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