Mutual Inductance (M)

Now consider two separate coils placed near each other, coil 1 and coil 2. A steady current $I_1$ flowing in coil 1 sets up a magnetic field $\vec{B}_1(x,y,z)$ throughout the surrounding region, including at coil 2; the resulting flux linked with coil 2's own area $s_2$ is $\Phi_{21}=\int_{s_2}\vec{B}_1\cdot d\vec{a}$. Since $\vec{B}_1$ (and hence $\Phi_{21}$) is proportional to $I_1$ as long as the coils' relative position is fixed, $\Phi_{21}=M_{21}I_1$, where the proportionality constant $M_{21}$ is the mutual inductance of coil 2 WITH RESPECT TO coil 1. If $I_1$ varies (slowly enough that the quasi-static field relation still holds), $\Phi_{21}$ varies in step, inducing an emf in coil 2: $e_{21}=-\frac{d\Phi_{21}}{dt}=-M_{21}\frac{dI_1}{dt}$.

By an identical argument with the roles reversed -- a current $I_2$ in coil 2 producing flux $\Phi_{12}=M_{12}I_2$ through coil 1, and hence an emf $e_{12}=-M_{12}\frac{dI_2}{dt}$ in coil 1 -- a remarkable symmetry emerges: $M_{12}=M_{21}=M$, regardless of the two coils' shapes, sizes or relative geometry. This lets M be defined equally validly as the flux linked with one circuit per unit current in the OTHER: $M=\frac{\Phi_{21}}{I_1}=\frac{\Phi_{12}}{I_2}$, or (more commonly, calling the current-carrying coil the PRIMARY and the other the SECONDARY) as the flux $\Phi_s$ linked with the secondary per unit current $I_p$ in the primary, giving the standard relation $e_s = M\frac{dI_p}{dt}$. The unit of M, like L, is the henry: a mutual inductance of 1 H means a current changing at 1 A/s in the primary induces exactly 1 V in the secondary. (An equivalent, less commonly used, definition treats M as the mutual potential energy $W=MI_1I_2$ of the two circuits per unit current flowing in each.)

The extent of coupling between the two coils is captured by the coefficient of coupling K (always $\le 1$), related to M and the two self-inductances by $M=K\sqrt{L_1L_2}$. K depends on the fraction of coil 1's flux that actually reaches coil 2: coils wound on a common iron core couple almost perfectly ($K\approx1$); air-core coils in typical use are tightly coupled if K > 0.5 and loosely coupled if K < 0.5 (K for radio-frequency coils, for instance, is typically only 0.001-0.05); and coupling is minimised (K small) by orienting the coils' axes at right angles and placing them as far apart as possible -- a trick deliberately used in appliances such as clothes dryers, where the heating coils are counter-wound specifically to cancel their combined field and keep any induced emf on the metal casing safely low, since (unlike in a transformer, where a LARGE M is exactly what's wanted) a large stray M is a hazard there. …