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	$M=\frac{N_2\Phi_{21}}{I_1}=\frac{N_1\Phi_{12}}{I_2}$
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:

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:

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 $L$ 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

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

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

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 $RL$ circuit—is connected to a voltage source, the time variation of the current is

(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

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 $LC$ circuit occurs at an angular frequency $q(t)=q_0\cos(\omega t+\phi),$
The charge and current in the circuit are given by

11.6 RLC Series Circuits

The underdamped solution for the capacitor charge in an $RLC$ circuit is

The angular frequency given in the underdamped solution for the

circuit is

Answers to Check Your Understanding

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.

11.6

11.8 a.

; b.

; c.

11.10 a.

; b.

; c.

11.11 a. overdamped; b.