Download POP4e: Ch. 23 Summary

Chapter 23 Problems
1, 2, 3 = straightforward, intermediate,
challenging
= full solution available in Student
Solutions Manual/Study Guide
= coached solution with
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= computer useful in solving problem
= paired numerical and symbolic
problems
= biomedical application
Section 23.1 Faraday’s Law of Induction
Section 23.3 Lenz’s Law
1.
A flat loop of wire consisting of a
single turn of cross-sectional area 8.00 cm2
is perpendicular to a magnetic field that
increases uniformly in magnitude from
0.500 T to 2.50 T in 1.00 s. What is the
resulting induced current if the loop has a
resistance of 2.00 Ω?
reaches zero in 20.0 ms. What is the current
induced in the coil?
4.
An aluminum ring of radius r1 and
resistance R is placed around the top of a
long air-core solenoid with n turns per
meter and smaller radius r2 as shown in
Figure P23.4. Assume that the axial
component of the field produced by the
solenoid over the area of the end of the
solenoid is one-half as strong as at the
solenoid’s center. Assume that the solenoid
produces negligible field outside its crosssectional area. The current in the solenoid is
increasing at a rate of ΔI/Δt. (a) What is the
induced current in the ring? (b) At the
center of the ring, what is the magnetic field
produced by the induced current in the
ring? (c) What is the direction of this field?
2.
A 25-turn circular coil of wire has
diameter 1.00 m. It is placed with its axis
along the direction of the Earth’s magnetic
field of 50.0 μT, and then in 0.200 s it is
flipped 180°. An average emf of what
magnitude is generated in the coil?
3.
A strong electromagnet
produces a uniform magnetic field of 1.60 T
over a cross-sectional area of 0.200 m2. We
place a coil having 200 turns and a total
resistance of 20.0 Ω around the
electromagnet. We then smoothly reduce
the current in the electromagnet until it
Figure P23.4
5.
(a) A loop of wire in the shape of a
rectangle of width w and length L and a
long, straight wire carrying a current I lie
on a tabletop as shown in Figure P23.5. (a)
Determine the magnetic flux through the
loop due to the current I. (b) Suppose the
current is changing with time according to I
= a + bt, where a and b are constants.
Determine the emf that is induced in the
loop if b = 10.0 A/s, h = 1.00 cm, w = 10.0 cm,
and L = 100 cm. What is the direction of the
induced current in the rectangle?
Figure P23.6
7.
A 30-turn circular coil of radius 4.00
cm and resistance 1.00 Ω is placed in a
magnetic field directed perpendicular to the
plane of the coil. The magnitude of the
magnetic field varies in time according to
the expression B = 0.010 0t + 0.040 0t2, where
t is in seconds and B is in teslas. Calculate
the induced emf in the coil at t = 5.00 s.
Figure P23.5 Problems 23.5 and 23.59.
6.
A coil of 15 turns and radius 10.0 cm
surrounds a long solenoid of radius 2.00 cm
and 1.00 × 103 turns/m (Fig. P23.6). The
current in the solenoid changes as I = (5.00
A) sin(120t). Find the induced emf in the 15turn coil as a function of time.
8.
An instrument based on induced
emf has been used to measure projectile
speeds up to 6 km/s. A small magnet is
imbedded in the projectile as shown in
Figure P23.8. The projectile passes through
two coils separated by a distance d. As the
projectile passes through each coil, a pulse
of emf is induced in the coil. The time
interval between pulses can be measured
accurately with an oscilloscope, and thus
the speed can be determined. (a) Sketch a
graph of ΔV versus t for the arrangement
shown. Consider a current that flows
counterclockwise as viewed from the
starting point of the projectile as positive.
On your graph, indicate which pulse is
from coil 1 and which is from coil 2. (b) If
the pulse separation is 2.40 ms and d = 1.50
m, what is the projectile speed?
Figure P23.8
9.
When a wire carries an AC current
with a known frequency, you can use a
Rogowski coil to determine the amplitude
Imax of the current without disconnecting the
wire to shunt the current in a meter. The
Rogowski coil, shown in Figure P23.9,
simply clips around the wire. It consists of a
toroidal conductor wrapped around a
circular return cord. The toroid has n turns
per unit length and a cross-sectional area A.
The current to be measured is given by I(t)
= Imax sin ωt. (a) Show that the amplitude of
the emf induced in the Rogowski coil is
ε
Figure P23.9
10.
A piece of insulated wire is shaped
into a figure eight as shown in Figure
P23.10. The radius of the upper circle is 5.00
cm and that of the lower circle is 9.00 cm.
The wire has a uniform resistance per unit
length of 3.00 W/m. A uniform magnetic
field is applied perpendicular to the plane
of the two circles, in the direction shown.
The magnetic field is increasing at a
constant rate of 2.00 T/s. Find the
magnitude and direction of the induced
current in the wire.
= μ0nAω Imax. (b) Explain why the wire
carrying the unknown current need not be
at the center of the Rogowski coil and why
the coil will not respond to nearby currents
that it does not enclose.
max
Figure P23.10
Section 23.2 Motional emf
Section 23.3 Lenz’s Law
Note: Problem 22.62 can be assigned with
this section.
11.
An automobile has a vertical radio
antenna 1.20 m long. The automobile
travels at 65.0 km/h on a horizontal road
where the Earth’s magnetic field is 50.0 μT
directed toward the north and downward
at an angle of 65.0° below the horizontal. (a)
Specify the direction that the automobile
should move to generate the maximum
motional emf in the antenna, with the top of
the antenna positive relative to the bottom.
(b) Calculate the magnitude of this induced
emf.
12.
Consider the arrangement shown in
Figure P23.12. Assume that R = 6.00 Ω, ℓ =
1.20 m, and a uniform 2.50-T magnetic field
is directed into the page. At what speed
should the bar be moved to produce a
current of 0.500 A in the resistor?
Figure P23.12 Problems 23.12, 23.13, 23.14,
and 23.15.
13.
Figure P23.12 shows a top view of a
bar that can slide without friction. The
resistor is 6.00 Ω, and a 2.50-T magnetic
field is directed perpendicularly
downward, into the paper. Let ℓ = 1.20 m.
(a) Calculate the applied force required to
move the bar to the right at a constant
speed of 2.00 m/s. (b) At what rate is energy
delivered to the resistor?
14.
A conducting rod of length ℓ moves
on two horizontal, frictionless rails as
shown in Figure P23.12. If a constant force
of 1.00 N moves the bar at 2.00 m/s through

a magnetic field B that is directed into the
page, (a) what is the current through the
8.00-Ω resistor R? (b) What is the rate at
which energy is delivered to the resistor?
(c) What is the mechanical power delivered

by the force Fapp ?
15.
A metal rod of mass m slides without
friction along two parallel horizontal rails,
separated by a distance ℓ and connected by
a resistor R, as shown in Figure P23.12. A
uniform vertical magnetic field of
magnitude B is applied perpendicular to
the plane of the paper. The applied force
shown in the figure acts only for a moment,
to give the rod a speed v. In terms of m, ℓ, R,
B, and v, find the distance the rod will then
slide as it coasts to a stop.
16.
Very large magnetic fields can be
produced using a procedure called flux
compression. A metallic cylindrical tube of
radius R is placed coaxially in a long
solenoid of somewhat larger radius. The
space between the tube and the solenoid is
filled with a highly explosive material.
When the explosive is set off, it collapses
the tube to a cylinder of radius r < R. If the
collapse happens very rapidly, induced
current in the tube maintains the magnetic
flux nearly constant inside the tube. If the
initial magnetic field in the solenoid is 2.50
T and R/r = 12.0, what maximum value of
magnetic field can be achieved?
17.
The homopolar generator, also called
the Faraday disk, is a low-voltage, highcurrent electric generator. It consists of a
rotating conducting disk with one
stationary brush (a sliding electrical
contact) at its axle and another at a point on
its circumference as shown in Figure
P23.17. A magnetic field is applied
perpendicular to the plane of the disk.
Assume that the field is 0.900 T, the angular
speed is 3 200 rev/min, and the radius of the
disk is 0.400 m. Find the emf generated
between the brushes. When
superconducting coils are used to produce
a large magnetic field, a homopolar
generator can have a power output of
several megawatts. Such a generator is
useful, for example, in purifying metals by
electrolysis. If a voltage is applied to the
output terminals of the generator, it runs in
reverse as a homopolar motor capable of
providing great torque, useful in ship
propulsion.
Figure P23.17
18.
Review problem. A flexible metallic
wire with linear density 3.00 × 10–3 kg/m is
stretched between two fixed clamps 64.0 cm
apart and held under tension 267 N. A
magnet is placed near the wire as shown in
Figure P23.18. Assume that the magnet
produces a uniform field of 4.50 mT over a
2.00-cm length at the center of the wire and
a negligible field elsewhere. The wire is set
vibrating at its fundamental (lowest)
frequency. The section of the wire in the
magnetic field moves with a uniform
amplitude of 1.50 cm. Find (a) the
frequency and (b) the amplitude of the
electromotive force induced between the
ends of the wire.
the current I in Figure P23.20c decreases
rapidly to zero? (d) A copper bar is moved
to the right while its axis is maintained in a
direction perpendicular to a magnetic field
as shown in Figure P31.28d. If the top of the
bar becomes positive relative to the bottom,
what is the direction of the magnetic field?
Figure P23.18
19.
A helicopter (Fig. P23.19) has blades
of length 3.00 m, extending out from a
central hub and rotating at 2.00 rev/s. If the
vertical component of the Earth’s magnetic
field is 50.0 μT, what is the emf induced
between the blade tip and the center hub?
Figure P23.20
(Ross Harrison/Getty Images)
Figure P23.19
20.
Use Lenz’s law to answer the
following questions concerning the
direction of induced currents. (a) What is
the direction of the induced current in
resistor R in Figure P23.20a when the bar
magnet is moved to the left? (b) What is the
direction of the current induced in the
resistor R immediately after the switch S in
Figure P23.20b is closed? (c) What is the
direction of the induced current in R when
21.
A conducting
rectangular loop of mass M, resistance R,
and dimensions w by ℓ falls from rest into a

magnetic field B as shown in Figure P23.21.
During the time interval before the top
edge of the loop reaches the field, the loop
approaches a terminal speed vT. (a) Show
that
vT 
MgR
B2w 2
(b) Why is vT proportional to R? (c) Why is
it inversely proportional to B2?
23.
A coil of area 0.100 m2
is rotating at 60.0 rev/s with the axis of
rotation perpendicular to a 0.200-T
magnetic field. (a) If the coil has 1 000 turns,
what is the maximum emf generated in it?
(b) What is the orientation of the coil with
respect to the magnetic field when the
maximum induced voltage occurs?
Figure P23.21
22.
A rectangular coil with resistance R
has N turns, each of length ℓ and width w as
shown in Figure P23.22. The coil moves into

a uniform magnetic field B with constant

velocity v . What are the magnitude and
direction of the total magnetic force on the
coil (a) as it enters the magnetic field, (b) as
it moves within the field, and (c) as it leaves
the field?
Figure P23.22
24.
A long solenoid, with its axis along
the x axis, consists of 200 turns per meter of
wire that carries a steady current of 15.0 A.
A coil is formed by wrapping 30 turns of
thin wire around a circular frame that has a
radius of 8.00 cm. The coil is placed inside
the solenoid and mounted on an axis that is
a diameter of the coil and that coincides
with the y axis. The coil is then rotated with
an angular speed of 4.00π rad/s. (The plane
of the coil is in the yz plane at t = 0.)
Determine the emf generated in the coil as a
function of time.
Section 23.4 Induced emfs and Electric
Fields
25.
A magnetic field directed into the
page changes with time according to B =
(0.030 0t2 + 1.40) T, where t is in seconds.
The field has a circular cross-section of
radius R = 2.50 cm (Fig. P23.25). What are
the magnitude and direction of the electric
field at point P1 when t = 3.00 s and r1 =
0.020 0 m?
30.
An emf of 24.0 mV is induced in a
500-turn coil at an instant when the current
is 4.00 A and is changing at the rate of 10.0
A/s. What is the magnetic flux through each
turn of the coil?
Figure P23.25 Problems 23.25 and 23.26.
26.
For the situation shown in Figure
P23.25, the magnetic field changes with
time according to the expression B = (2.00t3
– 4.00t2 + 0.800) T and r2 = 2R = 5.00 cm. (a)
Calculate the magnitude and direction of
the force exerted on an electron located at
point P2 when t = 2.00 s. (b) At what time is
this force equal to zero?
Section 23.5 Self-Inductance
27.
A coil has an inductance of 3.00 mH,
and the current in it changes from 0.200 A
to 1.50 A in a time interval of 0.200 s. Find
the magnitude of the average induced emf
in the coil during this time interval.
28.
A coiled telephone cord forms a
spiral with 70 turns, a diameter of 1.30 cm,
and an unstretched length of 60.0 cm.
Determine the self-inductance of one
conductor in the unstretched cord.
29.
A 10.0-mH inductor
carries a current I = Imax sin ωt, with Imax =
5.00 A and ω/2π = 60.0 Hz. What is the selfinduced emf as a function of time?
31.
The current in a 90.0-mH inductor
changes with time as I = 1.00t2 – 6.00t (in SI
units). Find the magnitude of the induced
emf at (a) t = 1.00 s and (b) t = 4.00 s. (c) At
what time is the emf zero?
32.
A toroid has a major radius R and a
minor radius r, and is tightly wound with N
turns of wire as shown in Figure P23.32. If
R >> r, the magnetic field in the region
enclosed by the wire of the torus, of crosssectional area A = πr2, is essentially the
same as the magnetic field of a solenoid
that has been bent into a large circle of
radius R. Modeling the field as the uniform
field of a long solenoid, show that the selfinductance of such a toroid is
approximately
L
0 N 2 A
2R
(An exact expression of the inductance of a
toroid with a rectangular cross-section is
derived in Problem 23.60.)
Figure P23.32
Section 23.6 RL Circuits
33.
A 12.0-V battery is connected into a
series circuit containing a 10.0-Ω resistor
and a 2.00-H inductor. In what time interval
will the current reach (a) 50.0% and (b)
90.0% of its final value?
34.
Show that I = Iie–t/τ is a solution of the
differential equation
IR  L
dI
0
dt
where τ = L/R and Ii is the current at t = 0.
35.
Consider the circuit in Figure P23.35,
taking ε = 6.00 V, L = 8.00 mH, and R = 4.00
Ω. (a) What is the inductive time constant
of the circuit? (b) Calculate the current in
the circuit 250 μs after the switch is closed.
(c) What is the value of the final steadystate current? (d) How long does it take the
current to reach 80.0% of its maximum
value?
Figure P23.35 Problems 23.35, 23.36, and
23.38.
36.
For the RL circuit shown in Figure
P23.35, let the inductance be 3.00 H, the
resistance 8.00 Ω, and the battery emf 36.0
V. (a) Calculate the ratio of the potential
difference across the resistor to the voltage
across the inductor when the current is 2.00
A. (b) Calculate the voltage across the
inductor when the current is 4.50 A.
37.
A circuit consists of a coil, a switch,
and a battery, all in series. The internal
resistance of the battery is negligible
compared with that of the coil. The switch
is originally open. It is thrown closed, and
after a time interval Δt, the current in the
circuit reaches 80.0% of its final value. The
switch remains closed for a time interval
much longer than Δt. Then the battery is
disconnected and the terminals of the coil
are connected together to form a short
circuit. (a) After an equal additional time
interval Δt elapses, the current is what
percentage of its maximum value? (b) At
the moment 2Δt after the coil is shortcircuited, the current in the coil is what
percentage of its maximum value?
38.
When the switch in Figure P23.35 is
closed, the current takes 3.00 ms to reach
98.0% of its final value. If R = 10.0 Ω, what
is the inductance?
39.
The switch in Figure P23.39 is open
for t < 0 and then closed at time t = 0. Find
the current in the inductor and the current
in the switch as functions of time thereafter.
41.
A 140-mH inductor
and a 4.90-Ω resistor are connected with a
switch to a 6.00-V battery as shown in
Figure P23.41. (a) If the switch is thrown to
the left (connecting the battery), how much
time elapses before the current reaches 220
mA? (b) What is the current in the inductor
10.0 s after the switch is closed? (c) Now the
switch is quickly thrown from a to b. How
much time elapses before the current falls
to 160 mA?
Figure P23.39
40.
One application of an RL circuit is
the generation of time-varying high voltage
from a low-voltage source as shown in
Figure P23.40. (a) What is the current in the
circuit a long time after the switch has been
in position a? (b) Now the switch is thrown
quickly from a to b. Compute the initial
voltage across each resistor and across the
inductor. (c) How much time elapses before
the voltage across the inductor drops to
12.0 V?
Figure P23.40
Figure P23.41
Section 23.7 Energy Stored in a Magnetic
Field
42.
The magnetic field inside a
superconducting solenoid is 4.50 T. The
solenoid has an inner diameter of 6.20 cm
and a length of 26.0 cm. Determine (a) the
magnetic energy density in the field and (b)
the energy stored in the magnetic field
within the solenoid.
43.
An air-core solenoid with 68 turns is
8.00 cm long and has a diameter of 1.20 cm.
How much energy is stored in its magnetic
field when it carries a current of 0.770 A?
44.
An RL circuit in which the
inductance is 4.00 H and the resistance is
5.00 Ω is connected to a 22.0-V battery at t =
0. (a) What energy is stored in the inductor
when the current is 0.500 A? (b) At what
rate is energy being stored in the inductor
when I = 1.00 A? (c) What power is being
delivered to the circuit by the battery when
I = 0.500 A?
45.
On a clear day at a certain location, a
100-V/m vertical electric field exists near
the Earth’s surface. At the same place, the
Earth’s magnetic field has a magnitude of
0.500 × 10–4 T. Compute the energy densities
of the two fields.
not induced in the opposite direction to
apply a downward force on the vehicle.)
The vehicle has a mass of 5 × 104 kg and
travels at a speed of 400 km/h. If the vehicle
has 100 loops carrying current at any
moment, what is the approximate
magnitude of the magnetic field required to
levitate the vehicle? Assume that the
magnetic force acts over the entire 10-cm
length of the horizontal wire. Your answer
should suggest that this design is
impractical for magnetic levitation.
Section 23.8 Context Connection—The
Repulsive Model for Magnetic Levitation
46.
The following is a crude model for
levitating a commercial transportation
vehicle using Faraday’s law. Assume that
magnets are used to establish regions of
horizontal magnetic field across the track as
shown in Figure P23.46. Rectangular loops
of wire are mounted on the vehicle so that
the lower 20 cm of each loop passes into
these regions of magnetic field. The upper
portion of each loop contains a 25-Ω
resistor. As the leading edge of a loop
passes into the magnetic field, a current is
induced in the loop as shown in the figure.
The magnetic force on this current in the
bottom of the loop, of length 10 cm, results
in an upward force on the vehicle. (By
electronic timing, a switch is opened in the
loop before the loop’s leading edge leaves
the region of magnetic field, so a current is
Figure P23.46
47.
The Meissner effect. Compare this
problem with Problem 20.76 on the force
attracting a perfect dielectric into a strong
electric field. A fundamental property of a
type I superconducting material is perfect
diamagnetism, or demonstration of the
Meissner effect, illustrated in the photograph
of the levitating magnet on page 693 and
described as follows. The superconducting

material has B  0 everywhere inside it. If a
sample of the material is placed into an
externally produced magnetic field or if it is
cooled to become superconducting while it
is in a magnetic field, electric currents
appear on the surface of the sample. The
currents have precisely the strength and
orientation required to make the total
magnetic field zero throughout the interior
of the sample. The following problem will
help you understand the magnetic force
that can then act on the superconducting
sample.
A vertical solenoid with a length of
120 cm and a diameter of 2.50 cm consists
of 1 400 turns of copper wire carrying a
counterclockwise current of 2.00 A as
shown in Figure P23.47a. (a) Find the
magnetic field in the vacuum inside the
solenoid. (b) Find the energy density of the
magnetic field. Note that the units J/m3 of
energy density are the same as the units
N/m2 of pressure. (c) Now a
superconducting bar 2.20 cm in diameter is
inserted partway into the solenoid. Its
upper end is far outside the solenoid, where
the magnetic field is negligible. The lower
end of the bar is deep inside the solenoid.
Identify the direction required for the
current on the curved surface of the bar so
that the total magnetic field is zero within
the bar. The field created by the
supercurrents is sketched in Figure P23.47b,
and the total field is sketched in Figure
P23.47c. (d) The field of the solenoid exerts
a force on the current in the
superconductor. Identify the direction of
the force on the bar. (e) Calculate the
magnitude of the force by multiplying the
energy density of the solenoid field times
the area of the bottom end of the
superconducting bar.
Figure P23.47
Additional Problems
48.
Figure P23.48 is a graph of the
induced emf versus time for a coil of N
turns rotating with angular speed ω in a
uniform magnetic field directed
perpendicular to the axis of rotation of the
coil. Copy this sketch (on a larger scale) and
on the same set of axes show the graph of
emf versus t (a) if the number of turns in
the coil is doubled, (b) if instead the angular
speed is doubled, and (c) if the angular
speed is doubled while the number of turns
in the coil is halved.
Figure P23.48
shown in Figure P23.52. A battery that
49.
A steel guitar string vibrates (Figure
23.6). The component of magnetic field
perpendicular to the area of a pickup coil
nearby is given by
B = 50.0 mT + (3.20 mT) sin(2π 523 t/s)
The circular pickup coil has 30 turns and
radius 2.70 mm. Find the emf induced in
the coil as a function of time.
50.
Strong magnetic fields are used in
such medical procedures as magnetic
resonance imaging. A technician wearing a
brass bracelet enclosing area 0.005 00 m2
places her hand in a solenoid whose
magnetic field is 5.00 T directed
perpendicular to the plane of the bracelet.
The electrical resistance around the
circumference of the bracelet is 0.020 0 Ω.
An unexpected power failure causes the
field to drop to 1.50 T in a time of 20.0 ms.
Find (a) the current induced in the bracelet
and (b) the power delivered to the bracelet.
(Note: As this problem implies, you should
not wear any metal objects when working
in regions of strong magnetic fields.)
51.
Suppose you wrap wire onto the
core from a roll of cellophane tape to make
a coil. Describe how you can use a bar
magnet to produce an induced voltage in
the coil. What is the order of magnitude of
the emf you generate? State the quantities
you take as data and their values.
52.
A bar of mass m, length d, and
resistance R slides without friction in a
horizontal plane, moving on parallel rails as
maintains a constant emf ε is connected
between the rails, and a constant magnetic

field B is directed perpendicularly to the
plane of the page. Assuming that the bar
starts from rest, show that at time t it moves
with a speed
v

1  e
Bd
 B 2 d 2 t / mR

Figure P23.52
53.
Review problem. A particle with a
mass of 2.00 × 10–16 kg and a charge of 30.0
nC starts from rest, is accelerated by a
strong electric field, and is fired from a
small source inside a region of uniform
constant magnetic field 0.600 T. The
velocity of the particle is perpendicular to
the field. The circular orbit of the particle
encloses a magnetic flux of 15.0 μWb. (a)
Calculate the speed of the particle. (b)
Calculate the potential difference through
which the particle accelerated inside the
source.
54.
An induction furnace uses
electromagnetic induction to produce eddy
currents in a conductor, thereby raising the
conductor’s temperature. Commercial units
operate at frequencies ranging from 60 Hz
to about 1 MHz and deliver powers from a
few watts to several megawatts. Induction
heating can be used for warming a metal
pan on a kitchen stove. It can also be used
for welding in a vacuum enclosure so as to
avoid oxidation and contamination of the
metal. At high frequencies, induced
currents occur only near the surface of the
conductor—this phenomenon is the “skin
effect.” By creating an induced current for a
short time interval at an appropriately high
frequency, one can heat a sample down to a
controlled depth. For example, the surface
of a farm tiller can be tempered to make it
hard and brittle for effective cutting while
keeping the interior metal soft and ductile
to resist breakage.
To explore induction heating,
consider a flat conducting disk of radius R,
thickness b, and resistivity ρ. A sinusoidal
magnetic field Bmax cos ωt is applied
perpendicular to the disk. Assume that the
field is uniform in space and that the
frequency is so low that the skin effect is
not important. Assume that the eddy
currents occur in circles concentric with the
disk. (a) Calculate the average power
delivered to the disk. By what factor does
the power change (b) when the amplitude
of the field doubles, (c) when the frequency
doubles, and (d) when the radius of the
disk doubles?
55.
The magnetic flux through a metal
ring varies with time t according to ΦB =
3(at3 – bt2)T · m2, with a = 2.00 s–3 and b = 6.00
s–2. The resistance of the ring is 3.00 Ω.
Determine the maximum current induced
in the ring during the interval from t = 0 to t
= 2.00 s.
56.
Figure P23.56 shows a stationary
conductor whose shape is similar to the
letter e. The radius of its circular portion is
a = 50.0 cm. It is placed in a constant
magnetic field of 0.500 T directed out of the
page. A straight conducting rod, 50.0 cm
long, is pivoted about point O and rotates
with a constant angular speed of 2.00 rad/s.
(a) Determine the induced emf in the loop
POQ. Note that the area of the loop is θa2/2.
(b) If all the conducting material has a
resistance per length of 5.00 Ω/m, what is
the induced current in the loop POQ at the
instant 0.250 s after point P passes point Q?
Figure P23.56
57.
A betatron accelerates electrons to
energies in the MeV range by means of
electromagnetic induction. Electrons in a
vacuum chamber are held in a circular orbit
by a magnetic field perpendicular to the
orbital plane. The magnetic field is
gradually increased to induce an electric
field around the orbit. (a) Show that the
electric field is in the correct direction to
make the electrons speed up. (b) Assume
that the radius of the orbit remains
constant. Show that the average magnetic
field over the area enclosed by the orbit
must be twice as large as the magnetic field
at the circumference of the circle.
58.
To monitor the breathing of a
hospital patient, a thin belt is girded around
the patient’s chest. The belt is a 200-turn
coil. When the patient inhales, the area
encircled by the coil increases by 39.0 cm2.
The magnitude of the Earth’s magnetic field
is 50.0 μT and makes an angle of 28.0° with
the plane of the coil. Assuming that a
patient takes 1.80 s to inhale, find the
average induced emf in the coil during this
time.
59.
A long, straight wire carries a
current that is given by I = Imax sin(ωt +  ).
The wire lies in the plane of a rectangular
coil of N turns of wire as shown in Figure
P23.5. The quantities Imax, ω, and  are all
constants. Determine the emf induced in
the coil by the magnetic field created by the
current in the straight wire. Assume that
Imax = 50.0 A, ω = 200 πs–1, N = 100, h = w =
5.00 cm, and L = 20.0 cm.
60.
The toroid in Figure P23.60 consists
of N turns and has a rectangular crosssection. Its inner and outer radii are a and b,
respectively. (a) Show that the inductance
of the toroid is
L
0 N 2 h b
ln
2
a
(b) Using this result, compute the selfinductance of a 500-turn toroid for which a
= 10.0 cm, b = 12.0 cm, and h = 1.00 cm. (c) In
Problem 23.32, an approximate expression
for the inductance of a toroid with R >> r
was derived. To get a feel for the accuracy
of that result, use the expression in Problem
23.32 to compute the approximate
inductance of the toroid described in part
(b). Compare the result with the answer to
part (b).
Figure P23.60
61.
(a) A flat, circular coil does not really
produce a uniform magnetic field in the
area it encloses. Nonetheless, estimate the
self-inductance of a flat, compact, circular
coil, with radius R and N turns, by
assuming that the field at its center is
uniform over its area. (b) A circuit on a
laboratory table consists of a 1.5-volt
battery, a 270-Ω resistor, a switch, and three
30-cm-long patch cords connecting them.
Suppose the circuit is arranged to be
circular. Think of it as a flat coil with one
turn. Compute the order of magnitude of its
self-inductance and (c) of the time constant
describing how fast the current increases
when you close the switch.
62.
To prevent damage from arcing in an
electric motor, a discharge resistor is
sometimes placed in parallel with the
armature. If the motor is suddenly
unplugged while running, this resistor
limits the voltage that appears across the
armature coils. Consider a 12.0-V DC motor
with an armature that has a resistance of
7.50 Ω and an inductance of 450 mH.
Assume that the magnitude of the selfinduced emf in the armature coils is 10.0 V
when the motor is running at normal speed.
(The equivalent circuit for the armature is
shown in Fig. P23.62.) Calculate the
maximum resistance R that limits the
voltage across the armature to 80.0 V when
the motor is unplugged.
Figure P23.62
Review problems. Problems 23.63 through
23.65 apply ideas from this chapter and
earlier chapters to some properties of
superconductors, which were introduced in
Section 21.3.
63.
The resistance of a superconductor. In
an experiment carried out by S. C. Collins
between 1955 and 1958, a current was
maintained in a superconducting lead ring
for 2.50 yr with no observed loss. If the
inductance of the ring was 3.14 × 10–8 H and
the sensitivity of the experiment was 1 part
in 109, what was the maximum resistance of
the ring? (Suggestion: Treat this problem as
a decaying current in an RL circuit and
recall that e–x ≈ 1 – x for small x.)
64.
A novel method of storing energy
has been proposed. A huge underground
superconducting coil, 1.00 km in diameter,
would be fabricated. It would carry a
maximum current of 50.0 kA through each
winding of a 150-turn Nb3Sn solenoid. (a) If
the inductance of this huge coil were 50.0
H, what would be the total energy stored?
(b) What would be the compressive force
per meter length acting between two
adjacent windings 0.250 m apart?
65.
Superconducting power transmission.
The use of superconductors has been
proposed for power transmission lines. A
single coaxial cable (Fig. P23.65) could carry
1.00 × 103 MW (the output of a large power
plant) at 200 kV, DC, over a distance of 1
000 km without loss. An inner wire of
radius 2.00 cm, made from the
superconductor Nb3Sn, carries the current I
in one direction. A surrounding
superconducting cylinder, of radius 5.00
cm, would carry the return current I. In
such a system, what is the magnetic field (a)
at the surface of the inner conductor and (b)
at the inner surface of the outer conductor?
(c) How much energy would be stored in
the space between the conductors in a
1 000-km superconducting line? (d) What is
the pressure exerted on the outer
conductor?
Figure P23.65
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