 # 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 self-inductance, 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.

self-inductance
effect of the device inducing emf in itself

## Key Equations

 Mutual inductance by flux Mutual inductance in circuits Self-inductance in terms of magnetic flux Self-inductance in terms of emf Self-inductance of a solenoid Self-inductance 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 Self-Inductance and Inductors

• Current changes in a device induce an emf in the device itself, called self-inductance,
• where is the self-inductance 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 self-inductance and inductance is the henry ( ), where • The self-inductance of a solenoid is
• where is its number of turns in the solenoid, is its cross-sectional area, is its length, and is the permeability of free space.
• The self-inductance 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 self-inductance 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

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 left-hand 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 left-hand side. To get a positive emf on the right-hand 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 right-hand 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 self-inductance, have the same units.

2. A

inductor carries a current of

Describe how a

emf can be induced across it.

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 Self-Inductance and Inductors

5. Does self-inductance 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 self-inductance of a

long, tightly wound solenoid differ from the self-inductance per meter of an infinite, but otherwise identical, solenoid?

7. Discuss how you might determine the self-inductance per unit length of a long, straight wire.

8. The self-inductance of a coil is zero if there is no current passing through the windings. True or false?

9. How does the self-inductance 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 steady-state 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 steady-state 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?

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 cross-sectional 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

single-turn 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 cross-sectional 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 a second smaller solenoid of

turns has length

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 Self-Inductance and Inductors

35. An emf of

is induced across a coil when the current through it changes uniformly from

to

in

What is the self-inductance 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 self-inductance 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 cross-sectional area of the coil is

What is the self-inductance of the solenoid?

41. A coil with a self-inductance 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

(a) Neglecting end effects, what is the self-inductance 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 self-inductance of

If

what is the ratio of its outer radius to its inner radius?

45. What is the self-inductance per meter of a coaxial cable whose inner 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 self-inductance of the coil?

47. Suppose that a rectangular toroid has

windings and a self-inductance 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 self-inductance of

and a resistance 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

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 steady-state 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 self-inductance and capacitance of an

circuit are

and

What is the angular frequency at which the circuit oscillates?

64. What is the self-inductance 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 self-inductance 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?

73. Show that the self-inductance 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 self-inductance 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 self-inductance of the solenoid is given by

where

is its length,

its cross-sectional 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 self-inductance of the filled solenoid?

78. A rectangular toroid with inner radius

height

and

is filled with an iron core of magnetic susceptibility

(a) What is the self-inductance 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 steady-state 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 self-inductance 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 self-inductance. (b) Let current in the inner cylinder be distributed uniformly over its cross-section and find the self-inductance. 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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