# Mutual Inductance

Mutual inductance.  When a changing current $i_1$ in one circuit causes a changing magnetic flux in a second circuit, an emf $\mathcal{E}_2$ is induced in the second circuit.  $\mathcal{E}_2=-M\frac{di_1}{dt}$ and $\mathcal{E}_1=-M\frac{di_2}{dt}$, $M=\frac{N_2\Phi_{B2}}{i_1}=\frac{N_1\Phi_{B1}}{i_2}$ – mutual inductance, $N_1$ – number of turns of coil of the first circuit, $\Phi_1$ – average magnetic flux through each turn of coil 1.

Self-inductance.  A changing current $i$ in any circuit causes a self-induced emf $\mathcal{E}=-L\frac{di}{dt}$$L=\frac{N\Phi_B}i$ – depends on the geometry of the circuit and the material surrounding it.

Magnetic field energy.  An inductor with inductance $L$ carrying current $I$ has energy $U$ associated with the inductor’s magnetic field: $U=\frac12LI^2$.  Magnetic energy density: $u=\frac{B^2}{2\mu}$.

R-L circuits.  In an R-L circuit the growth and decay of current are exponential with time constant $\tau=\frac LR$.

L-C circuits.  An L-C circuit undergoes electrical oscillations with an angular frequency $\omega=\sqrt{\frac1{LC}}$.

L-R-C circuits.  The frequency of damped oscillations $\omega'=\sqrt{\frac1{LC}-\frac{R^2}{4L^2}}$.