14.1 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 ${d I}_{1} / d t$ in one circuit induces an emf $(ε_{2})$ in the second:
$ε_{2} = - M \frac{d I 1}{d t},$
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 ${d I}_{2} / d t$ through the second circuit induces an emf $(ε_{1})$ in the first:
$ε_{1} = - M \frac{d I_{2}}{d t},$
where M is the same mutual inductance as in the reverse process.

14.2 Self-Inductance and Inductors

Current changes in a device induce an emf in the device itself, called self-inductance,
$ε = − L \frac{d I}{d t},$
where L is the self-inductance of the inductor and $d I / d t$ 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 (H), where $1 H = 1 Ω \cdot s$ .
The self-inductance of a solenoid is
$L = \frac{μ_{0} N^{2} A}{l},$
where N is its number of turns in the solenoid, A is its cross-sectional area, l is its length, and $μ_{0} = 4 π \times 10^{−7} T \cdot m/A$ is the permeability of free space.
The self-inductance of a toroid is
$L = \frac{μ_{0} N^{2} h}{2 π} ln \frac{R_{2}}{R_{1}},$
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 $μ_{0} = 4 π \times 10^{−7} T \cdot m/A$ is the permeability of free space.

The energy stored in an inductor U is
$U = \frac{1}{2} L I^{2} .$
The self-inductance per unit length of coaxial cable is
$\frac{L}{l} = \frac{μ_{0}}{2 π} ln \frac{R_{2}}{R_{1}} .$

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
$I (t) = \frac{ε}{R} (1 - e^{− R t / L}) = \frac{ε}{R} (1 - e^{− t / τ_{L}})$ (turning on),
where the initial current is $I_{0} = ε / R .$
The characteristic time constant $τ$ is $τ_{L} = L / R,$ where L is the inductance and R is the resistance.
In the first time constant $τ,$ the current rises from zero to $0.632 I_{0},$ and to 0.632 of the remainder in every subsequent time interval $τ .$
When the inductor is shorted through a resistor, current decreases as
$I (t) = \frac{ε}{R} e^{− t / τ_{L}}$ (turning off).
Current falls to $0.368 I_{0}$ in the first time interval $τ$ , and to 0.368 of the remainder toward zero in each subsequent time $τ .$

The energy transferred in an oscillatory manner between the capacitor and inductor in an LC circuit occurs at an angular frequency $ω = \sqrt{\frac{1}{L C}}$ .
The charge and current in the circuit are given by
$\begin{matrix} q (t) & = & q_{0} cos (ω t + ϕ), \\ i (t) & = & − ω q_{0} sin (ω t + ϕ) . \end{matrix}$

The underdamped solution for the capacitor charge in an RLC circuit is
$q (t) = q_{0} e^{− R t / 2 L} cos (ω ′ t + ϕ) .$
The angular frequency given in the underdamped solution for the RLC circuit is
$ω^{'} = \sqrt{\frac{1}{L C} - {(\frac{R}{2 L})}^{2}} .$