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PROBLEMS
1, 2, 3 = straightforward, intermediate, challenging
Manual/Study Guide
= biomedical application
Section 19.3 Magnetic Fields
1. An electron gun fires electrons into a
magnetic field that is directed straight
downward. Find the direction of the force
exerted by the field on an electron for each of
the following directions of the electron’s
velocity: (a) horizontal and due north; (b)
horizontal and 30° west of north; (c) due north,
but at 30° below the horizontal; (d) straight
upward. (Remember that an electron has a
negative charge.)
2. (a) Find the direction of the force on a proton
(a positively charged particle) moving through
the magnetic fields in Figure P19.2, as shown.
(b) Repeat part (a), assuming the moving
particle is an electron.
= full solution available in Student Solutions
direction of the magnetic force acting on it is as
indicated.
Figure P19.3 (Problems 3 and 12) For Problem 12,
replace the velocity vector with a current in that
direction.
4. Determine the initial direction of the
deflection of charged particles as they enter the
magnetic fields shown in Figure P19.4.
Figure P19.4
5. At the Equator near Earth’s surface, the
magnetic field is approximately 50.0 μT
northward and the electric field is about 100 N/C
downward in fair weather. Find the
gravitational, electric, and magnetic forces on an
electron with an instantaneous velocity of 6.00 ×
106 m/s directed to the east in this environment.
Figure P19.2 (Problems 2 and 13) For Problem 13,
replace the velocity vector with a current in that
direction.
3. Find the direction of the magnetic field acting
on the positively charged particle moving in the
various situations shown in Figure P19.3, if the
6. A proton travels with a speed of 3.0 × 106 m/s
at an angle of 37° with the direction of a
magnetic field of 0.30 T in the +y direction.
What are (a) the magnitude of the magnetic
force on the proton and (b) the proton’s
acceleration?
7. What velocity would a proton need to circle
Earth 1 000 km above the magnetic equator,
where Earth’s magnetic field is directed
horizontally north and has a magnitude of 4.00 ×
10–8 T?
8. An electron is accelerated through 2 400 V
from rest and then enters a region where there is
a uniform 1.70-T magnetic field. What are the
(a) maximum and (b) minimum magnitudes of
the magnetic force this charge can experience?
9. A proton moves perpendicularly to a uniform
magnetic field B at 1.0 × 107 m/s and
experiences an acceleration of 2.0 × 1013 m/s2 in
the +x direction when its velocity is in the +z
direction. Determine the magnitude and
direction of the field.
10. Sodium ions (Na+) move at 0.851 m/s
through a bloodstream in the arm of a person
standing near a large magnet. The magnetic field
has a strength of 0.254 T and makes an angle of
51.0° with the motion of the sodium ions. The
arm contains 100 cm3 of blood with 3.00 × 1020
Na+ ions per cubic centimeter. If no other ions
were present in the arm, what would be the
magnetic force on the arm?
Section 19.4 Magnetic Force on a CurrentCarrying Conductor
11. A current I = 15 A is directed along the
positive x axis and perpendicularly to a magnetic
field. The conductor experiences a magnetic
force per unit length of 0.12 N/m in the negative
y direction. Calculate the magnitude and
direction of the magnetic field in the region
through which the current passes.
12. In Figure P19.3, assume that in each case the
velocity vector shown is replaced with a wire
carrying a current in the direction of the velocity
vector. For each case, find the direction of the
magnetic field that will produce the magnetic
force shown.
13. In Figure P19.2, assume that in each case the
velocity vector shown is replaced with a wire
carrying a current in the direction of the velocity
vector. For each case, find the direction of the
magnetic force acting on the wire.
14. A wire carries a steady current of 2.40 A. A
straight section of the wire is 0.750 m long and
lies along the x axis within a uniform magnetic
field of magnitude 1.60 T in the positive z
direction. If the current is in the + x direction,
what is the magnetic force on the section of
wire?
15. A wire carries a current of 10.0 A in a
direction that makes an angle of 30.0° with the
direction of a magnetic field of strength 0.300 T.
Find the magnetic force on a 5.00-m length of
the wire.
16. At a certain location, Earth has a magnetic
field of 0.60 × 10–4 T pointing 75° below the
horizontal in a north-south plane. A 10.0-m-long
straight wire carries a 15-A current. (a) If the
current is directed horizontally toward the east,
what are the magnitude and direction of the
magnetic force on the wire? (b) What are the
magnitude and direction of the force if the
current is directed vertically upward?
17. A wire with a mass per unit length of 1.00
g/cm is placed on a horizontal surface with a
coefficient of friction of 0.200. The wire carries
a current of 1.50 A eastward and moves
horizontally to the north. What are the
magnitude and the direction of the smallest
vertical magnetic field that enables the wire to
move in this fashion?
18. A conductor suspended by two flexible wires
as shown in Figure P19.18 has a mass per unit
length of 0.040 0 kg/m. What current must exist
in the conductor for the tension in the supporting
wires to be zero when the magnetic field is 3.60
T into the page? What is the required direction
for the current?
Figure P19.18
19. An unusual message delivery system is
pictured in Figure P19.19. A 15-cm length of
conductor that is free to move is held in place
between two thin conductors. When a 5.0-A
current is directed as shown in the figure, the
wire segment moves upward at a constant
velocity. If the mass of the wire is 15 g, find the
magnitude and direction of the minimum
magnetic field that is required to move the wire.
(The wire slides without friction on the two
vertical conductors.)
Figure P19.21
Section 19.5 Torque on a Current Loop and
Electric Motors
22. A current of 17.0 mA is maintained in a
single circular loop with a circumference of 2.00
m. A magnetic field of 0.800 T is directed
parallel to the plane of the loop. What is the
magnitude of the torque exerted by the magnetic
field on the loop?
23. An 8-turn coil encloses an elliptical area
having a major axis of 40.0 cm and a minor axis
of 30.0 cm (Fig. P19.23). The coil lies in the
plane of the page and has a 6.00-A current
flowing clockwise around it. If the coil is in a
uniform magnetic field of 2.00 × 10–4 T, directed
toward the left of the page, what is the
magnitude of the torque on the coil? (Hint: The
area of an ellipse is A = πab, where a and b are
the semi-major and semi-minor axes of the
ellipse.)
Figure P19.19
20. A thin, horizontal copper rod is 1.00 m long
and has a mass of 50.0 g. What is the minimum
current in the rod that can cause it to float in a
horizontal magnetic field of 2.00 T?
21. In Figure P19.21, the cube is 40.0 cm on
each edge. Four straight segments of wire—ab,
bc, cd, and da—form a closed loop that carries a
current I = 5.00 A, in the direction shown. A
uniform magnetic field of magnitude B = 0.020
0 T is in the positive y direction. Determine the
magnitude and direction of the magnetic force
on each segment.
Figure P19.23
24. A rectangular loop consists of 100 closely
wrapped turns and has dimensions 0.40 m by
0.30 m. The loop is hinged along the y axis, and
the plane of the coil makes an angle of 30.0°
with the x axis (Fig. P19.24). What is the
magnitude of the torque exerted on the loop by a
uniform magnetic field of 0.80 T directed along
the x axis, when the current in the windings has
a value of 1.2 A in the direction shown? What is
the expected direction of rotation of the loop?
28. A cosmic-ray proton in interstellar space has
an energy of 10.0 MeV and executes a circular
orbit having a radius equal to that of Mercury’s
orbit around the Sun (5.80 × 1010 m). What is
the magnetic field in that region of space?
29. Figure P19.29a is a diagram of a device
called a velocity selector, in which particles of a
specific velocity pass through undeflected but
those with greater or lesser velocities are
deflected either upward or downward. An
electric field is directed perpendicularly to a
magnetic field. This produces on the charged
particle an electric force and a magnetic force
that can be equal in magnitude and opposite in
direction (Fig. P19.29b), and hence cancel.
Show that particles with a speed of v = E/B will
pass through undeflected.
Figure P19.24
25. A long piece of wire with a mass of 0.100 kg
and a total length of 4.00 m is used to make a
square coil with a side of 0.100 m. The coil is
hinged along a horizontal side, carries a 3.40-A
current, and is placed in a vertical magnetic field
with a magnitude of 0.010 0 T. (a) Determine
the angle that the plane of the coil makes with
the vertical when the coil is in equilibrium. (b)
Find the torque acting on the coil due to the
magnetic force at equilibrium.
26. A copper wire is 8.00 m long, and has a
cross-sectional area of 1.00 × 10–4 m2. This wire
forms a 1-turn loop in the shape of a square and
is then connected to a battery that applies a
potential difference of 0.100 V. If the loop is
placed in a uniform magnetic field of magnitude
0.400 T, what is the maximum torque that can
act on it? The resistivity of copper is 1.70 × 10–8
Ω · m.
Section 19.6 Motion of a Charged Particle in
a Magnetic Field
27. A particle with a +2.0 μC charge and a
kinetic energy of 0.090 J is fired into a uniform
magnetic field of magnitude 0.10 T. If the
particle moves in a circular path of radius 3.0 m,
determine its mass.
Figure P19.29
30. Consider the mass spectrometer shown
schematically in Figure P19.30. The electric
field between the plates of the velocity selector
is 950 V/m, and the magnetic fields in both the
velocity selector and the deflection chamber
have magnitudes of 0.930 T. Calculate the radius
of the path in the system for a singly charged ion
with mass m = 2.18 × 10–26 kg. (Hint: See
Problem 29.)
Figure P19.30 A mass spectrometer. Charged
particles are first sent through a velocity selector.
They then enter a region where a magnetic field B0
(inward) causes positive ions to move in a
semicircular path and strike a photographic film at P.
31. A singly charged positive ion has a mass of
2.50 × 10–26 kg. After being accelerated through
a potential difference of 250 V, the ion enters a
magnetic field of 0.500 T, in a direction
perpendicular to the field. Calculate the radius of
the ion’s path in the field.
32. A mass spectrometer is used to examine the
isotopes of uranium. Ions in the beam emerge
from the velocity selector at a speed of 3.00 ×
105 m/s and enter a uniform magnetic field of
0.600 T directed perpendicularly to the velocity
of the ions. What is the distance between the
impact points formed on the photographic plate
by singly charged ions of 235U and 238U?
33. An electron moves in a circular path
perpendicular to a constant magnetic field with a
magnitude of 1.00 mT. If the angular momentum
of the electron about the center of the circle is
4.00 × 10–25 J · s, determine (a) the radius of the
circular path and (b) the speed of the electron.
Section 19.7 Magnetic Field of a Long,
Straight Wire and Ampère’s Law
34. Find the direction of the current in the wire
in Figure P19.34 that would produce a magnetic
field directed as shown, in each case.
Figure P19.34
35. A lightning bolt may carry a current of 1.00
× 104 A for a short period of time. What is the
resulting magnetic field 100 m from the bolt?
Suppose that the bolt extends far above and
below the point of observation.
36. In 1962, measurements of the magnetic field
of a large tornado were made at the Geophysical
Observatory in Tulsa, Oklahoma. If the
tornado’s field was B = 1.50 × 10–8 T pointing
north when the tornado was 9.00 km east of the
observatory, what current was carried up or
down the funnel of the tornado? Model the
vortex as a long straight wire carrying a current.
37. At what distance from a long, straight wire
carrying a current of 5.0 A is the magnetic field
due to the wire equal to the strength of Earth’s
field, approximately 5.0 × 10–5 T?
38. The two wires shown in Figure P19.38 carry
currents of 5.00 A in opposite directions and are
separated by 10.0 cm. Find the direction and
magnitude of the net magnetic field (a) at a point
midway between the wires, (b) at point P1 (10.0
cm to the right of the wire on the right), and (c)
at point P2 (20.0 cm to the left of the wire on the
left).
Figure P19.40
Figure P19.38
39. Four long, parallel conductors carry equal
currents of I = 5.00 A. Figure P19.39 is an end
view of the conductors. The current direction is
into the page at points A and B (indicated by the
crosses) and out of the page at C and D
(indicated by the dots). Calculate the magnitude
and direction of the magnetic field at point P,
located at the center of the square of edge length
0.200 m.
41. 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 P19.41.
What is the magnetic field at point P located at x
= 4.00 m, y = 3.00 m?
Figure P19.41
Figure P19.39
40. The two wires in Figure P19.40 carry
currents of 3.00 A and 5.00 A in the direction
indicated. (a) Find the direction and magnitude
of the magnetic field at a point midway between
the wires. (b) Find the magnitude and direction
of the magnetic field at point P, located 20.0 cm
above the wire carrying the 5.00-A current.
42. A long, straight wire lies on a horizontal
table and carries a current of 1.20 μA. In a
vacuum, a proton moves parallel to the wire
(opposite the current) with a constant velocity of
2.30 × 104 m/s at a constant distance d above the
wire. Determine the value of d. You may ignore
the magnetic field due to Earth.
43. The magnetic field 40.0 cm away from a
long, straight wire carrying current 2.00 A is
1.00 μT. (a) At what distance is it 0.100 μT? (b)
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 3.00 mm 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?
Section 19.8 Magnetic Force Between Two
Parallel Conductors
44. Two parallel wires are 10.0 cm apart, and
each carries a current of 10.0 A. (a) If the
currents are in the same direction, find the force
per unit length exerted by one of the wires on
the other. Are the wires attracted or repelled? (b)
Repeat the problem with the currents in opposite
directions.
45. A wire with a weight per unit length of
0.080 N/m is suspended directly above a second
wire. The top wire carries a current of 30.0 A,
and the bottom wire carries a current of 60.0 A.
Find the distance of separation between the
wires so that the top wire will be held in place
by magnetic repulsion.
46. In Figure P19.46, the current in the long,
straight wire is I1 = 5.00 A, and the wire lies in
the plane of the rectangular loop, which carries
10.0 A. The dimensions are c = 0.100 m, a =
0.150 m, and  = 0.450 m. Find the magnitude
and direction of the net force exerted by the
magnetic field due to the straight wire on the
loop.
Section 19.10 Magnetic Field of a Solenoid
47. What current is required in the windings of a
long solenoid that has 1 000 turns uniformly
distributed over a length of 0.400 m in order to
produce a magnetic field of magnitude 1.00 ×
10–4 T at the center of the solenoid?
48. It is desired to construct a solenoid that has a
resistance of 5.00 Ω (at 20°C) and that produces
a magnetic field at its center of 4.00 × 10–2 T
when it carries a current of 4.00 A. 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
length the solenoid should have.
49. A single-turn square loop of wire, 2.00 cm
on a side, carries a counterclockwise 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 30 turns
per centimeter and carries a counterclockwise
current of 15.0 A. Find the force on each side of
the loop and the torque acting on it.
50. An electron moves at a speed of 1.0 × 104
m/s in a circular path of radius of 2.0 cm inside a
solenoid. The magnetic field of the solenoid is
perpendicular to the plane of the electron’s path.
Find (a) the strength of the magnetic field inside
the solenoid and (b) the current in the solenoid if
it has 25 turns per centimeter.
ADDITIONAL PROBLEMS
51. A circular coil consisting of a single loop of
wire has a radius of 30.0 cm and carries a
current of 25 A. It is placed in an external
magnetic field of 0.30 T. Find the torque on the
wire when the plane of the coil makes an angle
of 35° with the direction of the field.
Figure P19.46
52. An electron enters a region of magnetic field
of magnitude 0.010 0 T, traveling perpendicular
to the linear boundary of the region. The
direction of the field is perpendicular to the
velocity of the electron. (a) Determine the time it
takes for the electron to leave the “field-filled”
region, noting that its path is a semicircle. (b)
Find the kinetic energy of the electron if the
radius of its semicircular path is 2.00 cm.
53. Two long, straight wires cross each other at
right angles, as shown in Figure P19.53. (a) Find
the direction and magnitude of the magnetic
field at point P, which is in the same plane as the
two wires. (b) Find the magnetic field at a point
30.0 cm above the point of intersection (30.0 cm
out of the page, toward you).
Figure P19.56
Figure P19.53
54. A 0.200-kg metal rod carrying a current of
10.0 A glides on two horizontal rails 0.500 m
apart. What vertical magnetic field is required to
keep the rod moving at a constant speed if the
coefficient of kinetic friction between the rod
and rails is 0.100?
57. A heart surgeon monitors the flow rate of
blood through an artery using an
electromagnetic flowmeter (shown
schematically in Fig. P19.57). Electrodes A and
B make contact with the outer surface of the
blood vessel, which has interior diameter 3.00
mm. (a) For a magnetic field magnitude of 0.040
0 T, a potential difference of 160 μV appears
between the electrodes. Calculate the speed of
the blood. (b) Verify that electrode A is positive,
as shown. Does the sign of the emf depend on
whether the mobile ions in the blood are
predominantly positively or negatively charged?
Explain.
55. Two species of singly charged positive ions
of masses 20.0 × 10–27 kg and 23.4 × 10–27 kg
enter a magnetic field at the same location with a
speed of 1.00 × 105 m/s. If the strength of the
field is 0.200 T, and the ions move
perpendicularly to the field, find their distance
of separation after they complete one half of
their circular path.
56. Two parallel conductors carry currents in
opposite directions, as shown in Figure P19.56.
One conductor carries a current of 10.0 A. Point
A is the midpoint between the wires, and point C
is 5.00 cm to the right of the 10.0-A current. I is
adjusted so that the magnetic field at C is zero.
Find (a) the value of the current I and (b) the
value of the magnetic field at A.
Figure P19.57
58. Two circular loops are parallel, coaxial, and
almost in contact, 1.00 mm apart (Fig. P19.58).
Each loop is 10.0 cm in radius. The top loop
carries a clockwise current of 140 A. The bottom
loop carries a counterclockwise current of 140
A. (a) Calculate the magnetic force that the
bottom loop exerts on the top loop. (b) The
upper loop has a mass of 0.021 0 kg. Calculate
its acceleration, assuming that the only forces
acting on it are the force in part (a) and its
weight. (Hint: The distance between the loops is
small in comparison to the radius of curvature,
so the loops may be treated as long, straight,
parallel wires.)
Figure P19.58
62. A uniform horizontal wire with a linear mass
density of 0.50 g/m carries a 2.0-A current. It is
placed in a constant magnetic field, with a
strength of 4.0 × 10–3 T, that is horizontal and
perpendicular to the wire. As the wire moves
upward starting from rest, (a) what is its
acceleration and (b) how long does it take to rise
50 cm? Neglect the magnetic field of Earth.
63. Three long, parallel conductors carry
currents of I = 2.0 A. Figure P19.63 is an end
view of the conductors, with each current
coming out of the page. Given that a = 1.0 cm,
determine the magnitude and direction of the
magnetic field at points (a) A, (b) B, and (c) C.
59. A 1.00-kg ball having net charge Q = 5.00
μC is thrown out of a window horizontally at a
speed v = 20.0 m/s. The window is at a height h
= 20.0 m above the ground. A uniform
horizontal magnetic field of magnitude B =
0.010 0 T is perpendicular to the plane of the
ball’s trajectory. Find the magnitude of the
magnetic force acting on the ball just before it
hits the ground. (Hint: Ignore magnetic forces in
finding the ball’s final velocity.)
60. At the Fermilab accelerator in Batavia,
Illinois, protons having momentum 4.80 × 10–16
kg · m/s are held in a circular orbit of radius
1.00 km by an upward magnetic field. What is
the magnitude of this field?
61. Two long, parallel conductors carry currents
I1 = 3.00 A and I2 = 3.00 A, both directed into
the page in Figure P19.61. Determine the
magnitude and direction of the resultant
magnetic field at P.
Figure P19.63
64. Two long, parallel wires, each with a mass
per unit length of 40 g/m, are supported in a
horizontal plane by 6.0-cm long strings, as
shown in Figure P19.64. Each wire carries the
same current I, causing the wires to repel each
other so that the angle θ between the supporting
strings is 16°. (a) Are the currents in the same or
opposite directions? (b) Determine the
magnitude of each current.
Figure P19.64
Figure P19.61
65. Protons having a kinetic energy of 5.00 MeV
are moving in the positive x direction and enter a
magnetic field of 0.050 0 T in the z direction,
out of the plane of the page, and extending from
x = 0 to x = 1.00 m as in Figure P19.65. (a)
Calculate the y component of the protons’
momentum as they leave the magnetic field. (b)
Find the angle α between the initial velocity
vector of the proton beam and the velocity
vector after the beam emerges from the field.
(Hint: Neglect relativistic effects and note that 1
eV = 1.60 × 10–19 J.)
66. A straight wire of mass 10.0 g and length 5.0
cm is suspended from two identical springs that,
in turn, form a closed circuit (Fig. P19.66). The
springs stretch a distance of 0.50 cm under the
weight of the wire. The circuit has a total
resistance of 12 Ω. When a magnetic field is
turned on, directed out of the page (indicated by
the dots in Fig. P19.66), the springs are observed
to stretch an additional 0.30 cm. What is the
strength of the magnetic field? (The upper
portion of the circuit is fixed.)
Figure P19.65
Figure P19.66