# 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  circuit Time constant for a  circuit Charge oscillation in  circuits Angular frequency in  circuits Current oscillations in  circuits Charge as a function of time in  circuit Angular frequency in  circuit

# Summary

## 11.1Mutual 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.

## 11.2Self-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.

## 11.3Energy in a Magnetic Field

• The energy stored in an inductor  is

• The self-inductance per unit length of coaxial cable is

## 11.4RL 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

## 11.5Oscillations 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

## 11.6RLC 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

## 11.1Mutual 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.

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.

## 11.2Self-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?

## 11.3Energy in a Magnetic Field

10. Show that  has units of energy.

## 11.4RL 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.

## 11.5Oscillations 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?

## 11.6RLC 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

## 11.1Mutual 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 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?

## 11.2Self-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 has a radius of (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 and whose outer radius is ?

## 11.3Energy 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  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?

## 11.4RL 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 steady-state 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 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,

## 11.5Oscillations 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?

## 11.6RLC 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  outer 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