### Problems

## 14.1 Mutual Inductance

When the current in one coil changes at a rate of 5.6 A/s, an emf of $6.3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}\phantom{\rule{0.2em}{0ex}}\text{V}$ is induced in a second, nearby coil. What is the mutual inductance of the two coils?

An emf of $9.7\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}\phantom{\rule{0.2em}{0ex}}\text{V}$ is induced in a coil while the current in a nearby coil is decreasing at a rate of 2.7 A/s. What is the mutual inductance of the two coils?

Two coils close to each other have a mutual inductance of 32 mH. If the current in one coil decays according to $I={I}_{0}{e}^{\text{\u2212}\alpha t}$, where ${I}_{0}=5.0\phantom{\rule{0.2em}{0ex}}\text{A}$ and $\alpha =2.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}},$ what is the emf induced in the second coil immediately after the current starts to decay? At $t=1.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}\phantom{\rule{0.2em}{0ex}}\text{s}?$

A coil of 40 turns is wrapped around a long solenoid of cross-sectional area $7.5\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}.$ The solenoid is 0.50 m long and has 500 turns. (a) What is the mutual inductance of this system? (b) The outer coil is replaced by a coil of 40 turns whose radius is three times that of the solenoid. What is the mutual inductance of this configuration?

A 600-turn solenoid is 0.55 m long and 4.2 cm in diameter. Inside the solenoid, a small $(1.1\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}1.4\phantom{\rule{0.2em}{0ex}}\text{cm}),$ 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?

A toroidal coil has a mean radius of 16 cm and a cross-sectional area of ${0.25\phantom{\rule{0.2em}{0ex}}\text{cm}}^{2}$; it is wound uniformly with 1000 turns. A second toroidal coil of 750 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.

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?

## 14.2 Self-Inductance and Inductors

An emf of 0.40 V is induced across a coil when the current through it changes uniformly from 0.10 to 0.60 A in 0.30 s. What is the self-inductance of the coil?

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.

What is the rate at which the current though a 0.30-H coil is changing if an emf of 0.12 V is induced across the coil?

When a camera uses a flash, a fully charged capacitor discharges through an inductor. In what time must the 0.100-A current through a 2.00-mH inductor be switched on or off to induce a 500-V emf?

A coil with a self-inductance of 2.0 H carries a current that varies with time according to $I(t)=(2.0\phantom{\rule{0.2em}{0ex}}\text{A})\text{sin}\phantom{\rule{0.2em}{0ex}}120\pi t.$ Find an expression for the emf induced in the coil.

A solenoid 50 cm long is wound with 500 turns of wire. The cross-sectional area of the coil is ${2.0\phantom{\rule{0.2em}{0ex}}\text{cm}}^{2}$ What is the self-inductance of the solenoid?

A coil with a self-inductance of 3.0 H carries a current that decreases at a uniform rate $dI\text{/}dt=\mathrm{-0.050}\phantom{\rule{0.2em}{0ex}}\text{A/s}$. What is the emf induced in the coil? Describe the polarity of the induced emf.

The current *I(t)* through a 5.0-mH inductor varies with time, as shown below. The resistance of the inductor is $5.0\phantom{\rule{0.2em}{0ex}}\text{\Omega}.$ Calculate the voltage across the inductor at $t=2.0\phantom{\rule{0.2em}{0ex}}\text{ms},t=4.0\phantom{\rule{0.2em}{0ex}}\text{ms},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}t=8.0\phantom{\rule{0.2em}{0ex}}\text{ms}$.

A long, cylindrical solenoid with 100 turns per centimeter has a radius of 1.5 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 A/s, what is the emf induced per unit length?

Suppose that a rectangular toroid has 2000 windings and a self-inductance of 0.040 H. If $h=0.10\phantom{\rule{0.2em}{0ex}}\text{m}$, what is the ratio of its outer radius to its inner radius?

What is the self-inductance per meter of a coaxial cable whose inner radius is 0.50 mm and whose outer radius is 4.00 mm?

## 14.3 Energy in a Magnetic Field

At the instant a current of 0.20 A is flowing through a coil of wire, the energy stored in its magnetic field is $6.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}\phantom{\rule{0.2em}{0ex}}\text{J}.$ What is the self-inductance of the coil?

Suppose that a rectangular toroid has 2000 windings and a self-inductance of 0.040 H. If $h=0.10\phantom{\rule{0.2em}{0ex}}\text{m}$, what is the current flowing through a rectangular toroid when the energy in its magnetic field is $2.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-6}}\phantom{\rule{0.2em}{0ex}}\text{J}?$

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.

A 10-H inductor carries a current of 20 A. How much ice at $0\text{\xb0}\phantom{\rule{0.2em}{0ex}}\text{C}$ could be melted by the energy stored in the magnetic field of the inductor? (*Hint*: Use the value ${L}_{\text{f}}=334\phantom{\rule{0.2em}{0ex}}\text{J/g}$ for ice.)

A coil with a self-inductance of 3.0 H and a resistance of $100\phantom{\rule{0.2em}{0ex}}\text{\Omega}$ carries a steady current of 2.0 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?

A current of 1.2 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-m length of the cable?

## 14.4 RL Circuits

In Figure 14.12, $\epsilon =12\phantom{\rule{0.2em}{0ex}}\text{V}$, $L=20\phantom{\rule{0.2em}{0ex}}\text{mH}$, and $R=5.0\phantom{\rule{0.2em}{0ex}}\text{\Omega}$. 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 $t=2{\tau}_{L}.$

For the circuit shown below, $\epsilon =20\phantom{\rule{0.2em}{0ex}}\text{V}$, $L=4.0\phantom{\rule{0.2em}{0ex}}\text{mH,}$ and $R=5.0\phantom{\rule{0.2em}{0ex}}\text{\Omega}$. After steady state is reached with ${\text{S}}_{1}$ closed and ${\text{S}}_{2}$ open, ${\text{S}}_{2}$ is closed and immediately thereafter $(\text{at}\phantom{\rule{0.2em}{0ex}}t=0)$ ${\text{S}}_{1}$ is opened. Determine (a) the current through *L* at $t=0$, (b) the current through *L* at $t=4.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}\phantom{\rule{0.2em}{0ex}}\text{s}$, and (c) the voltages across *L* and ${R}_{2}$ at $t=4.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-4}}\phantom{\rule{0.2em}{0ex}}\text{s}$. ${R}_{1}={R}_{2}=R$.

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

How long after switch ${\text{S}}_{1}$ 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.

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 $\epsilon \text{/}R$ in one time constant.

The current in the *RL* circuit shown below reaches half its maximum value in 1.75 ms after the switch ${\text{S}}_{1}$ is thrown. Determine (a) the time constant of the circuit and (b) the resistance of the circuit if $L=250\phantom{\rule{0.2em}{0ex}}\text{mH}$.

Consider the circuit shown below. Find ${I}_{1},{I}_{2},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{I}_{3}$ when (a) the switch 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).

For the circuit shown below, $\epsilon =50\phantom{\rule{0.2em}{0ex}}\text{V}$, ${R}_{1}=10\phantom{\rule{0.2em}{0ex}}\text{\Omega ,}$, ${R}_{2}={R}_{3}=\phantom{\rule{0.2em}{0ex}}\text{19.4 \Omega ,}$, and $L=2.0\phantom{\rule{0.2em}{0ex}}\text{mH}$. Find the values of ${I}_{1}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{I}_{2}$ (a) immediately after switch S is closed, (b) a long time after S is closed, (c) immediately after S is reopened, and (d) a long time after S is reopened.

For the circuit shown below, find the current through the inductor $2.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}\phantom{\rule{0.2em}{0ex}}\text{s}$ after the switch is reopened.

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

## 14.5 Oscillations in an LC Circuit

A 5000-pF capacitor is charged to 100 V and then quickly connected to an 80-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.

The self-inductance and capacitance of an *LC* circuit are 0.20 mH and 5.0 pF. What is the angular frequency at which the circuit oscillates?

What is the self-inductance of an *LC* circuit that oscillates at 60 Hz when the capacitance is $10\phantom{\rule{0.2em}{0ex}}\mu \text{F}$?

In an oscillating *LC* circuit, the maximum charge on the capacitor is $2.0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-6}}\phantom{\rule{0.2em}{0ex}}\text{C}$ and the maximum current through the inductor is 8.0 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?

The self-inductance and capacitance of an oscillating *LC* circuit are $L=20\phantom{\rule{0.2em}{0ex}}\text{mH and}\phantom{\rule{0.2em}{0ex}}C=1.0\phantom{\rule{0.2em}{0ex}}\mu \text{F},$ respectively. (a) What is the frequency of the oscillations? (b) If the maximum potential difference between the plates of the capacitor is 50 V, what is the maximum current in the circuit?

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*.

In the circuit shown below, ${\text{S}}_{1}$ is opened and ${\text{S}}_{2}$ 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.

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

## 14.6 RLC Series Circuits

In an oscillating *RLC* circuit, $R=5.0\phantom{\rule{0.2em}{0ex}}\text{\Omega},L=5.0\phantom{\rule{0.2em}{0ex}}\text{mH},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}C=500\phantom{\rule{0.2em}{0ex}}\mu \text{F}.$ What is the angular frequency of the oscillations?

In an oscillating *RLC* circuit with $L=10\phantom{\rule{0.2em}{0ex}}\text{mH},C=1.5\phantom{\rule{0.2em}{0ex}}\mu \text{F},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}R=2.0\phantom{\rule{0.2em}{0ex}}\text{\Omega},$ how much time elapses before the amplitude of the oscillations drops to half its initial value?

What resistance *R* must be connected in series with a 200-mH inductor and a $10\mu \text{F}$ capacitor of the resulting *RLC* oscillating circuit is to decay to $50\text{\%}$ of its initial value of charge in 50 cycles? To $0.10\text{\%}$ of its initial value in 50 cycles?