Chapter 11 Review
Key Terms
henry (H)
unit of inductance,
; it is also expressed as a volt second per ampere
inductance
property of a device that tells how effectively it induces an emf in another device
inductive time constant
denoted by
, the characteristic time given by quantity
of a particular series
circuit
inductor
part of an electrical circuit to provide selfinductance, which is symbolized by a coil of wire
LC circuit
circuit composed of an ac source, inductor, and capacitor
magnetic energy density
energy stored per volume in a magnetic field
mutual inductance
geometric quantity that expresses how effective two devices are at inducing emfs in one another
RLC circuit
circuit with an ac source, resistor, inductor, and capacitor all in series.
selfinductance
effect of the device inducing emf in itself
Key Equations
Mutual inductance by flux  
Mutual inductance in circuits  
Selfinductance in terms of magnetic flux  
Selfinductance in terms of emf 

Selfinductance of a solenoid 

Selfinductance of a toroid 

Energy stored in an inductor 

Current as a function of time for a RL circuit 

Time constant for a RL circuit 

Charge oscillation in LC circuits 

Angular frequency in LC circuits 

Current oscillations in LC circuits 

Charge as a function of time in RLC circuit 

Angular frequency in RLC circuit 

Summary
11.1 Mutual Inductance
 Inductance is the property of a device that expresses how effectively it induces an emf in another device.
 Mutual inductance is the effect of two devices inducing emfs in each other.
 A change in current in one circuit induces an emf in the second:
 where is defined to be the mutual inductance between the two circuits and the minus sign is due to Lenz’s law.
 Symmetrically, a change in current through the second circuit induces an emf in the first:
where
is the same mutual inductance as in the reverse process.
11.2 SelfInductance and Inductors
 Current changes in a device induce an emf in the device itself, called selfinductance,
 where is the selfinductance of the inductor and is the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenz’s law. The unit of selfinductance and inductance is the henry (), where
 The selfinductance of a solenoid is
 where is its number of turns in the solenoid, is its crosssectional area, is its length, and is the permeability of free space.
 The selfinductance of a toroid is
 where is its number of turns in the toroid, and are the inner and outer radii of the toroid, is the height of the toroid, and is the permeability of free space.
11.3 Energy in a Magnetic Field
 The energy stored in an inductor is
 The selfinductance per unit length of coaxial cable is
11.4 RL Circuits
 When a series connection of a resistor and an inductor—an circuit—is connected to a voltage source, the time variation of the current is
 (turning on),where the initial current is
 The characteristic time constant is where is the inductance and is the resistance.
 In the first time constant the current rises from zero to and to of the remainder in every subsequent time interval
 When the inductor is shorted through a resistor, current decreases as
Current falls to
in the first time interval
and to
of the remainder toward zero in each subsequent time
11.5 Oscillations in an LC Circuit
 The energy transferred in an oscillatory manner between the capacitor and inductor in an circuit occurs at an angular frequency
 The charge and current in the circuit are given by
11.6 RLC Series Circuits
 The underdamped solution for the capacitor charge in an circuit is
The angular frequency given in the underdamped solution for the
circuit is
Answers to Check Your Understanding
11.1
11.2 a. decreasing; b. increasing; Since the current flows in the opposite direction of the diagram, in order to get a positive emf on the lefthand side of diagram (a), we need to decrease the current to the left, which creates a reinforced emf where the positive end is on the lefthand side. To get a positive emf on the righthand side of diagram (b), we need to increase the current to the left, which creates a reinforced emf where the positive end is on the righthand side.
11.3
11.4 a.
; b.
11.5 a.
b.
11.6
11.8 a.
; b.
; c.
11.10 a.
; b.
or
; c.
11.11 a. overdamped; b.
Conceptual Questions
11.1 Mutual Inductance
1. Show that
and
which are both expressions for selfinductance, have the same units.
2. A
inductor carries a current of
Describe how a
emf can be induced across it.
3. The ignition circuit of an automobile is powered by a
battery. How are we able to generate large voltages with this power source?
4. When the current through a large inductor is interrupted with a switch, an arc appears across the open terminals of the switch. Explain.
11.2 SelfInductance and Inductors
5. Does selfinductance depend on the value of the magnetic flux? Does it depend on the current through the wire? Correlate your answers with the equation
6. Would the selfinductance of a
long, tightly wound solenoid differ from the selfinductance per meter of an infinite, but otherwise identical, solenoid?
7. Discuss how you might determine the selfinductance per unit length of a long, straight wire.
8. The selfinductance of a coil is zero if there is no current passing through the windings. True or false?
9. How does the selfinductance per unit length near the centre of a solenoid (away from the ends) compare with its value near the end of the solenoid?
11.3 Energy in a Magnetic Field
10. Show that
has units of energy.
11.4 RL Circuits
11. Use Lenz’s law to explain why the initial current in the
circuit of Figure 11.4.1(b) is zero.
12. When the current in the
circuit of Figure 11.4.1(b) reaches its final value
what is the voltage across the inductor? Across the resistor?
13. Does the time required for the current in an
circuit to reach any fraction of its steadystate value depend on the emf of the battery?
14. An inductor is connected across the terminals of a battery. Does the current that eventually flows through the inductor depend on the internal resistance of the battery? Does the time required for the current to reach its final value depend on this resistance?
15. At what time is the voltage across the inductor of the
circuit of Figure 14.12(b) a maximum?
16. In the simple
circuit of Figure 11.4.1(b), can the emf induced across the inductor ever be greater than the emf of the battery used to produce the current?
17. If the emf of the battery of Figure 11.4.1(b) is reduced by a factor of
by how much does the steadystate energy stored in the magnetic field of the inductor change?
18. A steady current flows through a circuit with a large inductive time constant. When a switch in the circuit is opened, a large spark occurs across the terminals of the switch. Explain.
19. Describe how the currents through
and
shown below vary with time after switch
is closed.
20. Discuss possible practical applications of
circuits.
11.5 Oscillations in an LC Circuit
21. Do Kirchhoff’s rules apply to circuits that contain inductors and capacitors?
22. Can a circuit element have both capacitance and inductance?
23. In an
circuit, what determines the frequency and the amplitude of the energy oscillations in either the inductor or capacitor?
11.6 RLC Series Circuits
24. When a wire is connected between the two ends of a solenoid, the resulting circuit can oscillate like an
circuit. Describe what causes the capacitance in this circuit.
25. Describe what effect the resistance of the connecting wires has on an oscillating
circuit.
26. Suppose you wanted to design an
circuit with a frequency of
What problems might you encounter?
27. A radio receiver uses an
circuit to pick out particular frequencies to listen to in your house or car without hearing other unwanted frequencies. How would someone design such a circuit?
Problems
11.1 Mutual Inductance
28. When the current in one coil changes at a rate of
an emf of
is induced in a second, nearby coil. What is the mutual inductance of the two coils?
29. An emf of
is induced in a coil while the current in a nearby coil is decreasing at a rate of
What is the mutual inductance of the two coils?
30. Two coils close to each other have a mutual inductance of
If the current in one coil decays according to
where
and
what is the emf induced in the second coil immediately after the current starts to decay? At
?
31. A coil of
is wrapped around a long solenoid of crosssectional area
The solenoid is
long and has
(a) What is the mutual inductance of this system? (b) The outer coil is replaced by a coil of
whose radius is three times that of the solenoid. What is the mutual inductance of this configuration?
32. A
solenoid is
long and
in diameter. Inside the solenoid, a small
singleturn rectangular coil is fixed in place with its face perpendicular to the long axis of the solenoid. What is the mutual inductance of this system?
33. A toroidal coil has a mean radius of
and a crosssectional area of
; it is wound uniformly with
A second toroidal coil of
is wound uniformly over the first coil. Ignoring the variation of the magnetic field within a toroid, determine the mutual inductance of the two coils.
34. A solenoid of
turns has length
and radius
and a second smaller solenoid of
turns has length
and radius
The smaller solenoid is placed completely inside the larger solenoid so that their long axes coincide. What is the mutual inductance of the two solenoids?
11.2 SelfInductance and Inductors
35. An emf of
is induced across a coil when the current through it changes uniformly from
to
in
What is the selfinductance of the coil?
36. The current shown in part (a) below is increasing, whereas that shown in part (b) is decreasing. In each case, determine which end of the inductor is at the higher potential.
37. What is the rate at which the current though a
coil is changing if an emf of
is induced across the coil?
38. When a camera uses a flash, a fully charged capacitor discharges through an inductor. In what time must the
current through a
inductor be switched on or off to induce a
emf?
39. A coil with a selfinductance of
carries a current that varies with time according to
Find an expression for the emf induced in the coil.
40. A solenoid
long is wound with
of wire. The crosssectional area of the coil is
What is the selfinductance of the solenoid?
41. A coil with a selfinductance of
carries a current that decreases at a uniform rate
What is the emf induced in the coil? Describe the polarity of the induced emf.
42. The current
through a
inductor varies with time, as shown below. The resistance of the inductor is
Calculate the voltage across the inductor at
and
43. A long, cylindrical solenoid with
has a radius of
(a) Neglecting end effects, what is the selfinductance per unit length of the solenoid? (b) If the current through the solenoid changes at the rate
what is the emf induced per unit length?
44. Suppose that a rectangular toroid has
windings and a selfinductance of
If
what is the ratio of its outer radius to its inner radius?
45. What is the selfinductance per meter of a coaxial cable whose inner radius is
and whose outer radius is
?
11.3 Energy in a Magnetic Field
46. At the instant a current of
is flowing through a coil of wire, the energy stored in its magnetic field is
What is the selfinductance of the coil?
47. Suppose that a rectangular toroid has
windings and a selfinductance of
If
what is the current flowing through a rectangular toroid when the energy in its magnetic field is
?
48. Solenoid
is tightly wound while solenoid
has windings that are evenly spaced with a gap equal to the diameter of the wire. The solenoids are otherwise identical. Determine the ratio of the energies stored per unit length of these solenoids when the same current flows through each.
49. A
inductor carries a current of
How much ice at
could be melted by the energy stored in the magnetic field of the inductor? (Hint: Use the value
for ice.)
50. A coil with a selfinductance of
and a resistance of
carries a steady current of
(a) What is the energy stored in the magnetic field of the coil? (b) What is the energy per second dissipated in the resistance of the coil?
51. A current of
is flowing in a coaxial cable whose outer radius is five times its inner radius. What is the magnetic field energy stored in a
length of the cable?
11.4 RL Circuits
52. In Figure 11.4.1,
and
Determine (a) the time constant of the circuit, (b) the initial current through the resistor, (c) the final current through the resistor, (d) the current through the resistor when
and (e) the voltages across the inductor and the resistor when
53. For the circuit shown below,
and
After steady state is reached with
closed and
open,
is closed and immediately thereafter (at
)
is opened. Determine (a) the current through
at
(b) the current through
at
and (c) the voltages across
and
at
54. The current in the
circuit shown here increases to
of its steadystate value in
What is the time constant of the circuit?
55. How long after switch
is thrown does it take the current in the circuit shown to reach half its maximum value? Express your answer in terms of the time constant of the circuit.
56. Examine the circuit shown below in part (a). Determine
at the instant after the switch is thrown in the circuit of (a), thereby producing the circuit of (b). Show that if
were to continue to increase at this initial rate, it would reach its maximum
in one time constant.
57. The current in the
circuit shown below reaches half its maximum value in
after the switch
is thrown. Determine (a) the time constant of the circuit and (b) the resistance of the circuit if
58. Consider the circuit shown below. Find
and
when (a) the switch
is first closed, (b) after the currents have reached steadystate values, and (c) at the instant the switch is reopened (after being closed for a long time).
59. For the circuit shown below,
and
Find the values of
and
(a) immediately after switch
is closed, (b) a long time after
is closed, (c) immediately after
is reopened, and (d) a long time after
is reopened.
60. For the circuit shown below, find the current through the inductor
after the switch is reopened.
61. Show that for the circuit shown below, the initial energy stored in the inductor,
is equal to the total energy eventually dissipated in the resistor,
11.5 Oscillations in an LC Circuit
62. A
capacitor is charged to
and then quickly connected to an
inductor. Determine (a) the maximum energy stored in the magnetic field of the inductor, (b) the peak value of the current, and (c) the frequency of oscillation of the circuit.
63. The selfinductance and capacitance of an
circuit are
and
What is the angular frequency at which the circuit oscillates?
64. What is the selfinductance of an
circuit that oscillates at
when the capacitance is
?
65. In an oscillating
circuit, the maximum charge on the capacitor is
and the maximum current through the inductor is
(a) What is the period of the oscillations? (b) How much time elapses between an instant when the capacitor is uncharged and the next instant when it is fully charged?
66. The selfinductance and capacitance of an oscillating
circuit are
and
respectively. (a) What is the frequency of the oscillations? (b) If the maximum potential difference between the plates of the capacitor is
what is the maximum current in the circuit?
67. In an oscillating
circuit, the maximum charge on the capacitor is
Determine the charge on the capacitor and the current through the inductor when energy is shared equally between the electric and magnetic fields. Express your answer in terms of
and
68. In the circuit shown below,
is opened and
is closed simultaneously. Determine (a) the frequency of the resulting oscillations, (b) the maximum charge on the capacitor, (c) the maximum current through the inductor, and (d) the electromagnetic energy of the oscillating circuit.
69. An
circuit in an AM tuner (in a car stereo) uses a coil with an inductance of
and a variable capacitor. If the natural frequency of the circuit is to be adjustable over the range
to
(the AM broadcast band), what range of capacitance is required?
11.6 RLC Series Circuits
70. In an oscillating
circuit,
and
What is the angular frequency of the oscillations?
71. In an oscillating
circuit with
and
how much time elapses before the amplitude of the oscillations drops to half its initial value?
72. What resistance
must be connected in series with a
inductor of the resulting
oscillating circuit is to decay to
of its initial value of charge in
cycles? To
of its initial value in
cycles?
Additional Problems
73. Show that the selfinductance per unit length of an infinite, straight, thin wire is infinite.
74. Two long, parallel wires carry equal currents in opposite directions. The radius of each wire is
and the distance between the centres of the wires is
Show that if the magnetic flux within the wires themselves can be ignored, the selfinductance of a length
of such a pair of wires is
(Hint: Calculate the magnetic flux through a rectangle of length
between the wires and then use
)
75. A small, rectangular single loop of wire with dimensions
and
is placed, as shown below, in the plane of a much larger, rectangular single loop of wire. The two short sides of the larger loop are so far from the smaller loop that their magnetic fields over the smaller fields over the smaller loop can be ignored. What is the mutual inductance of the two loops?
76. Suppose that a cylindrical solenoid is wrapped around a core of iron whose magnetic susceptibility is
Using Equation 11.2.5, show that the selfinductance of the solenoid is given by
where
is its length,
its crosssectional area, and
its total number of turns.
77. The solenoid of the preceding problem is wrapped around an iron core whose magnetic susceptibility is
(a) If a current of
flows through the solenoid, what is the magnetic field in the iron core? (b) What is the effective surface current formed by the aligned atomic current loops in the iron core? (c) What is the selfinductance of the filled solenoid?
78. A rectangular toroid with inner radius
outer radius
height
and
is filled with an iron core of magnetic susceptibility
(a) What is the selfinductance of the toroid? (b) If the current through the toroid is
what is the magnetic field at the centre of the core? (c) For this same
current, what is the effective surface current formed by the aligned atomic current loops in the iron core?
79. The switch
of the circuit shown below is closed at
Determine (a) the initial current through the battery and (b) the steadystate current through the battery.
80. In an oscillating
circuit,
and
Initially, the capacitor has a charge of
and the current is zero. Calculate the charge on the capacitor (a) five cycles later and (b)
cycles later.
81. A
inductor has
of current turned off in
(a) What voltage is induced to oppose this? (b) What is unreasonable about this result? (c) Which assumption or premise is responsible?
Challenge Problems
82. A coaxial cable has an inner conductor of radius
and outer thin cylindrical shell of radius
A current
flows in the inner conductor and returns in the outer conductor. The selfinductance of the structure will depend on how the current in the inner cylinder tends to be distributed. Investigate the following two extreme cases. (a) Let current in the inner conductor be distributed only on the surface and find the selfinductance. (b) Let current in the inner cylinder be distributed uniformly over its crosssection and find the selfinductance. Compare with your results in (a).
83. In a damped oscillating circuit the energy is dissipated in the resistor. The
factor is a measure of the persistence of the oscillator against the dissipative loss. (a) Prove that for a lightly damped circuit the energy,
in the circuit decreases according to the following equation.
where
(b) Using the definition of the
factor as energy divided by the loss over the next cycle, prove that
factor of a lightly damped oscillator as defined in this problem is
(Hint: For (b), to obtain
divide
at the beginning of one cycle by the change
over the next cycle.)
84. The switch in the circuit shown below is closed at
Find currents through (a)
(b)
and (c) the battery as function of time.
85. A square loop of side
is placed
from a long wire carrying a current that varies with time at a constant rate of
as shown below. (a) Use Ampère’s law and find the magnetic field as a function of time from the current in the wire. (b) Determine the magnetic flux through the loop. (c) If the loop has a resistance of
how much induced current flows in the loop?
86. A rectangular copper ring, of mass
and resistance
is in a region of uniform magnetic field that is perpendicular to the area enclosed by the ring and horizontal to Earth’s surface. The ring is let go from rest when it is at the edge of the nonzero magnetic field region (see below). (a) Find its speed when the ring just exits the region of uniform magnetic field. (b) If it was let go at
what is the time when it exits the region of magnetic field for the following values:
and
? Assume the magnetic field of the induced current is negligible compared to
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Introduction to Electricity, Magnetism, and Circuits by Daryl Janzen is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.
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