### Summary

### 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 ${dI}_{1}\text{/}dt$ in one circuit induces an emf $\left({\epsilon}_{2}\right)$ in the second:

$${\epsilon}_{2}=-M\frac{dI1}{dt},$$

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 ${dI}_{2}\text{/}dt$ through the second circuit induces an emf $\left({\epsilon}_{1}\right)$ in the first:

$${\epsilon}_{1}=-M\frac{d{I}_{2}}{dt},$$

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,

$$\epsilon =\text{\u2212}L\frac{dI}{dt},$$

where*L*is the self-inductance of the inductor and $dI\text{/}dt$ 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 (H), where $1\phantom{\rule{0.2em}{0ex}}\text{H}=1\phantom{\rule{0.2em}{0ex}}\text{\Omega}\xb7\text{s}$. - The self-inductance of a solenoid is

$$L=\frac{{\mu}_{0}{N}^{2}A}{l},$$

where*N*is its number of turns in the solenoid,*A*is its cross-sectional area,*l*is its length, and ${\mu}_{0}=4\pi \phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-7}}\phantom{\rule{0.2em}{0ex}}\text{T}\xb7\text{m/A}$ is the permeability of free space. - The self-inductance of a toroid is

$$L=\frac{{\mu}_{0}{N}^{2}h}{2\pi}\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}\frac{{R}_{2}}{{R}_{1}},$$

where*N*is its number of turns in the toroid, ${R}_{1}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{R}_{2}$ are the inner and outer radii of the toroid,*h*is the height of the toroid, and ${\mu}_{0}=4\pi \phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-7}}\phantom{\rule{0.2em}{0ex}}\text{T}\xb7\text{m/A}$ is the permeability of free space.

### 14.3 Energy in a Magnetic Field

- 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

$$\frac{L}{l}=\frac{{\mu}_{0}}{2\pi}\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}\frac{{R}_{2}}{{R}_{1}}.$$

### 14.4 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

$I(t)=\frac{\text{\epsilon}}{R}(1-{e}^{\text{\u2212}Rt\text{/}L})=\frac{\text{\epsilon}}{R}(1-{e}^{\text{\u2212}t\text{/}{\tau}_{L}})$ (turning on),

where the initial current is ${I}_{0}=\epsilon \text{/}R.$ - The characteristic time constant $\tau $ is ${\tau}_{L}=L\text{/}R,$ where
*L*is the inductance and*R*is the resistance. - In the first time constant $\tau ,$ the current rises from zero to $0.632{I}_{0},$ and to 0.632 of the remainder in every subsequent time interval $\tau .$
- When the inductor is shorted through a resistor, current decreases as

$I(t)=\frac{\epsilon}{R}{e}^{\text{\u2212}t\text{/}{\tau}_{L}}$ (turning off).

Current falls to $0.368{I}_{0}$ in the first time interval $\tau $, and to 0.368 of the remainder toward zero in each subsequent time $\tau .$

### 14.5 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

$$\begin{array}{ccc}\hfill q\left(t\right)& =\hfill & {q}_{0}\phantom{\rule{0.2em}{0ex}}\text{cos}(\omega t+\varphi ),\hfill \\ \hfill i(t)& =\hfill & \text{\u2212}\omega {q}_{0}\phantom{\rule{0.2em}{0ex}}\text{sin}(\omega t+\varphi ).\hfill \end{array}$$

### 14.6 RLC Series Circuits

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

$$q(t)={q}_{0}{e}^{\text{\u2212}Rt\text{/}2L}\phantom{\rule{0.2em}{0ex}}\text{cos}(\omega \text{\u2032}t+\varphi ).$$ - The angular frequency given in the underdamped solution for the
*RLC*circuit is

$${\omega}^{\prime}=\sqrt{\frac{1}{LC}-{\left(\frac{R}{2L}\right)}^{2}}.$$