Induction and Displacement Current: Faraday and Ampère-Maxwell

Faraday's law

It describes how a varying magnetic field generates an electric field. In integral form, the circulation of E along a closed contour C equals the negative time derivative of the magnetic flux Phi_B through a surface bounded by C:

closed_integral(E . dl) = - d(Phi_B)/dt

Example: a coil in an increasing magnetic field develops an induced electromotive force; the minus sign reflects Lenz's law — the induced current opposes the change that produced it.

The generalized Ampère theorem

The original Ampère's law related the circulation of B to the enclosed current Ienc:

closed_integral(B . dl) = mu0 * Ienc

Maxwell noticed an inconsistency: between the plates of a charging capacitor, no real current flows, yet a field B exists. He added the displacement current, proportional to the rate of change of the electric flux Phi_E:

closed_integral(B . dl) = mu0 * Ienc + mu0 * epsilon0 * d(Phi_E)/dt

Example: while a parallel-plate capacitor is charging, the field E between the plates increases; this term exactly replaces the missing current and ensures the continuity of the field B around the circuit.

Common pitfall

Forgetting the displacement current between the plates of a capacitor amounts to believing that B depends only on real moving charges, which violates the consistency of the charge continuity equation.