Mutual Induction

When an electric current passing through one coil changes with time, an emf is induced in a NEIGHBOURING coil purely through their shared magnetic linkage -- this phenomenon is mutual induction, and the emf it produces is the mutually induced emf. Consider two coils placed near each other: sending current $i_1$ through coil 1 produces a magnetic field that is also partly linked with coil 2. If $\Phi_{21}$ is the flux linked with each turn of coil 2 (of $N_2$ turns) due to coil 1's current, the total flux linkage of coil 2, $N_2\Phi_{21}$, is proportional to $i_1$:

where the constant of proportionality $M_{21}$ is the mutual inductance (coefficient of mutual induction) of coil 2 with respect to coil 1, given by $M_{21}=N_2\Phi_{21}/i_1$. Setting $i_1=1$ A shows $M_{21}$ equals the flux linkage induced in coil 2 by unit current in coil 1.

If $i_1$ changes with time, Faraday's law gives the mutually induced emf in coil 2 as

with the negative sign again showing that this mutually induced emf opposes the change in $i_1$; setting $di_1/dt=-1$ A/s gives $M_{21}=-\varepsilon_2$, so mutual inductance also equals the opposing emf induced in coil 2 when the current through coil 1 changes at 1 A/s.

Exactly the same relations hold with the roles of the two coils exchanged: a changing current $i_2$ in coil 2 induces an emf $\varepsilon_1=-M_{12}\,di_2/dt$ in coil 1, where $M_{12}=N_1\Phi_{12}/i_2$ is coil 1's mutual inductance with respect to coil 2. In general, the mutual inductance between any pair of coils depends on their individual size and shape, the number of turns each carries, their relative orientation to one another, and the permeability of the medium between them -- but it can be PROVED (Long Answer Q11) that, for any given pair of coils, $M_{12}=M_{21}=M$, a single shared value regardless of which coil is treated as the source. …