38. A long, cylindrical conductor of radius R carries a cur-
rent I as shown in Figure P30.38. The current density
J, however, is not uniform over the cross section of the
conductor but rather is a function of the radius accord-
ing to J5 br, where b is a constant. Find an expression
for the magnetic field magnitude B (a) at a distance
, R and (b)at a distance r
. R, measured from the
center of the conductor.
39. Four long, parallel conductors carry equal currents of
I 5 5.00 A. Figure P30.39 is an end view of the conduc-
tors. The current direction is into the page at points
A and B and out of the page at points C and D. Cal-
culate (a) the magnitude and (b) the direction of the
magnetic field at point P, located at the center of the
square of edge length ,5 0.200 m.
Section 30.4 The Magnetic Field of a Solenoid
40. A certain superconducting magnet in the form of a
solenoid of length 0.500 m can generate a magnetic
field of 9.00 T in its core when its coils carry a current
of 75.0 A. Find the number of turns in the solenoid.
41. A long solenoid that has 1 000 turns uniformly dis-
tributed over a length of 0.400 m produces a magnetic
field of magnitude 1.00 3 1024 T at its center. What
current is required in the windings for that to occur?
42. You are given a certain volume of copper from which
you can make copper wire. To insulate the wire, you
can have as much enamel as you like. You will use the
wire to make a tightly wound solenoid 20 cm long hav-
ing the greatest possible magnetic field at the center
and using a power supply that can deliver a current
of 5 A. The solenoid can be wrapped with wire in one
or more layers. (a) Should you make the wire long
and thin or shorter and thick? Explain. (b) Should
you make the radius of the solenoid small or large?
43. A single-turn square loop of wire, 2.00 cm on each edge,
carries a clockwise current of 0.200 A. The loop is inside
a solenoid, with the plane of the loop perpendicular
to the magnetic field of the solenoid. The solenoid has
netic field inside the toroid along (a) the inner radius
and (b) the outer radius.
33. A long, straight wire lies on a horizontal table and car-
ries a current of 1.20 mA. In a vacuum, a proton moves
parallel to the wire (opposite the current) with a con-
stant speed of 2.30 3 104 m/s at a distance d above the
wire. Ignoring the magnetic field due to the Earth,
determine the value of d.
34. An infinite sheet of current lying in the yz plane car-
ries a surface current of linear density J
. The current
is in the positive z direction, and J
represents the cur-
rent per unit length measured along the y axis. Figure
P30.34 is an edge view of the sheet. Prove that the mag-
netic field near the sheet is parallel to the sheet and
perpendicular to the current direction, with magni-
(out of paper)
35. The magnetic field 40.0 cm away from a long, straight
wire carrying current 2.00 A is 1.00 mT. (a) At what dis-
tance is it 0.100 mT? (b) What If? At one instant, the
two conductors in a long household extension cord
carry equal 2.00-A currents in opposite directions. The
two wires are 3.00mm apart. Find the magnetic field
40.0 cm away from the middle of the straight cord, in
the plane of the two wires. (c)At what distance is it
one-tenth as large? (d) The center wire in a coaxial
cable carries current 2.00 A in one direction, and the
sheath around it carries current 2.00 A in the opposite
direction. What magnetic field does the cable create at
points outside the cable?
36. A packed bundle of 100 long, straight, insulated wires
forms a cylinder of radius R 5 0.500 cm. If each wire
carries 2.00 A, what are (a) the magnitude and (b) the
direction of the magnetic force per unit length acting
on a wire located 0.200 cm from the center of the bun-
dle? (c) What If? Would a wire on the outer edge of the
bundle experience a force greater or smaller than the
value calculated in parts (a) and (b)? Give a qualitative
argument for your answer.
37. The magnetic field created by a large current passing
through plasma (ionized gas) can force current-carrying
particles together. This pinch effect has been used in
designing fusion reactors. It can be demonstrated by
making an empty aluminum can carry a large cur-
rent parallel to its axis. Let R represent the radius of
the can and I the current, uniformly distributed over
the can’s curved wall. Determine the magnetic field
(a) just inside the wall and (b) just outside. (c) Deter-
mine the pressure on the wall.
chapter 30 Sources of the Magnetic Field
shown in Figure P30.48a. (b)Figure P30.48b shows an
enlarged end view of the same solenoid. Calculate the
flux through the tan area, which is an annulus with
an inner radius of a 5 0.400 cm and an outer radius
of b 5 0.800 cm.
Section 30.6 Magnetism in Matter
49. The magnetic moment of the Earth is approximately
8.003 1022 A ? m2. Imagine that the planetary mag-
netic field were caused by the complete magnetiza-
tion of a huge iron deposit with density 7 900 kg/m3
and approximately 8.50 3 1028 iron atoms/m3.
(a) How many unpaired electrons, each with a mag-
netic moment of 9.27 3 10224 A ? m2, would participate?
(b) At two unpaired electrons per iron atom, how many
kilograms of iron would be present in the deposit?
50. At saturation, when nearly all the atoms have their
magnetic moments aligned, the magnetic field is
equal to the permeability constant m
the magnetic moment per unit volume. In a sample of
iron, where the number density of atoms is approxi-
mately 8.50 3 1028 atoms/m3, the magnetic field can
reach 2.00 T. If each electron contributes a magnetic
moment of 9.27 3 10224 A ? m2 (1 Bohr magneton),
how many electrons per atom contribute to the satu-
rated field of iron?
51. A 30.0-turn solenoid of length 6.00 cm produces a
magnetic field of magnitude 2.00 mT at its center. Find
the current in the solenoid.
52. A wire carries a 7.00-A current along the x axis, and
another wire carries a 6.00-A current along the y axis,
as shown in Figure P30.52. What is the magnetic field
at point P, located at x 5 4.00 m, y 5 3.00 m?
(4.00, 3.00) m
30.0turns/cm and carries a clockwise current of 15.0 A.
Find (a) the force on each side of the loop and (b) the
torque acting on the loop.
44. A solenoid 10.0 cm in diameter and 75.0 cm long is
made from copper wire of diameter 0.100 cm, with very
thin insulation. The wire is wound onto a cardboard
tube in a single layer, with adjacent turns touching
each other. What power must be delivered to the sole-
noid if it is to produce a field of 8.00 mT at its center?
45. It is desired to construct a solenoid that will have a
resistance of 5.00 V (at 20.08C) and produce a mag-
netic field of 4.00 3 1022 T at its center when it carries
a current of 4.00A. The solenoid is to be constructed
from copper wire having a diameter of 0.500 mm. If
the radius of the solenoid is to be 1.00 cm, determine
(a) the number of turns of wire needed and (b) the
required length of the solenoid.
Section 30.5 Gauss’s Law in Magnetism
46. Consider the hemispherical closed surface in Figure
P30.46. The hemisphere is in a uniform magnetic
field that makes an angle u with the vertical. Calculate
the magnetic flux through (a) the flat surface S
(b)the hemispherical surfaceS
47. A cube of edge length , 5 2.50 cm is positioned as
shown in Figure P30.47. A uniform magnetic field
given by B
T exists throughout the
region. (a)Calculate the magnetic flux through the
shaded face. (b)What is the total flux through the six
48. A solenoid of radius r 5 1.25 cm and length , 5 30.0 cm
has 300 turns and carries 12.0 A. (a) Calculate the
flux through the surface of a disk-shaped area of
radius R 5 5.00 cm that is positioned perpendicu-
lar to and centered on the axis of the solenoid as
needle” is a magnetic compass mounted so that it can
rotate in a vertical north–south plane. At this location,
a dip needle makes an angle of 13.08 from the vertical.
What is the total magnitude of the Earth’s magnetic
field at this location?
59. A very large parallel-plate capacitor has uniform
charge per unit area 1s on the upper plate and 2s
on the lower plate. The plates are horizontal, and both
move horizontally with speed v to the right. (a) What
is the magnetic field between the plates? (b) What is
the magnetic field just above or just below the plates?
(c) What are the magnitude and direction of the mag-
netic force per unit area on the upper plate? (d) At
what extrapolated speed v will the magnetic force on a
plate balance the electric force on the plate? Suggestion:
Use Ampere’s law and choose a path that closes
between the plates of the capacitor.
60. Two circular coils of radius R, each with N turns, are
perpendicular to a common axis. The coil centers are
a distance R apart. Each coil carries a steady current
I in the same direction as shown in Figure P30.60.
(a) Show that the magnetic field on the axis at a dis-
tance x from the center of one coil is
(b) Show that dB/dx and d2B/dx2 are both zero at the
point midway between the coils. We may then conclude
that the magnetic field in the region midway between
the coils is uniform. Coils in this configuration are
called Helmholtz coils.
Problems 60 and 61.
61. Two identical, flat, circular coils of wire each have 100
turns and radius R 5 0.500 m. The coils are arranged
as a set of Helmholtz coils so that the separation dis-
tance between the coils is equal to the radius of the
coils (see Fig. P30.60). Each coil carries current I 5
10.0 A. Determine the magnitude of the magnetic field
at a point on the common axis of the coils and halfway
62. Two circular loops are parallel, coaxial, and almost in
contact, with their centers 1.00 mm apart (Fig. P30.62,
page 932). Each loop is 10.0 cm in radius. The top loop
carries a clockwise current of I 5 140 A. The bottom
loop carries a counterclockwise current of I 5 140 A.
(a) Calculate the magnetic force exerted by the bot-
tom loop on the top loop. (b)Suppose a student thinks
the first step in solving part (a) is to use Equation 30.7
to find the magnetic field created by one of the loops.
53. Suppose you install a compass on the center of a car’s
dashboard. (a) Assuming the dashboard is made
mostly of plastic, compute an order-of-magnitude esti-
mate for the magnetic field at this location produced
by the current when you switch on the car’s headlights.
(b) How does this estimate compare with the Earth’s
54. Why is the following situation impossible? The magnitude
of the Earth’s magnetic field at either pole is approxi-
mately 7.00 3 1025 T. Suppose the field fades away to
zero before its next reversal. Several scientists propose
plans for artificially generating a replacement mag-
netic field to assist with devices that depend on the
presence of the field. The plan that is selected is to lay
a copper wire around the equator and supply it with a
current that would generate a magnetic field of magni-
tude 7.00 3 1025 T at the poles. (Ignore magnetization
of any materials inside the Earth.) The plan is imple-
mented and is highly successful.
55. A nonconducting ring of radius 10.0 cm is uniformly
charged with a total positive charge 10.0 mC. The ring
rotates at a constant angular speed 20.0 rad/s about an
axis through its center, perpendicular to the plane of
the ring. What is the magnitude of the magnetic field
on the axis of the ring 5.00 cm from its center?
56. A nonconducting ring of radius R is uniformly charged
with a total positive charge q. The ring rotates at a con-
stant angular speed v about an axis through its cen-
ter, perpendicular to the plane of the ring. What is the
magnitude of the magnetic field on the axis of the ring
a distance 1
R from its center?
57. A very long, thin strip of metal of width w carries a
current I along its length as shown in Figure P30.57.
The current is distributed uniformly across the width
of the strip. Find the magnetic field at point P in the
diagram. Point P is in the plane of the strip at distance
b away from its edge.
58. A circular coil of five turns and a diameter of 30.0 cm
is oriented in a vertical plane with its axis perpendicu-
lar to the horizontal component of the Earth’s mag-
netic field. A horizontal compass placed at the coil’s
center is made to deflect 45.08 from magnetic north
by a current of 0.600A in the coil. (a) What is the
horizontal component of the Earth’s magnetic field?
(b) The current in the coil is switched off. A “dip
chapter 30 Sources of the Magnetic Field
ates a magnetic field (Section 30.1). (a) To understand
how a moving charge can also create a magnetic field,
consider a particle with charge q moving with velocity
. Define the position vector r
5rr^ leading from the
particle to some location. Show that the magnetic field
at that location is
(b) Find the magnitude of the magnetic field 1.00 mm
to the side of a proton moving at 2.00 3 107 m/s.
(c) Find the magnetic force on a second proton at this
point, moving with the same speed in the opposite direc-
tion. (d) Find the electric force on the second proton.
66. Review. Rail guns have been suggested for launch-
ing projectiles into space without chemical rockets.
A tabletop model rail gun (Fig. P30.66) consists of
two long, parallel, horizontal rails , 5 3.50 cm apart,
bridged by a bar of mass m 5 3.00 g that is free to slide
without friction. The rails and bar have low electric
resistance, and the current is limited to a constant
I 5 24.0 A by a power supply that is far to the left of
the figure, so it has no magnetic effect on the bar. Fig-
ure P30.66 shows the bar at rest at the midpoint of the
rails at the moment the current is established. We wish
to find the speed with which the bar leaves the rails
after being released from the midpoint of the rails.
(a) Find the magnitude of the magnetic field at a dis-
tance of 1.75 cm from a single long wire carrying a
current of 2.40 A. (b)For purposes of evaluating the
magnetic field, model the rails as infinitely long. Using
the result of part (a), find the magnitude and direc-
tion of the magnetic field at the midpoint of the bar.
(c) Argue that this value of the field will be the same
at all positions of the bar to the right of the midpoint
of the rails. At other points along the bar, the field is
in the same direction as at the midpoint, but is larger
in magnitude. Assume the average effective magnetic
field along the bar is five times larger than the field
at the midpoint. With this assumption, find (d) the
magnitude and (e) the direction of the force on the
bar. (f) Is the bar properly modeled as a particle under
constant acceleration? (g)Find the velocity of the bar
after it has traveled a distance d 5 130 cm to the end
of the rails.
67. Fifty turns of insulated wire 0.100 cm in diameter are
tightly wound to form a flat spiral. The spiral fills a
disk surrounding a circle of radius 5.00 cm and extend-
ing to a radius 10.00 cm at the outer edge. Assume the
wire carries a current I at the center of its cross section.
Approximate each turn of wire as a circle. Then a loop
How would you argue for or against this idea? (c) The
upper loop has a mass of 0.021 0kg. Calculate its accel-
eration, assuming the only forces acting on it are the
force in part (a) and the gravitational force.
63. Two long, straight wires cross each other perpendicu-
larly as shown in Figure P30.63. The wires are thin so
that they are effectively in the same plane but do not
touch. Find the magnetic field at a point 30.0 cm above
the point of intersection of the wires along the z axis;
that is, 30.0 cm out of the page, toward you.
64. Two coplanar and concentric circular loops of wire
carry currents of I
5 5.00 A and I
5 3.00 A in oppo-
site directions as in Figure P30.64. If r
5 12.0 cm and
5 9.00 cm, what are (a) the magnitude and (b) the
direction of the net magnetic field at the center of the
two loops? (c) Let r
remain fixed at 12.0 cm and let r
be a variable. Determine the value of r
such that the
net field at the center of the loops is zero.
of current exists at radius 5.05 cm, another at 5.15 cm,
and so on. Numerically calculate the magnetic field at
the center of the coil.
68. An infinitely long, straight wire carrying a current I
is partially surrounded by a loop as shown in Figure
P30.68. The loop has a length L and radius R, and
it carries a current I
. The axis of the loop coincides
with the wire. Calculate the magnetic force exerted on
69. Consider a solenoid of length , and radius a containing
N closely spaced turns and carrying a steady current
I. (a)In terms of these parameters, find the magnetic
field at a point along the axis as a function of posi-
tion x from the end of the solenoid. (b) Show that as ,
becomes very long, B approaches m
NI/2, at each end
of the solenoid.
70. We have seen that a long solenoid produces a uniform
magnetic field directed along the axis of a cylindrical
region. To produce a uniform magnetic field directed
parallel to a diameter of a cylindrical region, however,
one can use the saddle coils illustrated in Figure P30.70.
The loops are wrapped over a long, somewhat flat-
tened tube. Figure P30.70a shows one wrapping of wire
around the tube. This wrapping is continued in this
manner until the visible side has many long sections
of wire carrying current to the left in Figure P30.70a
and the back side has many lengths carrying current to
the right. The end view of the tube in Figure P30.70b
shows these wires and the currents they carry. By wrap-
ping the wires carefully, the distribution of wires can
take the shape suggested in the end view such that
the overall current distribution is approximately the
superposition of two overlapping, circular cylinders of
radius R (shown by the dashed lines) with uniformly
distributed current, one toward you and one away from
you. The current density J is the same for each cylinder.
The center of one cylinder is described by a position
relative to the center of the other cylinder.
Prove that the magnetic field inside the hollow tube is
Jd/2 downward. Suggestion: The use of vector meth-
ods simplifies the calculation.
71. A thin copper bar of length , 5 10.0 cm is supported
horizontally by two (nonmagnetic) contacts at its ends.
The bar carries a current of I
5 100 A in the negative
x direction as shown in Figure P30.71. At a distance
h 5 0.500 cm below one end of the bar, a long, straight
wire carries a current of I
5 200A in the positive z
direction. Determine the magnetic force exerted on
72. In Figure P30.72, both currents in the infinitely long
wires are 8.00 A in the negative x direction. The wires
are separated by the distance 2a 5 6.00cm. (a)Sketch
the magnetic field pattern in the yz plane. (b) What
is the value of the magnetic field at the origin? (c) At
(y 5 0, z S `)? (d)Find the magnetic field at points
along the z axis as a function of z. (e) At what distance
d along the positive z axis is the magnetic field a maxi-
mum? (f) What is this maximum value?
73. A wire carrying a current I is bent into the shape of
an exponential spiral, r 5 eu, from u 5 0 to u 5 2p as
suggested in Figure P30.73 (page 934). To complete a
loop, the ends of the spiral are connected by a straight
wire along the x axis. (a) The angle b between a radial
Wire lengths carrying
current out of the page
Wire lengths carrying
current into the page
Chapter 30 Sources of the Magnetic Field
line and its tangent line at any point on a curve r 5 f(u)
is related to the function by
Use this fact to show that b 5 p/4. (b) Find the mag-
netic field at the origin.
r = e
74. A sphere of radius R has a uniform
volume charge density r. When the
sphere rotates as a rigid object with
angular speed v about an axis through
its center (Fig. P30.74), determine
(a) the magnetic field at the center
of the sphere and (b)the magnetic
moment of the sphere.
75. A long, cylindrical conductor of radius
a has two cylindrical cavities each of diameter a through
its entire length as shown in the end view of Figure
P30.75. A current I is directed out of the page and is uni-
form through a cross section of the conducting material.
Find the magnitude and direction of the magnetic field
in terms of m
, I, r, and a at (a) point P
and (b) point P
76. A wire is formed into the shape of a square of edge
length L (Fig. P30.76). Show that when the current in
the loop is I, the magnetic field at point P a distance x
from the center of the square along its axis is
77. The magnitude of the force on a magnetic dipole m
aligned with a nonuniform magnetic field in the
positive x direction is F
0dB/dx. Suppose two flat
loops of wire each have radius R and carry a current I.
(a) The loops are parallel to each other and share the
same axis. They are separated by a variable distance
x .. R. Show that the magnetic force between them
varies as 1/x4. (b) Find the magnitude of this force,
taking I 5 10.0 A, R 5 0.500 cm, and x 5 5.00cm.
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