Download EOC_chapter22

Chapter 22 Problems
1, 2, 3 = straightforward, intermediate,
challenging
= full solution available in Student
Solutions Manual/Study Guide
= coached solution with
hints available at www.pop4e.com
= computer useful in solving problem
= paired numerical and symbolic
problems
= biomedical application
Section 22.2 The Magnetic Field
1.
Determine the initial
direction of the deflection of charged
particles as they enter the magnetic fields as
shown in Figure P22.1.
downward, (b) northward, (c) westward, or
(d) southeastward?
3.
A proton travels with a speed of 3.00
6
× 10 m/s at an angle of 37.0° with the
direction of a magnetic field of 0.300 T in
the +y direction. What are (a) the
magnitude of the magnetic force on the
proton and (b) its acceleration?
4.
An electron is accelerated through
2 400 V from rest and then enters a uniform
1.70-T magnetic field. What are (a) the
maximum and (b) the minimum values of
the magnetic force this charge can
experience?
5.
At the equator, near the surface of
the Earth, 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 in this
environment, assuming that the electron
has an instantaneous velocity of 6.00 × 106
m/s directed to the east.
6.
A proton moves with a velocity of

v  (2iˆ  4ˆj  kˆ ) m/s in a region in which

the magnetic field is B  (iˆ  2ˆj  3kˆ ) T.
What is the magnitude of the magnetic
force this charge experiences?
Figure P22.1
2.
Consider an electron near the Earth’s
equator. In which direction does it tend to
deflect if its velocity is directed (a)
Section 22.3 Motion of a Charged Particle
in a Uniform Magnetic Field
7.
Review problem. One electron
collides elastically with a second electron
initially at rest. After the collision, the radii
of their trajectories are 1.00 cm and 2.40 cm.
The trajectories are perpendicular to a
uniform magnetic field of magnitude 0.044
0 T. Determine the energy (in keV) of the
incident electron.
8.
Review problem. An electron moves
in a circular path perpendicular to a
constant magnetic field of magnitude 1.00
mT. 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.
9.
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?
Section 22.4 Applications Involving
Charged Particles Moving in a Magnetic
Field
10.
A velocity selector consists of electric
and magnetic fields described by the


expressions E  Ekˆ and B  Bˆj , with B =
15.0 mT. Find the value of E such that a 750eV electron moving along the positive x
axis is undeflected.
11.
Consider the mass spectrometer
shown schematically in Active Figure 22.12.
The magnitude of the electric field between
the plates of the velocity selector is 2 500
V/m, and the magnetic field in both the
velocity selector and the deflection chamber
has a magnitude of 0.035 0 T. Calculate the
radius of the path for a singly charged ion
having a mass m = 2.18 × 10–26 kg.
12.
A cyclotron designed to accelerate
protons has an outer radius of 0.350 m. The
protons are emitted nearly at rest from a
source at the center and are accelerated
through 600 V each time they cross the gap
between the dees. The dees are between the
poles of an electromagnet where the field is
0.800 T. (a) Find the cyclotron frequency. (b)
Find the speed at which protons exit the
cyclotron and (c) their maximum kinetic
energy. (d) How many revolutions does a
proton make in the cyclotron? (e) For what
time interval does one proton accelerate?
13.
The picture tube in a television uses
magnetic deflection coils rather than
electric deflection plates. Suppose an
electron beam is accelerated through a 50.0kV potential difference and then through a
region of uniform magnetic field 1.00 cm
wide. The screen is located 10.0 cm from the
center of the coils and is 50.0 cm wide.
When the field is turned off, the electron
beam hits the center of the screen. What
field magnitude is necessary to deflect the
beam to the side of the screen? Ignore
relativistic corrections.
14.
The Hall effect finds important
application in the electronics industry. It is
used to find the sign and density of the
carriers of electric current in semiconductor
chips. The arrangement is shown in Figure
P22.14. A semiconducting block of
thickness t and width d carries a current I in
the x direction. A uniform magnetic field B
is applied in the y direction. If the charge
carriers are positive, the magnetic force
deflects them in the z direction. Positive
charge accumulates on the top surface of
the sample and negative charge on the
bottom surface, creating a downward
electric field. In equilibrium, the downward
electric force on the charge carriers balances
the upward magnetic force and the carriers
move through the sample without
deflection. The Hall voltage ΔVH = Vc – Va
between the top and bottom surfaces is
measured, and the density of the charge
carriers can be calculated from it. (a)
Demonstrate that if the charge carriers are
negative the Hall voltage will be negative.
Hence, the Hall effect reveals the sign of the
charge carriers, so the sample can be
classified as p-type (with positive majority
charge carriers) or n-type (with negative).
(b) Determine the number of charge carriers
per unit volume n in terms of I, t, B, ΔVH,
and the magnitude q of the carrier charge.
Figure P22.14
Section 22.5 Magnetic Force on a CurrentCarrying Conductor
15.
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, B  1.60kˆ
T. If the current is in the +x direction, what
is the magnetic force on the section of wire?
16.
A wire 2.80 m in length carries a
current of 5.00 A in a region where a
uniform magnetic field has a magnitude of
0.390 T. Calculate the magnitude of the
magnetic force on the wire assuming that
the angle between the magnetic field and
the current is (a) 60.0°, (b) 90.0°, and (c)
120°.
17.
A nonuniform magnetic
field exerts a net force on a magnetic dipole. A
strong magnet is placed under a horizontal
conducting ring of radius r that carries
current I as shown in Figure P22.17. If the

magnetic field B makes an angle θ with the
vertical at the ring’s location, what are the
magnitude and direction of the resultant
force on the ring?
Section 22.6 Torque on a Current Loop in
a Uniform Magnetic Field
19.
A current of 17.0 mA is maintained
in a single circular loop of 2.00 m
circumference. A magnetic field of 0.800 T
is directed parallel to the plane of the loop.
(a) Calculate the magnetic moment of the
loop. (b) What is the magnitude of the
torque exerted by the magnetic field on the
loop?
Figure P22.17
18.
In Figure P22.18, 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 P22.18
20.
A current loop with magnetic dipole

moment  is placed in a uniform magnetic

field B , with its moment making angle θ
with the field. With the arbitrary choice of
U = 0 for θ = 90°, prove that the potential
energy of the dipole-field system is
 
U    B .
21.
A rectangular coil
consists of N = 100 closely wrapped turns
and has dimensions a = 0.400 m and b =
0.300 m. The coil is hinged along the y axis,
and its plane makes an angle θ = 30.0° with
the x axis (Fig. P22.21). What is the
magnitude of the torque exerted on the coil
by a uniform magnetic field B = 0.800 T
directed along the x axis when the current
is I = 1.20 A in the direction shown? What is
the expected direction of rotation of the
coil?
speed of 2.19 × 106 m/s. Compute the
magnitude of the magnetic field that this
motion produces at the location of the
proton.
24.
A lightning bolt may carry a current
of 1.00 × 104 A for a short time interval.
What is the resulting magnetic field 100 m
from the bolt? Assume that the bolt extends
far above and below the point of
observation.
Figure P22.21
22.
The rotor in a certain electric motor
is a flat, rectangular coil with 80 turns of
wire and dimensions 2.50 cm by 4.00 cm.
The rotor rotates in a uniform magnetic
field of 0.800 T. When the plane of the rotor
is perpendicular to the direction of the
magnetic field, it carries a current of 10.0
mA. In this orientation, the magnetic
moment of the rotor is directed opposite the
magnetic field. The rotor then turns
through one-half revolution. This process is
repeated to cause the rotor to turn steadily
at 3 600 rev/min. (a) Find the maximum
torque acting on the rotor. (b) Find the peak
power output of the motor. (c) Determine
the amount of work performed by the
magnetic field on the rotor in every full
revolution. (d) What is the average power
of the motor?
25.
Determine the
magnetic field at a point P located a
distance x from the corner of an infinitely
long wire bent at a right angle as shown in
Figure P22.25. The wire carries a steady
current I.
Figure P22.25
26.
Calculate the magnitude of the
magnetic field at a point 100 cm from a
long, thin conductor carrying a current of
1.00 A.
Section 22.7 The Biot–Savart Law
23.
In Niels Bohr’s 1913 model of the
hydrogen atom, an electron circles the
proton at a distance of 5.29 × 10–11 m with a
27.
A conductor consists of a circular
loop of radius R and two straight, long
sections as shown in Figure P22.27. The
wire lies in the plane of the paper and
carries a current I. Find an expression for
the vector magnetic field at the center of the
loop.
Figure P22.27
28.
Consider a flat, circular current loop
of radius R carrying current I. Choose the x
axis to be along the axis of the loop, with
the origin at the center of the loop. Plot a
graph of the ratio of the magnitude of the
magnetic field at coordinate x to that at the
origin, for x = 0 to x = 5R. It may be useful to
use a programmable calculator or a
computer to solve this problem.
29.
Two very long, straight, parallel
wires carry currents that are directed
perpendicular to the page as shown in
Figure P22.29. Wire 1 carries a current I1
into the page (in the –z direction) and
passes through the x axis at x = +a. Wire 2
passes through the x axis at x = –2a and
carries an unknown current I2. The total
magnetic field at the origin due to the
current-carrying wires has the magnitude
2μ0I1/(2πa). The current I2 can have either of
two possible values. (a) Find the value of I2
with the smaller magnitude, stating it in
terms of I1 and giving its direction. (b) Find
the other possible value of I2.
Figure P22.29
30.
One very long wire carries current
30.0 A to the left along the x axis. A second
very long wire carries current 50.0 A to the
right along the line (y = 0.280 m, z = 0). (a)
Where in the plane of the two wires is the
total magnetic field equal to zero? (b) A
particle with a charge of –2.00 μC is moving
with a velocity of 150 î Mm/s along the line
(y = 0.100 m, z = 0). Calculate the vector
magnetic force acting on the particle. (c) A
uniform electric field is applied to allow
this particle to pass through this region
unde- flected. Calculate the required vector
electric field.
31.
A current path shaped as shown in
Figure P22.31 produces a magnetic field at
P, the center of the arc. If the arc subtends
an angle of 30.0° and the radius of the arc is
0.600 m, what are the magnitude and
direction of the field produced at P if the
current is 3.00 A?
Figure P22.31
32.
Three long, parallel conductors carry
currents of I = 2.00 A. Figure P22.32 is an
end view of the conductors, with each
current coming out of the page. Taking a =
1.00 cm, determine the magnitude and
direction of the magnetic field at points A,
B, and C.
Figure P22.32
Figure P22.33
33.
In studies of the possibility of
migrating birds using the Earth’s magnetic
field for navigation, birds have been fitted
with coils as “caps” and “collars” as shown
in Figure P22.33. (a) If the identical coils
have radii of 1.20 cm and are 2.20 cm apart,
with 50 turns of wire apiece, what current
should they both carry to produce a
magnetic field of 4.50 × 10–5 T halfway
between them? (b) If the resistance of each
coil is 210 Ω, what voltage should the
battery supplying each coil have? (c) What
power is delivered to each coil?
Section 22.8 The Magnetic Force Between
Two Parallel Conductors
34.
Two long, parallel conductors,
separated by 10.0 cm, carry currents in the
same direction. The first wire carries
current I1 = 5.00 A and the second carries I2
= 8.00 A. (a) What is the magnitude of the
magnetic field created by I1 at the location
of I2? (b) What is the force per unit length
exerted by I1 on I2? (c) What is the
magnitude of the magnetic field created by
I2 at the location of I1? (d) What is the force
per length exerted by I2 on I1?
35.
In Figure P22.35, 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 the current I2 = 10.0 A.
The dimensions are c = 0.100 m, a = 0.150 m,
and ℓ = 0.450 m. Find the magnitude and
direction of the net force exerted on the
loop by the magnetic field created by the
wire.
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.
Figure P22.37
Figure P22.35
36.
Three long wires (wire 1, wire 2, and
wire 3) hang vertically. The distance
between wire 1 and wire 2 is 20.0 cm. On
the left, wire 1 carries an upward current of
1.50 A. To the right, wire 2 carries a
downward current of 4.00 A. Wire 3 is
located such that when it carries a certain
current, each wire experiences no net force.
Find (a) the position of wire 3 and (b) the
magnitude and direction of the current in
wire 3.
Section 22.9 Ampère’s Law
37.
Four long, parallel conductors carry
equal currents of I = 5.00 A. Figure P22.37 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).
38.
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 speed of 2.30 × 104 m/s at a
distance d above the wire. Determine the
value of d. You may ignore the magnetic
field due to the Earth.
39.
A packed bundle of 100
long, straight, insulated wires forms a
cylinder of radius R = 0.500 cm. (a) If each
wire carries 2.00 A, what are the magnitude
and direction of the magnetic force per unit
length acting on a wire located 0.200 cm
from the center of the bundle? (b) Would a
wire on the outer edge of the bundle
experience a force greater or smaller than
the value calculated in part (a)?
40.
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?
distributed over its curved wall. Determine
the magnetic field (a) just inside the wall
and (b) just outside. (c) Determine the
pressure on the wall.
41.
The magnetic coils of a tokamak
fusion reactor are in the shape of a toroid
having an inner radius of 0.700 m and an
outer radius of 1.30 m. The toroid has 900
turns of large-diameter wire, each of which
carries a current of 14.0 kA. Find the
magnitude of the magnetic field inside the
toroid along (a) the inner radius and (b) the
outer radius.
Section 22.10 The Magnetic Field of a
Solenoid
42.
Consider a column of electric current
in a plasma (ionized gas). Filaments of
current within the column are magnetically
attracted to one another. They can crowd
together to yield a very great current
density and a very strong magnetic field in
a small region. Sometimes the current can
be cut off momentarily by this pinch effect.
(In a metallic wire, a pinch effect is not
important because the current-carrying
electrons repel one another with electric
forces.) The pinch effect can be
demonstrated by making an empty
aluminum can carry a large current parallel
to its axis. Let R represent the radius of the
can and I the upward current, uniformly
43.
Niobium metal becomes a
superconductor when cooled below 9 K. Its
superconductivity is destroyed when the
surface magnetic field exceeds 0.100 T.
Determine the maximum current a 2.00mm-diameter niobium wire can carry and
remain superconducting, in the absence of
any external magnetic field.
44.
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 30 turns/cm and
carries a clockwise current of 15.0 A. Find
the force on each side of the loop and the
torque acting on the loop.
45.
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 to produce at the center of the
solenoid a magnetic field of magnitude 1.00
× 10–4 T?
46.
Consider a solenoid of length ℓ and
radius R, 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
distance a from the end of the solenoid. (b)
Show that as ℓ becomes very long, B
approaches μ0NI/2ℓ at each end of the
solenoid.
47.
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. To
produce a field of 8.00 mT at the center of
the solenoid, what power must be delivered
to the solenoid?
Section 22.11 Magnetism in Matter
48.
In Bohr’s 1913 model of the
hydrogen atom, the electron is in a circular
orbit of radius 5.29 × 10–11 m and its speed is
2.19 × 106 m/s. (a) What is the magnitude of
the magnetic moment due to the electron’s
motion? (b) If the electron moves in a
horizontal circle, counterclockwise as seen
from above, what is the direction of this
magnetic moment vector?
49.
The magnetic moment of the Earth is
approximately 8.00 × 1022 A · m2. (a) If it
were caused by the complete magnetization
of a huge iron deposit, how many unpaired
electrons would participate? (b) At two
unpaired electrons per iron atom, how
many kilograms of iron would that
correspond to? (Iron has a density of 7 900
kg/m3 and approximately 8.50 × 1028 iron
atoms/m3.)
Section 22.12 Context Connection—The
Attractive Model for Magnetic Levitation
50.
The following represents a crude
model for levitating a commercial
transportation vehicle. Suppose the
levitation is achieved by mounting small
electrically charged spheres below the
vehicle. The spheres pass through a
magnetic field established by permanent
magnets placed along the track. Let us
assume that the permanent magnets
produce a uniform magnetic field of 0.1 T at
the location of the spheres and that an
electronic control system maintains a
charge of 1 μC on each sphere. The vehicle
has a mass of 5 × 104 kg and travels at a
speed of 400 km/h. How many charged
spheres are required to support the weight
of the vehicle at this speed? Your answer
should suggest that this design would not
be practical as a means of magnetic
levitation.
51.
Data for the Transrapid maglev
system show that the input electric power
required to operate the vehicle is on the
order of 102 kW. (a) Assume that the
Transrapid vehicle moves at 400 km/h.
Approximately how much energy, in joules,
is used for each mile of travel for the
vehicle? (b) Calculate the energy per mile
used by an automobile that achieves 20
mi/gal. The energy available from gasoline
is approximately 40 MJ/kg, a typical
automobile engine efficiency is 20%, and
the density of gasoline is 754 kg/m3. (c)
Considering 1 passenger in the automobile
and 100 on the Transrapid vehicle, the
energy per mile necessary for each
passenger in the Transrapid is what fraction
of that for an automobile?
Additional Problems
52.
Consider a thin, straight wire
segment carrying a constant current I and
placed along the x axis as shown in Figure
P22.52. (a) Use the Biot–Savart law to show
that the total magnetic field at the point P,
located a distance a from the wire, is
B
magnetic field near the sheet. (Suggestion:
Use Ampère’s law and evaluate the line
integral for a rectangular path around the
sheet, represented by the dashed line in Fig.
P22.53.)
0 I
cos  1  cos  2 
4 a
(b) Assuming that the wire is infinitely
long, show that the result in part (a) gives a
magnetic field that agrees with that
obtained by using Ampère’s law in
Example 22.7.
Figure P22.53
Figure P22.52
53.
An infinite sheet of current lying in
the yz plane carries a surface current of

density J s . The current is in the y direction,
and Js represents the current per unit length
measured along the z axis. Figure P22.53 is
an edge view of the sheet. Find the
54.
Assume that the region to the right
of a certain vertical plane contains a vertical
magnetic field of magnitude 1.00 mT and
that the field is zero in the region to the left
of the plane. An electron, originally
traveling perpendicular to the boundary
plane, passes into the region of the field. (a)
Determine the time interval required for the
electron to leave the “field-filled” region,
noting that its path is a semicircle. (b) Find
the kinetic energy of the electron assuming
that the maximum depth of penetration
into the field is 2.00 cm.
55.
Heart–lung machines and artificial
kidney machines employ blood pumps. A
mechanical pump can mangle blood cells.
Figure P22.55 represents an electromagnetic
pump. The blood is confined to an
electrically insulating tube, cylindrical in
practice but represented as a rectangle of
width w and height h. The simplicity of
design makes the pump dependable. The
blood is easily kept uncontaminated; the
tube is simple to clean or cheap to replace.
Two electrodes fit into the top and bottom
of the tube. The potential difference
between them establishes an electric current
through the blood, with current density J
over a section of length L. A perpendicular
magnetic field exists in the same region. (a)
Explain why this arrangement produces on
the liquid a force that is directed along the
length of the pipe. (b) Show that the section
of liquid in the magnetic field experiences a
pressure increase JLB. (c) After the blood
leaves the pump, is it charged? Is it currentcarrying? Is it magnetized? The same
magnetic pump can be used for any fluid
that conducts electricity, such as liquid
sodium in a nuclear reactor.
Figure P22.55
56.
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 positive charge q = 3.20 × 10–19 C

moves with a velocity v  (2iˆ  3ˆj  kˆ ) m/s
through a region where both a uniform
magnetic field and a uniform electric field
exist. (a) Calculate the total force on the
moving charge (in unit-vector notation),

taking B  ( 2iˆ  4ˆj  kˆ ) T and

E  (4iˆ  ˆj  2kˆ ) V/m. (b) What angle does
the force vector make with the positive x
axis?
58.
Protons having a kinetic energy of
5.00 MeV are moving in the positive x
direction and enter a magnetic field

B  0.050 0kˆ T directed out of the plane of
the page and extending from x = 0 to x =
1.00 m as shown in Figure P22.58. (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. Ignore relativistic
effects and note that 1 eV = 1.60 × 10–19 J.
Figure P22.58
59.
A handheld electric mixer contains
an electric motor. Model the motor as a
single flat, compact, circular coil carrying
electric current in a region where a
magnetic field is produced by an external
permanent magnet. You need consider only
one instant in the operation of the motor.
(We will consider motors again in Chapter
23.) The coil moves because the magnetic
field exerts torque on the coil as described
in Section 22.6. Make order-of-magnitude
estimates of the magnetic field, the torque
on the coil, the current in it, its area, and the
number of turns in the coil, so that they are
related according to Equation 22.16. Note
that the input power to the motor is electric,
given by  = IΔV, and the useful output
power is mechanical,    .
60.
A cyclotron is sometimes used for
carbon dating as will be described in
Chapter 30. Carbon-14 and carbon-12 ions
are obtained from a sample of the material
to be dated and are accelerated in the
cyclotron. If the cyclotron has a magnetic
field of magnitude 2.40 T, what is the
difference in cyclotron frequencies for the
two ions?
61.
A uniform magnetic field of
magnitude 0.150 T is directed along the
positive x axis. A positron moving at 5.00 ×
106 m/s enters the field along a direction
that makes an angle of 85.0° with the x axis
(Fig. P22.61). The motion of the particle is
expected to be a helix as described in
Section 22.3. Calculate (a) the pitch p and
(b) the radius r of the trajectory.
Figure P22.61
62.
A heart surgeon monitors the flow
rate of blood through an artery using an
electromagnetic flowmeter (Fig. P22.62).
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, an
emf 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.
Figure P22.62
63.
A very long, thin strip of metal of
width w carries a current I along its length
as shown in Figure P22.63. Find the
magnetic field at the point P in the diagram.
The point P is in the plane of the strip at
distance b away from it.
field on the axis of the ring a distance R/2
from its center?
66.
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 P22.66. (a) Show that the magnetic
field on the axis at a distance x from the
center of one coil is

N 0 IR 2 
1
1
B

 2
2 3/2
2
2R 2  x 2  2 Rx3 / 2 
 R  x 
Figure P22.63
64.
The magnitude of the Earth’s
magnetic field at either pole is
approximately 7.00 × 10–5 T. Suppose the
field fades away, before its next reversal.
Scouts, sailors, and conservative politicians
around the world join together in a
program to replace the field. One plan is to
use a current loop around the equator,
without relying on magnetization of any
materials inside the Earth. Determine the
current that would generate such a field if
this plan were carried out. (Take the radius
of the Earth as RE = 6.37 × 106 m.)
65.
A nonconducting ring of radius R is
uniformly charged with a total positive
charge q. The ring rotates at a constant
angular speed ω about an axis through its
center, perpendicular to the plane of the
ring. What is the magnitude of the magnetic
(b) Show that dB/dx and d2B/dx2 are both
zero at the point midway between the coils.
Thus, the magnetic field in the region
midway between the coils is uniform. Coils
in this configuration are called Helmholtz
coils.
Figure P22.66
67.
Two circular loops are parallel,
coaxial, and almost in contact, 1.00 mm
apart (Fig. P22.67). 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 exerted by the
bottom loop 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 the
gravitational force. (Suggestion: Think about
how one loop looks to a bug perched on the
other loop.)
Figure P22.67
68.
Rail guns have been suggested for
launching projectiles into space without
chemical rockets and for ground-to-air
antimissile weapons of war. A tabletop
model rail gun (Fig. P22.68) consists of two
long, parallel, horizontal rails 3.50 cm apart,
bridged by a bar BD of mass 3.00 g. The bar
is originally at rest at the midpoint of the
rails and is free to slide without friction.
When the switch is closed, electric current
is quickly established in the circuit
ABCDEA. The rails and bar have low
© Copyright 2004 Thomson. All rights reserved.
electric resistance, and the current is limited
to a constant 24.0 A by the power supply.
(a) Find the magnitude of the magnetic
field 1.75 cm from a single very long,
straight wire carrying current 24.0 A. (b)
Find the magnitude and direction of the
magnetic field at point C in the diagram,
the midpoint of the bar, immediately after
the switch is closed. (Suggestion: Consider
what conclusions you can draw from the
Biot–Savart law.) (c) At other points along
the bar BD, the field is in the same direction
as at point C but is larger in magnitude.
Assume that the average effective magnetic
field along BD is five times larger than the
field at C. With this assumption, find the
magnitude and direction of the force on the
bar. (d) Find the acceleration of the bar
when it is in motion. (e) Does the bar move
with constant acceleration? (f) Find the
velocity of the bar after it has traveled 130
cm to the end of the rails.
Figure P22.68