Chapter 11 Review

Samuel J. Ling; Jeff Sanny; William Moebs; Daryl Janzen

Chapter 11 Review

Key Terms

henry (H)
unit of inductance, $1~\mathrm{H}=1~\Omega\cdot\mathrm{s}$ ; 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 $\tau$ , the characteristic time given by quantity $L/R$ of a particular series $RL$ 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	$M=\frac{N_2\Phi_{21}}{I_1}=\frac{N_1\Phi_{12}}{I_2}$
Mutual inductance in circuits	$\mathcal{E}_1=-M\frac{dI_2}{dt}$
Self-inductance in terms of magnetic flux	$N\Phi_{\mathrm{m}}=LI$
Self-inductance in terms of emf	$\mathcal{E}=-L\frac{dI}{dt}$
Self-inductance of a solenoid	$L_{\mathrm{solenoid}}=\frac{\mu_0N^2A}{l}$
Self-inductance of a toroid	$L_{\mathrm{toroid}}=\frac{\mu_0N^2h}{2\pi}\ln\frac{R_2}{R_1}$
Energy stored in an inductor	$U=\frac{1}{2}LI^2$
Current as a function of time for a $RL$ circuit	$I(t)=\frac{\mathcal{E}}{R}(1-e^{-t/\tau_L})$
Time constant for a $RL$ circuit	$\tau_L=L/R$
Charge oscillation in $LC$ circuits	$q(t)=q_0\cos(\omega t+\phi)$
Angular frequency in $LC$ circuits	$\omega=\sqrt{\frac{1}{LC}}$
Current oscillations in $LC$ circuits	$i(t)=-\omega q_0\sin(\omega t+\phi)$
Charge as a function of time in $RLC$ circuit	$q(t)=q_0e^{-Rt/2L\cos(\omega't+\phi)}$
Angular frequency in $RLC$ circuit	$\omega'=\sqrt{\frac{1}{LC}-\left(\frac{R}{2L}\right)^2}$

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 $dI_1/dt$ in one circuit induces an emf $(\mathcal{E}_2)$ in the second:
$\mathcal{E}_2=-M\frac{dI_1}{dt},$

where $M$ 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 $dI_2/dt$ through the second circuit induces an emf $(\mathcal{E}_1)$ in the first:
$\mathcal{E}_1=-M\frac{dI_2}{dt},$

where $M$ 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,
$\mathcal{E}=-L\frac{dI}{dt},$

where $L$ is the self-inductance of the inductor and $dI/dt$ 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 ( $\mathrm{H}$ ), where $1~\mathrm{H}=`~\Omega\cdot\mathrm{s}.$
The self-inductance of a solenoid is
$L=\frac{\mu_0N^2A}{l},$

where $N$ is its number of turns in the solenoid, $A$ is its cross-sectional area, $l$ is its length, and $\mu_0=4\pi\times10^{-7}~\mathrm{T}\cdot\mathrm{m/A}$ is the permeability of free space.
The self-inductance of a toroid is
$L=\frac{\mu_0N^2h}{2\pi}\ln\frac{R_2}{R_1},$

where $N$ is its number of turns in the toroid, $R_1$ and $R_2$ are the inner and outer radii of the toroid, $h$ is the height of the toroid, and $\mu_0=4\pi\times10^{-7}~\mathrm{T}\cdot\mathrm{m/A}$ is the permeability of free space.

11.3 Energy in a Magnetic Field

The energy stored in an inductor $U$ is
$U=\frac{1}{2}LI^2.$
The self-inductance per unit length of coaxial cable is
$\frac{L}{l}=\frac{\mu_0}{2\pi}\ln\frac{R_2}{R_1}.$

11.4 RL Circuits

When a series connection of a resistor and an inductor—an $RL$ circuit—is connected to a voltage source, the time variation of the current is
$I(t)=\frac{\mathcal{E}}{R}(1-e^{-Rt/L})=\frac{\mathcal{E}}{R}(1-e^{-t/\tau_L})~~~(\mathrm{turning~on})$
(turning on),where the initial current is $I_0=\mathcal{E}{R}.$
The characteristic time constant $\tau$ is $\tau_L=L/R,$ where $L$ is the inductance and $R$ is the resistance.
In the first time constant $\tau,$ the current rises from zero to $0.632I_0,$ and to $0.632$ of the remainder in every subsequent time interval $\tau.$
When the inductor is shorted through a resistor, current decreases as
$I(t)=\frac{\mathcal{E}}{R}e^{-t/\tau_L}~~~(\mathrm{turning~off}).$

Current falls to $0.368I_0$ in the first time interval $\tau,$ and to $0.368$ of the remainder toward zero in each subsequent time $\tau.$

11.5 Oscillations in an LC Circuit

The energy transferred in an oscillatory manner between the capacitor and inductor in an $LC$ circuit occurs at an angular frequency $\omega=\sqrt{\frac{1}{LC}}.$
The charge and current in the circuit are given by
$q(t)=q_0\cos(\omega t+\phi),$

$i(t)=-\omega q_0\sin(\omega t+\phi).$

11.6 RLC Series Circuits

The underdamped solution for the capacitor charge in an $RLC$ circuit is
$q(t)=q_0e^{-Rt/2L}\cos(\omega't+\phi).$
The angular frequency given in the underdamped solution for the $RLC$ circuit is
$\omega'=\sqrt{\frac{1}{LC}-\left(\frac{R}{2L}\right)^2}.$

Answers to Check Your Understanding

11.1 $4.77\times10^{-2}~\mathrm{V}$

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 $40~\mathrm{A/s}$

11.4 a. $4.5\times10^{-5}~\mathrm{H}$ ; b. $4.5\times10^{-3}~\mathrm{V}$

11.5 a. $2.4\times10^{-7}~\mathrm{Wb}$ b. $6.4\times10^{-5}~\mathrm{m}^2$

11.6 $0.50~\mathrm{J}$

11.8 a. $2.2~\mathrm{s}$ ; b. $43~\mathrm{H}$ ; c. $1.0~\mathrm{s}$

11.10 a. $2.5~\mu\mathrm{F}$ ; b. $\pi/2~\mathrm{rad}$ or $3\pi/2~\mathrm{rad}$ ; c. $1.4\times10^3~\mathrm{rad/s}$

11.11 a. overdamped; b. $0.75~\mathrm{J}$

Conceptual Questions

11.1 Mutual Inductance

1. Show that $N\Phi_{\mathrm{m}}/I$ and $\mathcal{E}/(dI/dt)$ which are both expressions for self-inductance, have the same units.

2. A $10{\text -}\mathrm{H}$ inductor carries a current of $20~\mathrm{A}.$ Describe how a $50{\text -}\mathrm{V}$ emf can be induced across it.

3. The ignition circuit of an automobile is powered by a $12{\text -}\mathrm{V}$ 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 $N\Phi_{\mathrm{m}}=LI.$

6. Would the self-inductance of a $1.0~\mathrm{m}$ 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 $LI^2/2$ has units of energy.

11.4 RL Circuits

11. Use Lenz’s law to explain why the initial current in the $RL$ circuit of Figure 11.4.1(b) is zero.

12. When the current in the $RL$ circuit of Figure 11.4.1(b) reaches its final value $\mathcal{E}/R,$ what is the voltage across the inductor? Across the resistor?

13. Does the time required for the current in an $RL$ 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 $RL$ circuit of Figure 14.12(b) a maximum?

16. In the simple $RL$ 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 $2,$ 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 $R_1$ and $R_2$ shown below vary with time after switch $\mathrm{S}$ is closed.

Figure shows a circuit with resistor R1 connected in series with battery epsilon, through open switch S. R1 is parallel to resistor R2 and inductor L.

20. Discuss possible practical applications of $RL$ 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 $LC$ 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 $RLC$ circuit. Describe what causes the capacitance in this circuit.

25. Describe what effect the resistance of the connecting wires has on an oscillating $LC$ circuit.

26. Suppose you wanted to design an $LC$ circuit with a frequency of $0.01~\mathrm{Hz}.$ What problems might you encounter?

27. A radio receiver uses an $RLC$ 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 $5.6~\mathrm{A/s},$ an emf of $6.3\times10^{-3}~\mathrm{V}$ is induced in a second, nearby coil. What is the mutual inductance of the two coils?

29. An emf of $9.7\times10^{-3}~\mathrm{V}$ is induced in a coil while the current in a nearby coil is decreasing at a rate of $2.7~\mathrm{A/s}.$ What is the mutual inductance of the two coils?

30. Two coils close to each other have a mutual inductance of $32~\mathrm{mH}.$ If the current in one coil decays according to $I=I_0e^{-\alpha t},$ where $I_0=5.0~\mathrm{A}$ and $2.0\times10^3~\mathrm{s}^{-1},$ what is the emf induced in the second coil immediately after the current starts to decay? At $t=1.0\times10^{-3}~\mathrm{s}$ ?

31. A coil of $40~\mathrm{turns}$ is wrapped around a long solenoid of cross-sectional area $7.5\times10^{-3}~\mathrm{m}^2.$ The solenoid is $0.50~\mathrm{m}$ long and has $500~\mathrm{turns}.$ (a) What is the mutual inductance of this system? (b) The outer coil is replaced by a coil of $40~\mathrm{turns}$ whose radius is three times that of the solenoid. What is the mutual inductance of this configuration?

32. A $600{\text -}\mathrm{turn}$ solenoid is $0.55~\mathrm{m}$ long and $4.2~\mathrm{cm}$ in diameter. Inside the solenoid, a small $(1.1~\mathrm{cm}\times1.4\mathrm{cm}),$ 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 $16~\mathrm{cm}$ and a cross-sectional area of $0.25~\mathrm{cm}^2$ ; it is wound uniformly with $1000~\mathrm{turns}.$ A second toroidal coil of $750~\mathrm{turns}$ 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 $N_1$ turns has length $l_1$ and radius $R_1,$ and a second smaller solenoid of $N_2$ turns has length $l_2$ and radius $R_2.$ 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 $0.40~\mathrm{V}$ is induced across a coil when the current through it changes uniformly from $0.10$ to $0.60~\mathrm{A}$ in $0.30~\mathrm{s}.$ 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.

Figure a shows current flowing through a coil from left to right. Figure b shows current flowing through a coil from right to left.

37. What is the rate at which the current though a $0.30{\text -}\mathrm{H}$ coil is changing if an emf of $0.12~\mathrm{V}$ 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 $0.100{\text -}\mathrm{A}$ current through a $2.00{\text -}\mathrm{mH}$ inductor be switched on or off to induce a $500{\text -}\mathrm{V}$ emf?

39. A coil with a self-inductance of $2.0~\mathrm{H}$ carries a current that varies with time according to $I(t)=(2.0~\mathrm{A})\sin(120\pi t).$ Find an expression for the emf induced in the coil.

40. A solenoid $50~\mathrm{cm}$ long is wound with $500~\mathrm{turns}$ of wire. The cross-sectional area of the coil is $2.0~\mathrm{cm^2}.$ What is the self-inductance of the solenoid?

41. A coil with a self-inductance of $3.0~\mathrm{H}$ carries a current that decreases at a uniform rate $dI/dt=-0.050~\mathrm{A/s}.$ What is the emf induced in the coil? Describe the polarity of the induced emf.

42. The current $I(t)$ through a $5.0{\text -}\mathrm{mH}$ inductor varies with time, as shown below. The resistance of the inductor is $5.0~\Omega.$ Calculate the voltage across the inductor at $t=2.0~\mathrm{ms},$ $t=4.0~\mathrm{ms},$ and $t=8.0~\mathrm{ms}.$

The graph of current in amperes versus time in milliseconds. The current starts from 0 at 0 milliseconds, increases with time and reaches just over 6 amperes at roughly 3 milliseconds. It decreases sharply till about 6 milliseconds, then decreases at a slightly slower rate till it reaches 0 at 12 milliseconds.

43. A long, cylindrical solenoid with $100~\mathrm{turns/cm}$ has a radius of $1.5~\mathrm{cm}.$ (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 $5.0~\mathrm{A/s},$ what is the emf induced per unit length?

44. Suppose that a rectangular toroid has $2000$ windings and a self-inductance of $0.040~\mathrm{H}.$ If $h=0.10~\mathrm{m},$ 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 $0.50~\mathrm{mm}$ and whose outer radius is $4.00~\mathrm{mm}$ ?

11.3 Energy in a Magnetic Field

46. At the instant a current of $0.20~\mathrm{A}$ is flowing through a coil of wire, the energy stored in its magnetic field is $6.0\times10^{-3}~\mathrm{J}.$ What is the self-inductance of the coil?

47. Suppose that a rectangular toroid has $2000$ windings and a self-inductance of $0.040~\mathrm{H}.$ If $h=0.10~\mathrm{m},$ what is the current flowing through a rectangular toroid when the energy in its magnetic field is $2.0\times10^{-6}~\mathrm{J}$ ?

48. Solenoid $A$ is tightly wound while solenoid $B$ 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 $10{\text -}\mathrm{H}$ inductor carries a current of $20~\mathrm{A}.$ How much ice at $0~^{\circ}\mathrm{C}$ could be melted by the energy stored in the magnetic field of the inductor? (Hint: Use the value $L_{\mathrm{f}}=334~\mathrm{J/g}$ for ice.)

50. A coil with a self-inductance of $3.0~\mathrm{H}$ and a resistance of $100~\Omega$ carries a steady current of $2.0~\mathrm{A}.$ (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 $1.2~\mathrm{A}$ is flowing in a coaxial cable whose outer radius is five times its inner radius. What is the magnetic field energy stored in a $3.0{\text -}\mathrm{m}$ length of the cable?

11.4 RL Circuits

52. In Figure 11.4.1, $\mathcal{E}=12~\mathrm{V},$ $L=20~\mathrm{mH},$ and $R=5.0~\Omega.$ 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 $t=2\tau_L,$ and (e) the voltages across the inductor and the resistor when $t=2\tau_L.$

53. For the circuit shown below, $\mathcal{E}=20~\mathrm{V},$ $L=4.0~\mathrm{mH},$ and $R=5.0~\Omega.$ After steady state is reached with $S_1$ closed and $S_2$ open, $S_2$ is closed and immediately thereafter (at $t=0$ ) $S_1$ is opened. Determine (a) the current through $L$ at $t=0,$ (b) the current through $L$ at $t=4.0\times10^{-4}~\mathrm{s},$ and (c) the voltages across $L$ and $R$ at $t=4.0\times10^{-4}~\mathrm{s}.$

Figure shows a circuit with R and L connected in series with battery epsilon through closed switch S. L is connected in parallel with another resistor R through open switch S2.

54. The current in the $RL$ circuit shown here increases to $40\%$ of its steady-state value in $2.0~\mathrm{s}.$ What is the time constant of the circuit?

55. How long after switch $S_1$ 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.

Figure shows a circuit with R and L in series with a battery, epsilon and a switch S1 which is open.

56. Examine the circuit shown below in part (a). Determine $dI/dt$ at the instant after the switch is thrown in the circuit of (a), thereby producing the circuit of (b). Show that if $I$ were to continue to increase at this initial rate, it would reach its maximum $\mathcal{E}/R$ in one time constant.

Figure a shows a circuit with R and L in series with a battery, epsilon and a switch S1 which is open. Figure b shows a circuit with R and L in series with a battery, epsilon. The end of L that is connected to the positive terminal of the battery is at positive potential. Current flows through L from the positive end to the negative one.

57. The current in the $RL$ circuit shown below reaches half its maximum value in $1.75~\mathrm{ms}$ after the switch $S_1$ is thrown. Determine (a) the time constant of the circuit and (b) the resistance of the circuit if $L=250~\mathrm{mH}.$

Figure shows a circuit with R and L in series with a battery, epsilon and a switch S1 which is open.

58. Consider the circuit shown below. Find $I_1,$ $I_2,$ and $I_3$ when (a) the switch $\mathrm{S}$ 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).

Figure shows a circuit with R1 and L connected in series with a battery epsilon and a closed switch S. R2 is connected in parallel with L. The currents through R1, L and R2 are I1, I2 and I3 respectively.

59. For the circuit shown below, $\mathcal{E}=50~\mathrm{V},$ $R_1=10~\Omega,$ and $L=2.0~\mathrm{mH}.$ Find the values of $I_1$ and $I_2$ (a) immediately after switch $\mathrm{S}$ is closed, (b) a long time after $\mathrm{S}$ is closed, (c) immediately after $\mathrm{S}$ is reopened, and (d) a long time after $\mathrm{S}$ is reopened.

Figure shows a circuit with R1 and R2 connected in series with a battery, epsilon and a closed switch S. R2 is connected in parallel with L and R3. The currents through R1 and R2 are I1 and I2 respectively.

60. For the circuit shown below, find the current through the inductor $2.0\times10^{-5}~\mathrm{s}$ after the switch is reopened.

Figure shows a circuit with R1 and R2 connected in series with a battery, epsilon and a closed switch S. R2 is connected in parallel with L and R3. The currents through R1 and R2 are I1 and I2 respectively.

61. Show that for the circuit shown below, the initial energy stored in the inductor, $LI^2(0)/2,$ is equal to the total energy eventually dissipated in the resistor, $\int_0^{\infty}I^2(t)Rdt.$

11.5 Oscillations in an LC Circuit

62. A $5000{\text -}\mathrm{pF}$ capacitor is charged to $100~\mathrm{V}$ and then quickly connected to an $80{\text -}\mathrm{mH}$ 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 $LC$ circuit are $0.20~\mathrm{mH}$ and $5.0~\mathrm{pF}.$ What is the angular frequency at which the circuit oscillates?

64. What is the self-inductance of an $LC$ circuit that oscillates at $60~\mathrm{Hz}$ when the capacitance is $10~\mu\mathrm{F}$ ?

65. In an oscillating $LC$ circuit, the maximum charge on the capacitor is $2.0\times10^{-6}\mathrm{C}$ and the maximum current through the inductor is $8.0~\mathrm{mA}.$ (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 $LC$ circuit are $L=20~\mathrm{mH}$ and $C=1.0~\mu\mathrm{F},$ respectively. (a) What is the frequency of the oscillations? (b) If the maximum potential difference between the plates of the capacitor is $50~\mathrm{V},$ what is the maximum current in the circuit?

67. In an oscillating $LC$ circuit, the maximum charge on the capacitor is $q_m.$ 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 $q_m,$ $L,$ and $C.$

68. In the circuit shown below, $S_1$ is opened and $S_2$ 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.

A 12 volt battery is connected to a 4 microfarad capacitor and a 100 millihenry inductor which are both connected in parallel with each other. There are two switches in the circuit. Switch S1 is closed. If opened, it would open the whole circuit. Switch S2 is open and hence the inductor is currently disconnected.

69. An $LC$ circuit in an AM tuner (in a car stereo) uses a coil with an inductance of $2.5~\mathrm{mH}$ and a variable capacitor. If the natural frequency of the circuit is to be adjustable over the range $540$ to $1600~\mathrm{kHz}$ (the AM broadcast band), what range of capacitance is required?

11.6 RLC Series Circuits

70. In an oscillating $RLC$ circuit, $R=5.0~\Omega,$ $L=5.0~\mathrm{mH},$ and $C=500~\mu\mathrm{F}.$ What is the angular frequency of the oscillations?

71. In an oscillating $RLC$ circuit with $L=10~\mathrm{mH},$ $C=1.5~\mu\mathrm{F},$ and $R=2.0~\Omega,$ how much time elapses before the amplitude of the oscillations drops to half its initial value?

72. What resistance $R$ must be connected in series with a $200{\text -}\mathrm{mH}$ inductor of the resulting $RLC$ oscillating circuit is to decay to $50\%$ of its initial value of charge in $50$ cycles? To $0.10\%$ of its initial value in $50$ cycles?

Additional Problems

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 $a,$ and the distance between the centres of the wires is $d.$ Show that if the magnetic flux within the wires themselves can be ignored, the self-inductance of a length $l$ of such a pair of wires is

$L=\frac{\mu_0 l}{\pi}\ln\frac{d-a}{a}.$

(Hint: Calculate the magnetic flux through a rectangle of length $l$ between the wires and then use $L=N\Phi/I.$ )

75. A small, rectangular single loop of wire with dimensions $l,$ and $a$ 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?

The figure shows a rectangular loop of wire. The length of the rectangle is l and width is a. On both sides of the rectangle are wires parallel to its length. They are a distance d away from the rectangle. Current I1 flows through both in opposites directions.

76. Suppose that a cylindrical solenoid is wrapped around a core of iron whose magnetic susceptibility is $\mathbf{x}.$ Using Equation 11.2.5, show that the self-inductance of the solenoid is given by $L=\frac{(1+x)\mu_0N^2A}{l},$ where $l$ is its length, $A$ its cross-sectional area, and $N$ its total number of turns.

77. The solenoid of the preceding problem is wrapped around an iron core whose magnetic susceptibility is $4.0\times10^3.$ (a) If a current of $2.0~\mathrm{A}$ 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 $R_1=7.0~\mathrm{cm},$ outer radius $R_2=9.0~\mathrm{cm},$ height $h=3.0~\mathrm{cm},$ and $N=3000~\mathrm{turns}$ is filled with an iron core of magnetic susceptibility $5.2\times10^3.$ (a) What is the self-inductance of the toroid? (b) If the current through the toroid is $2.0~\mathrm{A},$ what is the magnetic field at the centre of the core? (c) For this same $2.0{\text -}\mathrm{A}$ current, what is the effective surface current formed by the aligned atomic current loops in the iron core?

79. The switch $\mathrm{S}$ of the circuit shown below is closed at $t=0.$ Determine (a) the initial current through the battery and (b) the steady-state current through the battery.

A 12 volt battery is connected in series with a 5 ohm resistor, a 1 Henry inductor, a 3 ohm resistor and an open switch S. Parallel to the 3 ohm resistor is a 2 Henry inductor.

80. In an oscillating $RLC$ circuit, $R=7.0~\Omega,$ $L=10~\mathrm{mH},$ and $C=3.0~\mu\mathrm{F}.$ Initially, the capacitor has a charge of $8.0~\mu\mathrm{C}$ and the current is zero. Calculate the charge on the capacitor (a) five cycles later and (b) $50$ cycles later.

81. A $25.0{\text -}\mathrm{H}$ inductor has $100~\mathrm{A}$ of current turned off in $1.00~\mathrm{ms}.$ (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 $a,$ and outer thin cylindrical shell of radius $b.$ A current $I$ 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 $Q$ -factor is a measure of the persistence of the oscillator against the dissipative loss. (a) Prove that for a lightly damped circuit the energy, $U,$ in the circuit decreases according to the following equation.

$\frac{dU}{dt}=-2\beta U,$

where $\beta=\frac{R}{2L}.$

(b) Using the definition of the $Q$ -factor as energy divided by the loss over the next cycle, prove that $Q$ -factor of a lightly damped oscillator as defined in this problem is

$Q\equiv\frac{U_{\mathrm{begin}}}{\Delta U_{\mathrm{one~cycle}}}=\frac{1}{R}\sqrt{\frac{L}{C}}.$

(Hint: For (b), to obtain $Q,$ divide $E$ at the beginning of one cycle by the change $\Delta E$ over the next cycle.)

84. The switch in the circuit shown below is closed at $t=0~\mathrm{s}.$ Find currents through (a) $R_1,$ (b) $R_2,$ and (c) the battery as function of time.

A 12 volt battery is connected to a 6 ohm resistor and a switch S, which is open at time t=0. Connected in parallel with the 6 ohm resistor are another 6 ohm resistor and a 24 Henry inductor.

85. A square loop of side $2~\mathrm{cm}$ is placed $1~\mathrm{cm}$ from a long wire carrying a current that varies with time at a constant rate of $3~\mathrm{A/s}$ 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 $3~\Omega,$ how much induced current flows in the loop?

86. A rectangular copper ring, of mass $100~\mathrm{g}$ and resistance $0.2~\Omega,$ 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 $t=0,$ what is the time when it exits the region of magnetic field for the following values: $a=25~\mathrm{cm},$ $b=50~\mathrm{cm},$ $B=3~\mathrm{T},$ and $g=9.8~\mathrm{m/s}^2$ ? Assume the magnetic field of the induced current is negligible compared to $3~\mathrm{T}.$

Figure a shows a box with crosses in it. It is labeled t=0. An area within it is demarcated with breadth equal to a and length equal to b. Figure b shows the same box with crosses in it. It is labeled, “when ring exits”. The demarcated are from figure a is now below the box. There are two downward arrows labeled g and v.

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