Chapter 32
Chapter 32
Chapter 32
PROBLEMS
1, 2, 3 = straightforward, intermediate, challenging = full solution available in the Student Solutions Manual and Study Guide
= coached solution with hints available at http://www.pse6.com = computer useful in solving problem
= paired numerical and symbolic problems
Section 32.1 Self-Inductance 12. A toroid has a major radius R and a minor radius r, and is
1. A coil has an inductance of 3.00 mH, and the current in it tightly wound with N turns of wire, as shown in Figure
changes from 0.200 A to 1.50 A in a time of 0.200 s. Find P32.12. If R -- r, the magnetic field in the region
the magnitude of the average induced emf in the coil enclosed by the wire of the torus, of cross-sectional area
during this time. A " )r 2, is essentially the same as the magnetic field of a
solenoid that has been bent into a large circle of radius R.
2. A coiled telephone cord forms a spiral with 70 turns, a
Modeling the field as the uniform field of a long solenoid,
diameter of 1.30 cm, and an unstretched length of 60.0 cm.
show that the self-inductance of such a toroid is approxi-
Determine the self-inductance of one conductor in the
mately
unstretched cord.
3. A 2.00-H inductor carries a steady current of 0.500 A. ' 0N 2A
L(
When the switch in the circuit is opened, the current is 2)R
effectively zero after 10.0 ms. What is the average induced
emf in the inductor during this time? (An exact expression of the inductance of a toroid with a
rectangular cross section is derived in Problem 64.)
4. Calculate the magnetic flux through the area enclosed by
a 300-turn, 7.20-mH coil when the current in the coil is
10.0 mA.
5. A 10.0-mH inductor carries a current I " I max sin 1t,
with I max " 5.00 A and 1/2) " 60.0 Hz. What is the back
emf as a function of time?
6. An emf of 24.0 mV is induced in a 500-turn coil at an
R Area
instant when the current is 4.00 A and is changing at the r
A
rate of 10.0 A/s. What is the magnetic flux through each
turn of the coil?
7. An inductor in the form of a solenoid contains 420 turns, is
16.0 cm in length, and has a cross-sectional area of 3.00 cm2.
What uniform rate of decrease of current through the
inductor induces an emf of 175 'V? Figure P32.12
8. The current in a 90.0-mH inductor changes with time as
I " 1.00t 2 # 6.00t (in SI units). Find the magnitude of the 13. A self-induced emf in a solenoid of inductance L changes
induced emf at (a) t " 1.00 s and (b) t " 4.00 s. (c) At in time as ! " !0 e#kt. Find the total charge that passes
what time is the emf zero? through the solenoid, assuming the charge is finite.
9. A 40.0-mA current is carried by a uniformly wound air-core
solenoid with 450 turns, a 15.0-mm diameter, and 12.0-cm
length. Compute (a) the magnetic field inside the sole- Section 32.2 RL Circuits
noid, (b) the magnetic flux through each turn, and (c) the 14. Calculate the resistance in an RL circuit in which L " 2.50 H
inductance of the solenoid. (d) What If? If the current and the current increases to 90.0% of its final value in
were different, which of these quantities would change? 3.00 s.
10. A solenoid has 120 turns uniformly wrapped around a 15. A 12.0-V battery is connected into a series circuit contain-
wooden core, which has a diameter of 10.0 mm and a ing a 10.0-. resistor and a 2.00-H inductor. How long will
length of 9.00 cm. (a) Calculate the inductance of the sole- it take the current to reach (a) 50.0% and (b) 90.0% of its
noid. (b) What If? The wooden core is replaced with a soft final value?
iron rod that has the same dimensions, but a magnetic
16. Show that I " I 0 e#t/+ is a solution of the differential
permeability 'm " 800'0. What is the new inductance?
equation
11. A piece of copper wire with thin insulation, 200 m long
and 1.00 mm in diameter, is wound onto a plastic tube to dI
IR , L "0
form a long solenoid. This coil has a circular cross section dt
and consists of tightly wound turns in one layer. If the where + " L/R and I 0 is the current at t " 0.
current in the solenoid drops linearly from 1.80 A to
17. Consider the circuit in Figure P32.17, taking ! " 6.00 V,
zero in 0.120 seconds, an emf of 80.0 mV is induced in
L " 8.00 mH, and R " 4.00 .. (a) What is the inductive
the coil. What is the length of the solenoid, measured
time constant of the circuit? (b) Calculate the current in
along its axis?
1026 C H A P T E R 3 2 • Inductance
S I (t )
10.0 A
200 µ s
I (t )
ε L
100 Ω 10.0 mH
R
Figure P32.25
Figure P32.17 Problems 17, 18, 19, and 22.
S S
Figure P32.23
ε L
are connected in series, show that they are equivalent to a 38. Assume that the magnitude of the magnetic field outside a
single ideal inductor having L eq " L 1 , L 2 . (b) Assuming sphere of radius R is B " B 0(R/r)2, where B 0 is a constant.
these same two inductors are connected in parallel, show Determine the total energy stored in the magnetic field
that they are equivalent to a single ideal inductor having outside the sphere and evaluate your result for B 0 "
1/L eq " 1/L 1 , 1/L 2. (c) What If? Now consider two 5.00 ( 10#5 T and R " 6.00 ( 106 m, values appropriate
inductors L 1 and L 2 that have nonzero internal resistances for the Earth’s magnetic field.
R 1 and R 2, respectively. Assume they are still far apart so
that their mutual inductance is zero. Assuming these induc- Section 32.4 Mutual Inductance
tors are connected in series, show that they are equivalent to
a single inductor having L eq " L 1 , L 2 and R eq " 39. Two coils are close to each other. The first coil carries a time-
R 1 , R 2. (d) If these same inductors are now connected in varying current given by I (t) " (5.00 A) e#0.025 0t sin(377t ).
parallel, is it necessarily true that they are equivalent to a At t " 0.800 s, the emf measured across the second coil is
single ideal inductor having 1/L eq " 1/L 1 , 1/L 2 and # 3.20 V. What is the mutual inductance of the coils?
1/R eq " 1/R 1 , 1/R 2 ? Explain your answer. 40. Two coils, held in fixed positions, have a mutual induc-
tance of 100 'H. What is the peak voltage in one when a
sinusoidal current given by I (t ) " (10.0 A) sin(1 000t) is
Section 32.3 Energy in a Magnetic Field in the other coil?
29. Calculate the energy associated with the magnetic field of 41. An emf of 96.0 mV is induced in the windings of a coil
a 200-turn solenoid in which a current of 1.75 A produces when the current in a nearby coil is increasing at the rate of
a flux of 3.70 ( 10#4 Wb in each turn. 1.20 A/s. What is the mutual inductance of the two coils?
30. The magnetic field inside a superconducting solenoid is 42. On a printed circuit board, a relatively long straight con-
4.50 T. The solenoid has an inner diameter of 6.20 cm and ductor and a conducting rectangular loop lie in the same
a length of 26.0 cm. Determine (a) the magnetic energy plane, as shown in Figure P31.9. Taking h " 0.400 mm,
density in the field and (b) the energy stored in the w " 1.30 mm, and L " 2.70 mm, find their mutual
magnetic field within the solenoid. inductance.
31. An air-core solenoid with 68 turns is 8.00 cm long and has 43. Two solenoids A and B, spaced close to each other and
a diameter of 1.20 cm. How much energy is stored in its sharing the same cylindrical axis, have 400 and 700 turns,
magnetic field when it carries a current of 0.770 A? respectively. A current of 3.50 A in coil A produces an aver-
32. At t " 0, an emf of 500 V is applied to a coil that has an age flux of 300 'Wb through each turn of A and a flux of
inductance of 0.800 H and a resistance of 30.0 .. (a) Find 90.0 'Wb through each turn of B. (a) Calculate the
the energy stored in the magnetic field when the current mutual inductance of the two solenoids. (b) What is the
reaches half its maximum value. (b) After the emf is self-inductance of A? (c) What emf is induced in B when
connected, how long does it take the current to reach this the current in A increases at the rate of 0.500 A/s?
value? 44. A large coil of radius R 1 and having N 1 turns is coaxial
33. On a clear day at a certain location, a 100-V/m verti- with a small coil of radius R 2 and having N 2 turns. The
cal electric field exists near the Earth’s surface. At the centers of the coils are separated by a distance x that is
same place, the Earth’s magnetic field has a magnitude of much larger than R 1 and R 2. What is the mutual induc-
0.500 ( 10#4 T. Compute the energy densities of the two tance of the coils? Suggestion: John von Neumann proved
fields. that the same answer must result from considering the flux
through the first coil of the magnetic field produced by
34. Complete the calculation in Example 32.4 by proving that
the second coil, or from considering the flux through the
!0
0
e #2Rt/L dt "
L
2R
second coil of the magnetic field produced by the first
coil. In this problem it is easy to calculate the flux through
the small coil, but it is difficult to calculate the flux
35. An RL circuit in which L " 4.00 H and R " 5.00 . is con- through the large coil, because to do so you would have to
nected to a 22.0-V battery at t " 0. (a) What energy is know the magnetic field away from the axis.
stored in the inductor when the current is 0.500 A? (b) At 45. Two inductors having self-inductances L 1 and L 2 are con-
what rate is energy being stored in the inductor when nected in parallel as shown in Figure P32.45a. The mutual
I " 1.00 A? (c) What power is being delivered to the inductance between the two inductors is M. Determine the
circuit by the battery when I " 0.500 A? equivalent self-inductance Leq for the system (Figure
36. A 10.0-V battery, a 5.00-. resistor, and a 10.0-H inductor are P32.45b).
connected in series. After the current in the circuit has
reached its maximum value, calculate (a) the power being
supplied by the battery, (b) the power being delivered to the I (t ) I (t )
resistor, (c) the power being delivered to the inductor, and L1 M L2 Leq
(d) the energy stored in the magnetic field of the inductor.
37. A uniform electric field of magnitude 680 kV/m through-
out a cylindrical volume results in a total energy of 3.40 'J.
What magnetic field over this same region stores the same (a) (b)
total energy? Figure P32.45
1028 C H A P T E R 3 2 • Inductance
Section 32.5 Oscillations in an LC Circuit (a) the energy stored in the capacitor; (b) the energy stored
46. A 1.00-'F capacitor is charged by a 40.0-V power supply. in the inductor; (c) the total energy in the circuit.
The fully charged capacitor is then discharged through a
10.0-mH inductor. Find the maximum current in the
resulting oscillations. Section 32.6 The RLC Circuit
47. An LC circuit consists of a 20.0-mH inductor and a 54. In Figure 32.21, let R " 7.60 ., L " 2.20 mH, and C "
0.500-'F capacitor. If the maximum instantaneous current 1.80 'F. (a) Calculate the frequency of the damped oscilla-
is 0.100 A, what is the greatest potential difference across tion of the circuit. (b) What is the critical resistance?
the capacitor? 55. Consider an LC circuit in which L " 500 mH and
48. In the circuit of Figure P32.48, the battery emf is 50.0 V, C " 0.100 'F. (a) What is the resonance frequency 1 0 ?
the resistance is 250 ., and the capacitance is 0.500 'F. (b) If a resistance of 1.00 k. is introduced into this circuit,
The switch S is closed for a long time and no voltage is what is the frequency of the (damped) oscillations? (c) What
measured across the capacitor. After the switch is opened, is the percent difference between the two frequencies?
the potential difference across the capacitor reaches 56. Show that Equation 32.28 in the text is Kirchhoff’s loop
a maximum value of 150 V. What is the value of the rule as applied to the circuit in Figure 32.21.
inductance? 57. The energy of an RLC circuit decreases by 1.00% during
each oscillation when R " 2.00 .. If this resistance is
removed, the resulting LC circuit oscillates at a frequency
R
of 1.00 kHz. Find the values of the inductance and the
capacitance.
58. Electrical oscillations are initiated in a series circuit con-
ε L C
taining a capacitance C, inductance L, and resistance R.
S (a) If R ** √4L/C (weak damping), how much time
elapses before the amplitude of the current oscillation falls
Figure P32.48 off to 50.0% of its initial value? (b) How long does it take
the energy to decrease to 50.0% of its initial value?
49. A fixed inductance L " 1.05 'H is used in series with a
variable capacitor in the tuning section of a radiotele-
Additional Problems
phone on a ship. What capacitance tunes the circuit to the
signal from a transmitter broadcasting at 6.30 MHz? 59. Review problem. This problem extends the reasoning of
Section 26.4, Problem 26.37, Example 30.6, and Section
50. Calculate the inductance of an LC circuit that oscillates at
32.3. (a) Consider a capacitor with vacuum between its
120 Hz when the capacitance is 8.00 'F.
large, closely spaced, oppositely charged parallel plates.
51. An LC circuit like the one in Figure 32.16 contains an Show that the force on one plate can be accounted for by
82.0-mH inductor and a 17.0-'F capacitor that initially car- thinking of the electric field between the plates as exerting
ries a 180-'C charge. The switch is open for t * 0 and a “negative pressure” equal to the energy density of the
then closed at t " 0. (a) Find the frequency (in hertz) of electric field. (b) Consider two infinite plane sheets carry-
the resulting oscillations. At t " 1.00 ms, find (b) the ing electric currents in opposite directions with equal lin-
charge on the capacitor and (c) the current in the circuit. ear current densities Js . Calculate the force per area acting
52. The switch in Figure P32.52 is connected to point a for on one sheet due to the magnetic field created by the
a long time. After the switch is thrown to point b, what other sheet. (c) Calculate the net magnetic field between
are (a) the frequency of oscillation of the LC circuit, the sheets and the field outside of the volume between
(b) the maximum charge that appears on the capacitor, them. (d) Calculate the energy density in the magnetic
(c) the maximum current in the inductor, and (d) the field between the sheets. (e) Show that the force on one
total energy the circuit possesses at t " 3.00 s? sheet can be accounted for by thinking of the magnetic
field between the sheets as exerting a positive pressure
equal to its energy density. This result for magnetic
10.0 Ω 0.100 H pressure applies to all current configurations, not just to
a b
S
sheets of current.
60. Initially, the capacitor in a series LC circuit is charged. A
12.0 V 1.00 µ
µF
switch is closed at t " 0, allowing the capacitor to
discharge, and at time t the energy stored in the capacitor
is one fourth of its initial value. Determine L, assuming C
Figure P32.52
is known.
53. An LC circuit like that in Figure 32.16 consists of a 61. A 1.00-mH inductor and a 1.00-'F capacitor are connected
3.30-H inductor and an 840-pF capacitor, initially carrying a in series. The current in the circuit is described by
105-'C charge. The switch is open for t * 0 and then closed I " 20.0t, where t is in seconds and I is in amperes. The
at t " 0. Compute the following quantities at t " 2.00 ms: capacitor initially has no charge. Determine (a) the
Problems 1029
voltage across the inductor as a function of time, (b) the be used for connections. (a) How many turns of this wire
voltage across the capacitor as a function of time, and can be wrapped around the rod? For an accurate answer
(c) the time when the energy stored in the capacitor first you should add the diameter of the wire to the diameter of
exceeds that in the inductor. the rod in determining the circumference of each turn.
62. An inductor having inductance L and a capacitor having Also note that the wire spirals diagonally along the surface
capacitance C are connected in series. The current in the of the rod. (b) What is the resistance of this inductor?
circuit increases linearly in time as described by I " Kt, (c) What is its inductance?
where K is a constant. The capacitor is initially uncharged. 67. A wire of nonmagnetic material, with radius R, carries
Determine (a) the voltage across the inductor as a current uniformly distributed over its cross section. The
function of time, (b) the voltage across the capacitor as a total current carried by the wire is I. Show that the mag-
function of time, and (c) the time when the energy stored netic energy per unit length inside the wire is '0I 2/16).
in the capacitor first exceeds that in the inductor. 68. An 820-turn wire coil of resistance 24.0 . is placed around
a 12 500-turn solenoid 7.00 cm long, as shown in Figure
63. A capacitor in a series LC circuit has an initial charge Q P32.68. Both coil and solenoid have cross-sectional areas of
and is being discharged. Find, in terms of L and C, the flux 1.00 ( 10#4 m2. (a) How long does it take the solenoid
through each of the N turns in the coil, when the charge current to reach 63.2% of its maximum value? Determine
on the capacitor is Q /2. (b) the average back emf caused by the self-inductance of
64. The toroid in Figure P32.64 consists of N turns and has a the solenoid during this time interval, (c) the average rate
rectangular cross section. Its inner and outer radii are a of change in magnetic flux through the coil during this
and b, respectively. (a) Show that the inductance of the time interval, and (d) the magnitude of the average
toroid is induced current in the coil.
' 0 N 2h b
L" ln
2) a 14.0 Ω
S
h
Figure P32.68
a
b
Figure P32.64 69. At t " 0, the open switch in Figure P32.69 is closed. By
using Kirchhoff’s rules for the instantaneous currents and
voltages in this two-loop circuit, show that the current in
65. (a) A flat circular coil does not really produce a uniform the inductor at time t - 0 is
magnetic field in the area it encloses, but estimate the self-
inductance of a flat, compact circular coil, with radius R I (t ) "
! [1 # e #(R 3/L)t ]
and N turns, by assuming that the field at its center is uni- R1
form over its area. (b) A circuit on a laboratory table
consists of a 1.5-volt battery, a 270-. resistor, a switch, and where R 3 " R 1R 2/(R 1 , R 2).
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 R1
self-inductance and (c) of the time constant describing
how fast the current increases when you close the switch. S
66. A soft iron rod ('m " 800'0) is used as the core of a sole-
noid. The rod has a diameter of 24.0 mm and is 10.0 cm
ε R2 L
70. In Figure P32.69 take ! " 6.00 V, R 1 " 5.00 ., and Armature
R 2 " 1.00 .. The inductor has negligible resistance. When
the switch is opened after having been closed for a
long time, the current in the inductor drops to 0.250 A in
0.150 s. What is the inductance of the inductor? 7.50 Ω
2.00 kΩ
74. An air-core solenoid 0.500 m in length contains 1 000
turns and has a cross-sectional area of 1.00 cm2. (a) Ignor-
R1
ing end effects, find the self-inductance. (b) A secondary
S a winding wrapped around the center of the solenoid has
100 turns. What is the mutual inductance? (c) The sec-
6.00 kΩ R2
ε 18.0 V
L 0.400 H ondary winding carries a constant current of 1.00 A, and
the solenoid is connected to a load of 1.00 k.. The
b
constant current is suddenly stopped. How much charge
flows through the load resistor?
75. The lead-in wires from a television antenna are often con-
Figure P32.71
structed in the form of two parallel wires (Fig. P32.75).
(a) Why does this configuration of conductors have an
72. The open switch in Figure P32.72 is closed at t " 0. Before inductance? (b) What constitutes the flux loop for this
the switch is closed, the capacitor is uncharged, and all configuration? (c) Ignoring any magnetic flux inside the
currents are zero. Determine the currents in L, C, and R wires, show that the inductance of a length x of this type of
and the potential differences across L, C, and R (a) at the lead-in is
instant after the switch is closed, and (b) long after it is
closed.
L"
'0x
)
ln " w #a a #
L where a is the radius of the wires and w is their center-to-
center separation.
R TV set
I
TV antenna
C
ε0
S
I
Figure P32.72 Figure P32.75
field, and note that the units J/m3 of energy density are
the same as the units N/m2 of pressure. (c) Now a super-
I conducting bar 2.20 cm in diameter is inserted partway
a = 2.00 cm into the solenoid. Its upper end is far outside the solenoid,
where the magnetic field is negligible. The lower end of
I b = 5.00 cm the bar is deep inside the solenoid. Identify the direction
a
required for the current on the curved surface of the bar,
b
so that the total magnetic field is zero within the bar. The
field created by the supercurrents is sketched in Figure
P32.79b, and the total field is sketched in Figure P32.79c.
(d) The field of the solenoid exerts a force on the current
Figure P32.78
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
79. The Meissner effect. Compare this problem with Problem 65 in the area of the bottom end of the superconducting bar.
Chapter 26, 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 demon- Answers to Quick Quizzes
stration of the Meissner effect, illustrated in Figure 30.35, and
32.1 (c), (f). For the constant current in (a) and (b), there is
described as follows. The superconducting material has
no potential difference across the resistanceless inductor.
B " 0 everywhere inside it. If a sample of the material is
In (c), if the current increases, the emf induced in the
placed into an externally produced magnetic field, or if it is
inductor is in the opposite direction, from b to a, making
cooled to become superconducting while it is in a magnetic
a higher in potential than b. Similarly, in (f), the decreas-
field, electric currents appear on the surface of the sample.
ing current induces an emf in the same direction as the
The currents have precisely the strength and orientation
current, from b to a, again making the potential higher at
required to make the total magnetic field zero throughout
a than b.
the interior of the sample. The following problem will help
you to understand the magnetic force that can then act on 32.2 (b), (d). As the switch is closed, there is no current, so
the superconducting sample. there is no voltage across the resistor. After a long time,
A vertical solenoid with a length of 120 cm and a diam- the current has reached its final value, and the inductor
eter of 2.50 cm consists of 1 400 turns of copper wire carry- has no further effect on the circuit.
ing a counterclockwise current of 2.00 A, as in Figure 32.3 (b). When the iron rod is inserted into the solenoid,
P32.79a. (a) Find the magnetic field in the vacuum inside the inductance of the coil increases. As a result, more
the solenoid. (b) Find the energy density of the magnetic potential difference appears across the coil than before.
1032 C H A P T E R 3 2 • Inductance
Consequently, less potential difference appears across the (b), the change in cross-sectional area has no effect on
bulb, so the bulb is dimmer. the magnetic field. In (c), increasing the length but keep-
32.4 (b). Figure 32.10 shows that circuit B has the greater time ing n fixed has no effect on the magnetic field. Increasing
constant because in this circuit it takes longer for the the current in (d) increases the magnetic field in the
current to reach its maximum value and then longer for solenoid.
this current to decrease to zero after the switch is thrown 32.6 (a). M12 increases because the magnetic flux through coil
to position b. Equation 32.8 indicates that, for equal resis- 2 increases.
tances R A and R B, the condition +B - +A means that 32.7 (b). If the current is at its maximum value, the charge on
LA * LB. the capacitor is zero.
32.5 (a), (d). Because the energy density depends on the mag- 32.8 (c). If the current is zero, this is the instant at which the
nitude of the magnetic field, to increase the energy den- capacitor is fully charged and the current is about to
sity, we must increase the magnetic field. For a solenoid, reverse direction.
B " '0nI, where n is the number of turns per unit length.
In (a), we increase n to increase the magnetic field. In