Walker, Physics, 3rd Edition
Chapter 22
** Can skip to Problems: 6, 7, 9, 15, 19, 28, 29, 35, 36, 39, 42 **
Conceptual Questions
(Answers to odd-numbered Conceptual Questions can be found in the back of the book, beginning on page ANSxxx.)
1.
Two charged particles move at right angles to a magnetic field and deflect in opposite directions. Can one
conclude that the particles have opposite charges?
2.
An electron moves with constant velocity through a region of space that is free of magnetic fields. Can one
conclude that the electric field is zero in this region? Explain.
3.
An electron moves with constant velocity through a region of space that is free of electric fields. Can one
conclude that the magnetic field is zero in this region? Explain.
4.
Describe how the motion of a charged particle can be used to distinguish between an electric and a magnetic
field.
5.
Explain how a charged particle moving in a circle of small radius can take the same amount of time to complete
an orbit as an identical particle orbiting in a circle of large radius.
6.
A current-carrying wire is placed in a region with a uniform magnetic field. The wire experiences zero
magnetic force. Explain.
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Conceptual Exercises
(Answers to odd-numbered Conceptual Exercises can be found in the back of the book, beginning on page ANSxxx.)
1.
Proton 1 moves with a speed v from the east coast to the west coast in the continental United States; proton 2
moves with the same speed from the southern United States toward Canada. Is the magnitude of the magnetic
force experienced by proton 2 greater than, less than, or equal to the force experienced by proton 1? Explain.
2.
An electron moves west to east in the continental United States. Does the magnetic force experienced by the
electron point in a direction that is generally north, south, east, west, upward, or downward? Explain.
3.
Suppose particles A, B, and C in Figure 22–31 have identical masses and charges of the same magnitude. Rank
the particles in order of increasing speed. Indicate ties where appropriate.
4.
At a point near the equator, the Earth’s magnetic field is horizontal and points to the north. If an electron is
moving vertically upward at this point, does the magnetic force acting on it point north, south, east, west,
upward, or downward? Explain.
5.
A proton is to orbit the Earth at the equator using the Earth’s magnetic field to supply part of the necessary
centripetal force. Should the proton move eastward or westward? Explain.
6.
Referring to Figure 22–31, what is the sign of the charge for each of the three particles? Explain.
7.
Suppose the three particles in Figure 22–31 have the same mass and speed. Rank the particles in order of
increasing magnitude of their charge. Indicate ties where appropriate.
8.
A velocity selector is to be constructed using a magnetic field in the positive y direction. If positively charged
particles move through the selector in the positive z direction, (a) what must be the direction of the electric
field? (b) Repeat part (a) for the case of negatively charged particles.
9.
An electron moving in the positive x direction, at right angles to a magnetic field, experiences a magnetic force
in the positive y direction. What is the direction of the magnetic field?
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10. A positively charged particle moves through a region with a uniform electric field pointing toward the top of
the page and a uniform magnetic field pointing into the page. The particle can have one of the four velocities
shown in Figure 22–32. (a) Rank the four possibilities in order of increasing magnitude of the net force
( F1 , F2 , F3 , and F4 ) the particle experiences. Indicate ties where appropriate. (b) Which of the four velocities
could potentially result in zero net force?
11. Suppose the fields in Figure 22–32 are interchanged, with the magnetic field pointing toward the top of the
page and the electric field pointing into the page. (a) Rank the four possibilities in order of increasing
magnitude of the net force ( F1 , F2 , F3 , and F4 ) the particle experiences. Indicate ties where appropriate. (b)
Which of the four velocities could potentially result in zero net force?
12. The accompanying photograph shows an electron beam whose initial direction of motion is horizontal, from
right to left. A magnetic field deflects the beam downward. What is the direction of the magnetic field?
13. A proton follows the path shown in Figure 22–33 as it moves through three regions with different uniform
magnetic fields, B1 , B2 , and B3 . In each region the proton completes a half-circle. (a) Rank the three fields in
order of increasing magnitude. Indicate ties where appropriate. (b) Give the direction of each of the fields.
14. Suppose the speed of the proton in Figure 22–33 is increased. (a) Does the radius of each half-circular path
segment increase, decrease, or stay the same? Explain. (b) Does the time spent in each of the field regions
increase, decrease, or stay the same? Explain.
15. The three wires shown in Figure 22–34 are long and straight, and they each carry a current of the same
magnitude, I. The currents in wires 1 and 3 are out of the page; the current in wire 2 is into the page. What is
the direction of the magnetic force experienced by wire 3?
16. Each of the current-carrying wires in Figure 22–34 is long and straight, and carries the current I either into or
out of the page, as shown. What is the direction of the net magnetic field produced by these three wires at the
center of the triangle?
17. The four wires shown in Figure 22–35 are long and straight, and they each carry a current of the same
magnitude, I. The currents in wires 1, 2, and 3 are out of the page; the current in wire 4 is into the page. What is
the direction of the magnetic force experienced by wire 2?
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18. Each of the current-carrying wires in Figure 22–35 is long and straight, and carries the current I either into or
out of the page, as shown. What is the direction of the net magnetic field produced by these four wires at the
center of the square?
19. For each of the square current loops shown in Figure 22–36, indicate whether there will be a tendency for the
loop to rotate clockwise, counterclockwise, or not at all, when viewed from above along the indicated axis.
20. A loop of wire is connected to the terminals of a battery, as indicated in Figure 22–37. If the loop is to attract
the bar magnet, which of the terminals, A or B, should be the positive terminal of the battery? Explain.
21. The number of turns in a solenoid is doubled, and at the same time its length is doubled. Does the magnetic
field within the solenoid increase, decrease, or stay the same? Explain.
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Problems: 6, 7, 9, 15, 19, 28, 29, 35, 36, 39, 42
Note: IP denotes an integrated conceptual/quantitative problem. BIO identifies problems of biological or medical
interest. Red bullets (•, ••, •••) are used to indicate the level of difficulty of each problem.
Section 22–2 The Magnetic Force on Moving Charges
1.
• What is the acceleration of a proton moving with a speed of 9.5 m/s at right angles to a magnetic field of 1.6
T?
2.
• An electron moves at right angles to a magnetic field of 0.12 T. What is its speed if the force exerted on it is
8.9 × 10 −15 N?
3.
• A negatively charged ion moves due north with a speed of 1.5 × 10 6 m s at the Earth’s equator. What is the
magnetic force exerted on this ion?
4.
• A proton high above the equator approaches the Earth moving straight downward with a speed of 355 m/s.
Find the acceleration of the proton, given that the magnetic field at its altitude is 4.05 × 10 −5 T.
5.
•• A 0.32 - μC particle moves with a speed of 16 m/s through a region where the magnetic field has a strength of
0.95 T. At what angle to the field is the particle moving if the force exerted on it is (a) 4.8 × 10 −6 N, (b)
3.0 × 10 −6 N, or (c) 1.0 × 10 −7 N?
6.
•• A particle with a charge of 14 μC experiences a force of 2.2 × 10 −4 N when it moves at right angles to a
magnetic field with a speed of 27 m/s. What force does this particle experience when it moves with a speed of
6.3 m/s at an angle of 25° relative to the magnetic field?
7.
•• An ion experiences a magnetic force of 6.2 × 10 −16 N when moving in the positive x direction but no
magnetic force when moving in the positive y direction. What is the magnitude of the magnetic force exerted on
the ion when it moves in the x-y plane along the line x = y ? Assume that the ion’s speed is the same in all
cases.
8.
•• An electron moving with a speed of 9.1 × 10 5 m s in the positive x direction experiences zero magnetic force.
When it moves in the positive y direction, it experiences a force of 2.0 × 10 −13 N that points in the negative z
direction. What is the direction and magnitude of the magnetic field?
9.
•• IP Two charged particles with different speeds move one at a time through a region of uniform magnetic
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field. The particles move in the same direction and experience equal magnetic forces. (a) If particle 1 has four
times the charge of particle 2, which particle has the greater speed? Explain. (b) Find the ratio of the speeds,
v1 v2 .
10. •• A 6.60 - μC particle moves through a region of space where an electric field of magnitude 1250 N/C points in
the positive x direction, and a magnetic field of magnitude 1.02 T points in the positive z direction. If the net
force acting on the particle is 6.23 × 10 −3 N in the positive x direction, find the magnitude and direction of the
particle’s velocity. Assume the particle’s velocity is in the x-y plane.
11. ••• When at rest, a proton experiences a net electromagnetic force of magnitude 8.0 × 10 −13 N pointing in the
positive x direction. When the proton moves with a speed of 1.5 × 10 6 m s in the positive y direction, the net
electromagnetic force on it decreases in magnitude to 7.5 × 10 −13 N, still pointing in the positive x direction.
Find the magnitude and direction of (a) the electric field and (b) the magnetic field.
Section 22–3 The Motion of Charged Particles in a Magnetic Field
12. • Find the radius of an electron’s orbit when it moves perpendicular to a magnetic field of 0.45 T with a speed
of 6.27 × 10 5 m s .
13. • Find the radius of a proton’s orbit when it moves perpendicular to a magnetic field of 0.45 T with a speed of
6.27 × 10 5 m s .
14. • Charged particles pass through a velocity selector with electric and magnetic fields at right angles to each
other, as shown in Figure 22–38. If the electric field has a magnitude of 450 N/C and the magnetic field has a
magnitude of 0.18 T, what speed must the particles have to pass through the selector undeflected?
15. • The velocity selector in Figure 22–39 is designed to allow charged particles with a speed of 4.5 × 10 3 m s to
pass through undeflected. Find the direction and magnitude of the required electric field, given that the
magnetic field has a magnitude of 0.96 T.
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16. •• IP BIO The artery in Figure 22–11 has an inside diameter of 2.75 mm and passes through a region where the
magnetic field is 0.065 T. (a) If the voltage difference between the electrodes is 195 μ V, what is the speed of
the blood? (b) Which electrode is at the higher potential? Does your answer depend on the sign of the ions in
the blood? Explain.
17. •• An electron accelerated from rest through a voltage of 410 V enters a region of constant magnetic field. If the
electron follows a circular path with a radius of 17 cm, what is the magnitude of the magnetic field?
18. •• A 12.5- μC particle with a mass of 2.80 × 10 −5 kg moves perpendicular to a 1.01-T magnetic field in a
circular path of radius 26.8 m. (a) How fast is the particle moving? (b) How long will it take the particle to
complete one orbit?
19. •• IP When a charged particle enters a region of uniform magnetic field, it follows a circular path, as indicated
in Figure 22–40. (a) Is this particle positively or negatively charged? Explain. (b) Suppose that the magnetic
field has a magnitude of 0.180 T, the particle’s speed is 6.0 × 10 6 m s , and the radius of its path is 52.0 cm.
Find the mass of the particle, given that its charge has a magnitude of 1.60 × 10 −19 C. Give your result in atomic
mass units, u, where 1 u = 1.67 × 10 −27 kg.
20. •• A proton with a kinetic energy of 4.9 × 10 −16 J moves perpendicular to a magnetic field of 0.26 T. What is
the radius of its circular path?
21. •• IP An alpha particle (the nucleus of a helium atom) consists of two protons and two neutrons, and has a mass
of 6.64 × 10 −27 kg. A horizontal beam of alpha particles is injected with a speed of 1.3 × 10 5 m s into a region
with a vertical magnetic field of magnitude 0.155 T. (a) How long does it take for an alpha particle to move
halfway through a complete circle? (b) If the speed of the alpha particle is doubled, does the time found in part
(a) increase, decrease, or stay the same? Explain. (c) Repeat part (a) for alpha particles with a speed of
2.6 × 10 5 m s .
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22. ••• An electron and a proton move in circular orbits in a plane perpendicular to a uniform magnetic field B.
Find the ratio of the radii of their circular orbits when the electron and the proton have (a) the same momentum
and (b) the same kinetic energy.
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Section 22–4 The Magnetic Force Exerted on a Current-Carrying Wire
23. • What is the magnetic force exerted on a 2.15-m length of wire carrying a current of 0.695 A perpendicular to
a magnetic field of 0.720 T?
24. • A wire with a current of 2.8 A is at an angle of 36.0° relative to a magnetic field of 0.88 T. Find the force
exerted on a 2.25-m length of the wire.
25. • The magnetic force exerted on a 1.2-m segment of straight wire is 1.6 N. The wire carries a current of 3.0 A in
a region with a constant magnetic field of 0.50 T. What is the angle between the wire and the magnetic field?
26. • Consider a current loop immersed in a magnetic field, as in Figure 22–36 (a) (Conceptual Exercise 19). It is
given that B = 0.34 T and I = 9.5 A. In addition, the loop is a square 0.46 m on a side. Find the magnitude of
the magnetic force exerted on each side of the loop.
27. •• A 0.45-m copper rod with a mass of 0.17 kg carries a current of 11 A in the positive x direction. What are the
magnitude and direction of the minimum magnetic field needed to levitate the rod?
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28. •• The long, thin wire shown in Figure 22–41 is in a region of constant magnetic field B. The wire carries a
current of 6.2 A and is oriented at an angle of 7.5° to the direction of the magnetic field. (a) If the magnetic
force exerted on this wire per meter is 0.033 N, what is the magnitude of the magnetic field? (b) At what angle
will the force exerted on the wire per meter be equal to 0.015 N?
29. •• A wire with a length of 2.7 m and a mass of 0.75 kg is in a region of space with a magnetic field of 0.84 T.
What is the minimum current needed to levitate the wire?
30. •• A high-voltage power line carries a current of 110 A at a location where the Earth’s magnetic field has a
magnitude of 0.59 G and points to the north, 72° below the horizontal. Find the direction and magnitude of the
magnetic force exerted on a 250-m length of wire if the current in the wire flows (a) horizontally toward the
east or (b) horizontally toward the south.
31. ••• A metal bar of mass m and length L is suspended from two conducting wires, as shown in Figure 22–42. A
uniform magnetic field of magnitude B points vertically downward. Find the angle θ the suspending wires
make with the vertical when the bar carries a current I.
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Section 22–5 Loops of Current and Magnetic Torque
32. • A rectangular loop of 260 turns is 22 cm wide and 16 cm high. What is the current in this loop if the
maximum torque in a field of 0.48 T is 23 N ⋅ m?
33. • A single circular loop of radius 0.23 m carries a current of 2.6 A in a magnetic field of 0.95 T. What is the
maximum torque exerted on this loop?
34. •• In the previous problem, find the angle the plane of the loop must make with the field if the torque is to be
half its maximum value.
35. •• Consider a current loop in a region of uniform magnetic field, as shown in Figure 22–36 (a) (Conceptual
Exercise 19). Find the magnitude of the torque exerted on the loop about the vertical axis of rotation, using the
data given in Problem 26.
36. •• IP Two current loops, one square the other circular, have one turn made from wires of the same length. (a) If
these loops carry the same current and are placed in magnetic fields of equal magnitude, is the maximum torque
of the square loop greater than, less than, or the same as the maximum torque of the circular loop? Explain. (b)
Calculate the ratio of the maximum torques, τ square τ circle .
37. ••• IP Each of the 10 turns of wire in a vertical, rectangular loop carries a current of 0.22 A. The loop has a
height of 8.0 cm and a width of 15 cm. A horizontal magnetic field of magnitude 0.050 T is oriented at an angle
of θ = 65º relative to the normal to the plane of the loop, as indicated in Figure 22–43. Find (a) the magnetic
force on each side of the loop, (b) the net magnetic force on the loop, and (c) the magnetic torque on the loop.
(d) If the loop can rotate about a vertical axis with only a small amount of friction, will it end up with an
orientation given by θ = 0, θ = 90 º , or θ = 180º ? Explain.
Section 22–6 Electric Currents, Magnetic Fields, and Ampère’s Law
38. • Find the magnetic field 6.25 cm from a long, straight wire that carries a current of 5.81 A.
39. • A long, straight wire carries a current of 7.2 A. How far from this wire is the magnetic field it produces equal
to the Earth’s magnetic field, which is approximately 5.0 × 10 −5 T?
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40. • You travel to the north magnetic pole of the Earth, where the magnetic field points vertically downward.
There, you draw a circle on the ground. Applying Ampère’s law to this circle, show that zero current passes
through its area.
41. • Two power lines, each 270 m in length, run parallel to each other with a separation of 25 cm. If the lines carry
parallel currents of 110 A, what is the magnitude and direction of the magnetic force each exerts on the other?
42. • BIO Pacemaker Switches Some pacemakers employ magnetic reed switches to enable doctors to change
their mode of operation without surgery. A typical reed switch can be switched from one position to another
with a magnetic field of 5.0 × 10 −4 T. What current must a wire carry if it is to produce a 5.0 × 10 −4 T field at a
distance of 0.50 m?
43. •• IP Consider the long, straight, current-carrying wires shown in Figure 22–44. One wire carries a current of
6.2 A in the positive y direction; the other wire carries a current of 4.5 A in the positive x direction. (a) At
which of the two points, A or B, do you expect the magnitude of the net magnetic field to be greater? Explain.
(b) Calculate the magnitude of the net magnetic field at points A and B.
44. •• IP Repeat Problem 43 for the case where the 6.2-A current is reversed in direction.
45. •• In Oersted’s experiment, suppose that the compass was 0.25 m from the current-carrying wire. If a magnetic
field of half the Earth’s magnetic field of 5.0 × 10 −5 T was required to give a noticeable deflection of the
compass needle, what current must the wire have carried?
46. •• IP Two long, straight wires are separated by a distance of 12.2 cm. One wire carries a current of 2.75 A, the
other carries a current of 4.33 A. (a) Find the force per meter exerted on the 2.75-A wire. (b) Is the force per
meter exerted on the 4.33-A wire greater than, less than, or the same as the force per meter exerted on the 2.75A wire? Explain.
47. ••• Two long, straight wires are oriented perpendicular to the page, as shown in Figure 22–45. The current in
one wire is I1 = 3.0 A, pointing into the page, and the current in the other wire is I2 = 4.0 A, pointing out of the
page. Find the magnitude and direction of the net magnetic field at point P.
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Section 22–7 Current Loops and Solenoids
48. • It is desired that a solenoid 38 cm long and with 430 turns produce a magnetic field within it equal to the
Earth’s magnetic field (5.0 × 10 −5 T). What current is required?
49. • A solenoid that is 75 cm long produces a magnetic field of 1.3 T within its core when it carries a current of 8.4
A. How many turns of wire are contained in this solenoid?
50. • The maximum current in a superconducting solenoid can be as large as 3.75 kA. If the number of turns per
meter in such a solenoid is 3250, what is the magnitude of the magnetic field it produces?
51. •• To construct a solenoid, you wrap insulated wire uniformly around a plastic tube 12 cm in diameter and 55
cm in length. You would like a 2.0-A current to produce a 2.5-kG magnetic field inside your solenoid. What is
the total length of wire you will need to meet these specifications?
General Problems
52. • A stationary proton (q = 1.60 × 10 −19 C) is located between the poles of a horseshoe magnet, where the
magnetic field is 0.15 T. What is the magnitude of the magnetic force acting on the proton?
53. • BIO Brain Function and Magnetic Fields Experiments have shown that thought processes in the brain can
be affected if the parietal lobe is exposed to a magnetic field with a strength of 1.0 T. How much current must a
long, straight wire carry if it is to produce a 1.0-T magnetic field at a distance of 0.50 m? (For comparison, a
typical lightning bolt carries a current of about 20,000 A, which would melt most wires.)
54. • BIO Magnetoencephalography Magnetoencephalography (MEG) is a noninvasive method for studying
electrical activity in the brain. It is based on the principle that an electric current creates a magnetic field,
whether the current is in a wire or in the neurons of a brain. Thus, your very thoughts at this moment are
generating measurable magnetic fields outside your head. With today’s technology, an MEG can detect
magnetic fields as small as 1.0 × 10 −15 T. Approximating a neuron by a straight wire, what electric current is
needed to produce this magnetic field strength at a distance of 5.0 cm?
55. • A mixture of two isotopes is injected into a mass spectrometer. One isotope follows a curved path of radius
R1 = 48.9 cm; the other follows a curved path of radius R2 = 51.7 cm. Find the mass ratio, m1 m2 , assuming
that the two isotopes have the same charge and speed.
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56. • High above the surface of the Earth, charged particles (such as electrons and protons) can become trapped in
the Earth’s magnetic field in regions known as Van Allen belts. A typical electron in a Van Allen belt has an
energy of 45 keV and travels in a roughly circular orbit with an average radius of 220 m. What is the magnitude
of the Earth’s magnetic field where such an electron orbits?
57. • Credit-card Magnetic Strips Experiments carried out on the television show Mythbusters determined that a
magnetic field of 1000 gauss is needed to corrupt the information on a credit card’s magnetic strip. (They also
busted the myth that a credit card can be demagnetized by an electric eel or an eelskin wallet.) Suppose a long,
straight wire carries a current of 2.5 A. How close can a credit card be held to this wire without damaging its
magnetic strip?
58. • Superconducting Solenoid Cryomagnetics, Inc. advertises a high-field, superconducting solenoid that
produces a magnetic field of 17 T with a current of 105 A. What is the number of turns per meter in this
solenoid?
59. •• BIO Magnetic Resonance Imaging An MRI (magnetic resonance imaging) solenoid produces a magnetic
field of 1.5 T. The solenoid is 2.5 m long, 1.0 m in diameter, and wound with insulated wires 2.2 mm in
diameter. Find the current that flows in the solenoid. (Your answer should be rather large. A typical MRI
solenoid uses niobium–titanium wire kept at liquid helium temperatures, where it is superconducting.)
60. •• IP A long, straight wire carries a current of 14 A. Next to the wire is a square loop with sides 1.0 m in length,
as shown in Figure 22–46. The loop carries a current of 2.5 A in the direction indicated. (a) What is the
direction of the net force exerted on the loop? Explain. (b) Calculate the magnitude of the net force acting on
the loop.
61. •• Suppose the 14-A current in the straight wire in Figure 22–46 is reversed in direction, but the current in the
loop is unchanged. (a) Calculate the magnitude and direction of the net force acting on the loop. (b) If the loop
is extended in the horizontal direction, so that it is 1.0 m high and 2.0 m wide, does the net force exerted on the
loop increase or decrease? By what factor? Explain. (c) If, instead, the loop is extended in the vertical direction,
so it is 2.0 m high and 1.0 m wide, does the net force exerted on the loop increase or decrease? Explain.
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62. •• A charged particle moves through a region of space containing both electric and magnetic fields. The
r
r
velocity of the particle is V = ( 4.4 × 10 3 m s)x$ + (2.7 × 10 3 m s)y$ and the magnetic field is B = (0.73 T)z$ . Find
r
the electric field vector E necessary to yield zero net force on the particle. (Note: You may wish to use cross
products in this problem. They are discussed in Appendix A.)
63. •• IP Medical X-rays An electron in a medical X-ray machine is accelerated from rest through a voltage of
10.0 kV. (a) Find the maximum force a magnetic field of 0.957 T can exert on this electron. (b) If the voltage
of the X-ray machine is increased, does the maximum force found in part (a) increase, decrease, or stay the
same? Explain. (c) Repeat part (a) for an electron accelerated through a potential of 25.0 kV.
64. •• A particle with a charge of 34 μC moves with a speed of 62 m/s in the positive x direction. The magnetic
field in this region of space has a component of 0.40 T in the positive y direction, and a component of 0.85 T in
the positive z direction. What are the magnitude and direction of the magnetic force on the particle?
65. •• IP A beam of protons with various speeds is directed in the positive x direction. The beam enters a region
with a uniform magnetic field of magnitude 0.52 T pointing in the negative z direction, as indicated in Figure
22–47. It is desired to use a uniform electric field (in addition to the magnetic field) to select from this beam
only those protons with a speed of 1.42 × 10 5 m s — that is, only these protons should be undeflected by the two
fields. (a) Determine the magnitude and direction of the electric field that yields the desired result. (b) Suppose
the electric field is to be produced by a parallel-plate capacitor with a plate separation of 2.5 cm. What potential
difference is required between the plates? (c) Which plate in Figure 22–47 (top or bottom) should be positively
charged? Explain.
66. •• IP A charged particle moves in a horizontal plane with a speed of 8.70 × 10 6 m s . When this particle
encounters a uniform magnetic field in the vertical direction it begins to move on a circular path of radius 15.9
cm. (a) If the magnitude of the magnetic field is 1.21 T, what is the charge-to-mass ratio (q/m) of this particle?
(b) If the radius of the circular path were greater than 15.9 cm, would the corresponding charge to mass ratio be
greater than, less than, or the same as that found in part (a)? Explain. (Assume that the magnetic field remains
the same.)
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67. •• Two parallel wires, each carrying a current of 2.2 A in the same direction, are shown in Figure 22–48. Find
the direction and magnitude of the net magnetic field at points A, B, and C.
68. •• Repeat Problem 67 for the case where the current in wire 1 is reversed in direction.
69. •• Lightning Bolts A powerful bolt of lightning can carry a current of 225 kA. (a) Treating a lightning bolt as a
long, thin wire, calculate the magnitude of the magnetic field produced by such a bolt of lightning at a distance
of 35 m. (b) If two such bolts strike simultaneously at a distance of 35 m from each other, what is the magnetic
force per meter exerted by one bolt on the other?
70. •• IP Consider the two current-carrying wires shown in Figure 22–49. The current in wire 1 is 3.7 A; the
current in wire 2 is adjusted to make the net magnetic field at point A equal to zero. (a) Is the magnitude of the
current in wire 2 greater than, less than, or the same as that in wire 1? Explain. (b) Find the magnitude and
direction of the current in wire 2.
71. •• IP Consider the physical system shown in Figure 22–49, which consists of two current-carrying wires each
with a length of 71 cm. (a) If the net magnetic field at the point A is out of the page, is the force between the
wires attractive or repulsive? Explain. (b) Calculate the magnitude of the force exerted by each wire on the
other wire, given that the magnetic field at point A is out of the page with a magnitude of 2.1 T × 10 −6 T.
72. •• Magnetars The astronomical object 4 U0142 + 61 has the distinction of creating the most powerful magnetic
field ever observed. This object is referred to as a “magnetar” (a subclass of pulsars), and its magnetic field is
1.3 × 1015 times greater than the Earth’s magnetic field. (a) Suppose a 2.5-m straight wire carrying a current of
1.1 A is placed in this magnetic field at an angle of 65° to the field lines. What force does this wire experience?
(b) A field this strong can significantly change the behavior of an atom. To see this, consider an electron
moving with a speed of 2.2 × 10 6 m s . Compare the maximum magnetic force exerted on the electron to the
electric force a proton exerts on an electron in a hydrogen atom. The radius of the hydrogen atom is
5.29 × 10 −11 m.
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73. •• Consider a system consisting of two concentric solenoids, as illustrated in Figure 22–50. The current in the
outer solenoid is I1 = 1.25 A, and the current in the inner solenoid is I2 = 2.17 A. Given that the number of
turns per centimeter is 105 for the outer solenoid and 125 for the inner solenoid, find the magnitude and
direction of magnetic field (a) between the solenoids and (b) inside the inner solenoid.
74. •• IP A long, straight wire on the x axis carries a current of 3.12 A in the positive x direction. The magnetic
field produced by the wire combines with a uniform magnetic field of 1.45 × 10 −6 T that points in the positive z
direction. (a) Is the net magnetic field of this system equal to zero at a point on the positive y axis or at a point
on the negative y axis? Explain. (b) Find the distance from the wire to the point where the field vanishes.
75. •• Find the angle between the plane of a loop and the magnetic field for which the magnetic torque acting on the
loop is equal to x times its maximum value, where 0 ≤ x ≤ 1.
76. •• Solenoids produce magnetic fields that are relatively intense for the amount of current they carry. To make a
direct comparison, consider a solenoid with 55 turns per centimeter, a radius of 1.05 cm, and a current of 0.124
A. (a) Find the magnetic field at the center of the solenoid. (b) What current must a long, straight wire carry to
have the same magnetic field as that found in part (a)? Let the distance from the wire be the same as the radius
of the solenoid, 1.05 cm.
77. •• The current in a solenoid with 22 turns per centimeter is 0.50 A. The solenoid has a radius of 1.5 cm. A long,
straight wire runs along the axis of the solenoid, carrying a current of 16 A. Find the magnitude of the net
magnetic field a radial distance of 0.75 cm from the straight wire.
78. •• IP BIO Transcranial Magnetic Stimulation A recently developed method to study brain function is to
produce a rapidly changing magnetic field within the brain. When this technique, known as transcranial
magnetic stimulation (TMS), is applied to the prefrontal cortex, for example, it can reduce a person’s ability to
conjugate verbs, though other thought processes are unaffected. The rapidly varying magnetic field is produced
with a circular coil of 21 turns and a radius of 6.0 cm placed directly on the head. The current in this loop
increases at the rate of 1.2 × 10 7 A s (by discharging a capacitor). (a) At what rate does the magnetic field at
the center of the coil increase? (b) Suppose a second coil with half the area of the first coil is used instead.
Would your answer to part (a) increase, decrease, or stay the same? By what factor?
15
r
79. ••• An electron with a velocity given by v = (1.5 × 10 5 m s)x$ + (0.67 × 10 4 m s)y$ moves through a region of
r
r
space with a magnetic field B = (0.25 T) x$ − (0.11 T)z$ and an electric field E = (220 N C)x$ . Using cross
products, find the magnitude of the net force acting on the electron. (Cross products are discussed in Appendix
A.)
80. ••• A thin ring of radius R and charge per length λ rotates with an angular speed ω about an axis perpendicular
to its plane and passing through its center. Find the magnitude of the magnetic field at the center of the ring.
81. ••• A solenoid is made from a 25-m length of wire of resistivity 2.3 × 10 −8 Ω ⋅ m. The wire, whose radius is 2.1
mm, is wrapped uniformly onto a plastic tube 4.5 cm in diameter and 1.65 m long. Find the emf to which the
ends of the wire must be connected to produce a magnetic field of 0.015 T within the solenoid.
82. ••• IP A single current-carrying circular loop of radius R is placed next to a long, straight wire, as shown in
Figure 22–51. The current in the wire points to the right and is of magnitude I. (a) In which direction must
current flow in the loop to produce zero magnetic field at its center? Explain. (b) Calculate the magnitude of the
current in part (a).
83. ••• Magnetic Fields in the Bohr Model In the Bohr model of the hydrogen atom, the electron moves in a
circular orbit of radius 5.29 × 10 −11 m about the nucleus. Given that the charge on the electron is
−1.60 × 10 −19 C, and that its speed is 2.2 × 10 6 m s, find the magnitude of the magnetic field the electron
produces at the nucleus of the atom.
84. ••• A single-turn square loop carries a current of 18 A. The loop is 15 cm on a side and has a mass of 0.035 kg.
Initially the loop lies flat on a horizontal tabletop. When a horizontal magnetic field is turned on, it is found that
only one side of the loop experiences an upward force. Find the minimum magnetic field, Bmin , necessary to
start tipping the loop up from the table.
85. ••• Consider the physical system shown in Figure 22–45. (a) Find the net magnetic field (direction and
magnitude) at an arbitrary point on the bottom side of the square, a distance 0 < x < 5.0 cm to the right of wire
1. (b) Find the magnitude of the net magnetic field at an arbitrary point on the left side of the square, a distance
0 < y < 5.0 cm above wire 1.
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Interactive Problems
86. •• IP Referring to Example 22–3 Suppose the speed of the isotopes is doubled. (a) Does the separation
distance, d, increase, decrease, or stay the same? Explain. (b) Find the separation distance for this case.
87. •• IP Referring to Example 22–3 Suppose we change the initial speed of
238
U, leaving everything else the
same. (a) If we want the separation distance to be zero, should the initial speed of
238
U be increased or
decreased? Explain. (b) Find the required initial speed.
88. •• Referring to Active Example 22–2 The current I1 is adjusted until the magnetic field halfway between the
wires has a magnitude of 2.5 × 10 −6 T and points into the page. Everything else in the system remains the same
as in Active Example 22–2. Find the magnitude and direction of I1 .
89. •• Referring to Active Example 22–2 The current I2 is adjusted until the magnetic field 5.5 cm below wire 2
has a magnitude of 2.5 × 10 −6 T and points out of page. Everything else in the system remains the same as in
Active Example 22–2. Find the magnitude and direction of I2 .
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