
where q is the ring’s charge and m is the electron’s mass.
73
The electric field in an xy plane produced by a posiSSM
v ϭ A
eq
4p´ 0 mR
3
what time does the hour hand point in the same direction as the
electric field vector at the center of the dial? (Hint: Use symmetry.)
80 Calculate the electric dipole moment of an electron and a
proton 4.30 nm apart.
81 An electric field with an average magnitude of about 150 N/C
points downward in the atmosphere near Earth’s surface. We wish
to “float” a sulfur sphere weighing 4.4 N in this field by charging
the sphere. (a) What charge (both sign and magnitude) must be
used? (b) Why is the experiment impractical?
82 A circular rod has a radius of curvature R ϭ 9.00 cm and a
uniformly distributed positive charge Q ϭ 6.25 pC and subtends
an angle u ϭ 2.40 rad. What is the magnitude of the electric field
that Q produces at the center of curvature?
83
An electric dipole with dipole moment
is in an electric field
(a) What is the potential
energy of the electric dipole? (b) What is the torque acting on it?
(c) If an external agent turns the dipole until its electric dipole
moment is
how much work is done by the
agent?
84 In Fig. 22-68, a uniform, upward
electric field
of magnitude 2.00 ϫ
10
3
N/C has been set up between two
horizontal plates by charging the
lower plate positively and the upper
plate negatively. The plates have
length L ϭ 10.0 cm and separation d
ϭ 2.00 cm. An electron is then shot
between the plates from the left edge of the lower plate. The initial
velocity of the electron makes an angle u ϭ 45.0Њ with the lower
v
:
0
E
:
p
: ϭ (Ϫ4.00i ˆ ϩ 3.00j ˆ )(1.24 ϫ 10
Ϫ30
C иm),
E
: ϭ (4000 N/C)i ˆ .
p
: ϭ (3.00i ˆ ϩ 4.00j ˆ )(1.24 ϫ 10
Ϫ30
Cиm)
SSM
E
:
tively charged particle is
at the point
(3.0, 3.0) cm and
at the point (2.0, 0) cm. What are the (a)
x and (b) y coordinates of the particle? (c) What is the charge of
the particle?
74 (a) What total (excess) charge q must the disk in Fig. 22-15
have for the electric field on the surface of the disk at its center to
have magnitude 3.0 ϫ 10
6
N/C, the E value at which air breaks
down electrically, producing sparks? Take the disk radius as 2.5 cm.
(b) Suppose each surface atom has an effective cross-sectional
area of 0.015 nm
2
. How many atoms are needed to make up the
disk surface? (c) The charge calculated in (a) results from some of
the surface atoms having one excess electron. What fraction of
these atoms must be so charged?
75 In Fig. 22-66, particle 1 (of
charge ϩ1.00 mC), particle 2 (of
charge ϩ1.00 mC), and particle 3
(of charge Q) form an equilateral
triangle of edge length a. For what
value of Q (both sign and magnitude) does the net electric field produced by the particles at the center
of the triangle vanish?
76 In Fig. 22-67, an electric dipole swings
from an initial orientation i (u i ϭ 20.0Њ) to a
final orientation f (u f ϭ 20.0Њ) in a uniform
external electric field . The electric dipole
moment is 1.60 ϫ 10
Ϫ27
C иm; the field magnitude is 3.00 ϫ 10
6
N/C. What is the change in
the dipole’s potential energy?
77 A particle of charge Ϫq 1 is at the origin of
an x axis. (a) At what location on the axis should
a particle of charge Ϫ4q 1 be placed so that the
net electric field is zero at x ϭ 2.0 mm on the
axis? (b) If, instead, a particle of charge ϩ4q 1 is
placed at that location, what is the direction (relative to the positive
direction of the x axis) of the net electric field at x ϭ 2.0 mm?
78 Two particles, each of positive charge q, are fixed in place on a
y axis, one at y ϭ d and the other at y ϭ Ϫd. (a) Write an expression that gives the magnitude E of the net electric field at points on
the x axis given by x ϭ ad. (b) Graph E versus a for the range 0 Ͻ
a Ͻ 4. From the graph, determine the values of a that give (c) the
maximum value of E and (d) half the maximum value of E.
79 A clock face has negative point charges Ϫq, Ϫ2q, Ϫ3q, . . . ,
Ϫ12q fixed at the positions of the corresponding numerals. The
clock hands do not perturb the net field due to the point charges. At
E
:
100i ˆ N/C
7.2(4.0i ˆ ϩ 3.0j ˆ ) N/C
a
a
a
1
2
3
y
x
Figure 22-66 Problems 75
and 86.
f
p f
p i
E
θ
i
θ
Figure 22-67
Problem 76.
L
θ
d
v 0
E
Figure 22-68 Problem 84.
plate and has a magnitude of 6.00 ϫ 10
6
m/s. (a) Will the electron
strike one of the plates? (b) If so, which plate and how far horizontally from the left edge will the electron strike?
85 For the data of Problem 70, assume that the charge q on the
drop is given by q ϭ ne, where n is an integer and e is the elementary charge. (a) Find n for each given value of q. (b) Do a linear regression fit of the values of q versus the values of n and then use
that fit to find e.
86 In Fig. 22-66, particle 1 (of charge ϩ2.00 pC), particle 2 (of
charge Ϫ2.00 pC), and particle 3 (of charge ϩ5.00 pC) form an
equilateral triangle of edge length a ϭ 9.50 cm. (a) Relative to
the positive direction of the x axis, determine the direction of
the force
on particle 3 due to the other particles by sketching
electric field lines of the other particles. (b) Calculate the magnitude of .
87 In Fig. 22-69, particle 1 of charge q 1 ϭ 1.00 pC and particle 2
of charge q 2 ϭ Ϫ2.00 pC are fixed at a distance d ϭ 5.00 cm apart.
In unit-vector notation, what is the net electric field at points (a) A,
(b) B, and (c) C? (d) Sketch the electric field lines.
F
:
3
F
:
3