Problems

When the current in one coil changes at a rate of 5.6 A/s, an emf of $6.3\times {10}^{\mathrm{-3}}\text{V}$ is induced in a second, nearby coil. 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 $I=I_0{e}^{\text{−}\alpha t}$, where $I_0=5.0\text{A}$ and $\alpha =2.0\times {10}^3{\text{s}}^{\mathrm{-1}},$ what is the emf induced in the second coil immediately after the current starts to decay? At $t=1.0\times {10}^{\mathrm{-3}}\text{s}?$

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

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.

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

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

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

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

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 coil with a self-inductance of 3.0 H and a resistance of $100\text{Ω}$ 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?

In Figure 14.12, $\epsilon =12\text{V}$, $L=20\text{mH}$, and $R=5.0\text{Ω}$. 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 $t=2{\tau }_L.$

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?

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.

Consider the circuit shown below. Find $I_1,I_2,\text{and}{I}_3$ 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, find the current through the inductor $2.0\times {10}^{\mathrm{-5}}\text{s}$ after the switch is reopened.

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.

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

The self-inductance and capacitance of an oscillating LC circuit are $L=20\text{mH and}C=1.0\mu \text{F},$ 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 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.

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

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?