Faraday's law of induction

Remarks

This is a law of physics, found by experiment, and it is one of Maxwell's equations. By Stokes' theorem the left side is the flux of $\nabla \times E$ through $S$, where $\nabla \times$ is the curl. Since $S$ is arbitrary, the law is equivalent to the differential form $\nabla \times E = - \partial B / \partial t$. The minus sign is Lenz's law: the induced current makes a field that opposes the change of flux.

For a circuit that moves or deforms, the electromotive force is the line integral of $E + v \times B$, where $v$ is the velocity of the wire. With this definition the flux rule $\mathcal{E} = -d\Phi/dt$ holds also for moving circuits, as long as the circuit is a well-defined closed curve moving smoothly.

Example: let a flat loop of area $a$ rotate with angular speed $\omega$ in a uniform field of size $B_0$. If the axis lies in the plane of the loop and is perpendicular to the field, the flux is $\Phi(t) = B_0 a \cos \omega t$. So the induced electromotive force is $B_0 a \omega \sin \omega t$, which is how an alternating current generator works.

Going deeper

For a student

In Maxwell's theory Faraday's law says that a time-varying magnetic field produces an electric field with nonzero curl. So in electrodynamics $E$ is no longer the gradient of a scalar potential, and one writes $E = -\nabla \phi - \partial_t A$ with the magnetic vector potential $A$. Combined with the Ampère–Maxwell law it gives the electromagnetic wave equation. With Gauss's law for magnetism it forms the homogeneous half of Maxwell's equations, which in the covariant formulation becomes the closedness of the field strength $2$-form.