EMF Induced in a Coupled Coil

This is Faraday's law with the changing flux supplied by a neighbour. A current $I_1$ in the first coil makes flux; some fraction of that flux threads the second coil; change $I_1$ and the flux through coil 2 changes, so a voltage appears across coil 2 whether or not it is connected to anything. All the geometry — turns, core, spacing, alignment — is bundled into the single number $M$, which leaves $\varepsilon_2 = -M\,dI_1/dt$: the induced voltage depends on how fast the primary current changes, and on nothing about its size.

A pair with $M = 200\ \text{mH}$, with the primary current swinging 4 A in 0.1 s, induces $0.2 \times 40 = 8\ \text{V}$. Now interrupt that same current in 1 µs instead, as a relay contact opening does: $dI/dt$ is $4 \times 10^6$ A/s and the induced voltage is 800 kV — which of course never appears, because the air arcs over long before. That inequality is the whole reason for flyback diodes, snubbers and contact-suppression networks.

The minus sign is Lenz's law, and it is a conservation statement rather than a bookkeeping one: the induced voltage drives a current whose own field opposes the change that created it. If it did not, the induced current would reinforce the change, which would induce more current, and you would have a machine that made energy from nothing. This page uses the magnitude form and drops the sign, so read the answer as a size and take the polarity from the winding sense and the dot convention on the schematic.

Two limits deserve stating. The $\Delta I/\Delta t$ written here is an average over the interval, and it equals the instantaneous $dI/dt$ only for a linear ramp; feed it a sine wave over half a cycle and you get the average slope, not the peak, which is $\pi/2$ times higher. And $M$ is only constant while the magnetic circuit is: drive a core into saturation and $M$ collapses, which is why the induced voltage in a real transformer stops following this equation exactly where the designer stopped wanting it to. Finally, mutual induction is not always a component — it is the mechanism of crosstalk between cables, of the noise a contactor coil injects into a signal pair beside it, and of every current transformer ever fitted.