displacement current
Ampere's original law said a magnetic field circulates around a current. But Maxwell spotted that it was quietly broken — it failed whenever the current was not continuous, as at the gap of a charging capacitor. His fix, the displacement current, is one of the most consequential corrections in the history of physics: it completed the equations and let light exist.
The displacement current is not a current of moving charge at all, but the term J_d = epsilon_0 dE/dt that Maxwell added to the right-hand side of Ampere's law, turning it into the Ampere-Maxwell law curl B = mu_0 J + mu_0 epsilon_0 dE/dt. In words: a changing electric field sources a magnetic field exactly as a real current does. In a charging capacitor no conduction current crosses the gap, yet the electric field between the plates is rising; the displacement current epsilon_0 dE/dt through the gap exactly matches the conduction current in the wire, so the magnetic circulation is continuous whichever surface you cap the loop with.
Why it had to be there: take the divergence of the uncorrected curl B = mu_0 J and you get div J = 0, which contradicts charge conservation whenever charge piles up. Adding epsilon_0 dE/dt fixes this precisely — the divergence now reproduces the continuity equation. And because a changing E makes B while a changing B makes E, the two can sustain each other and propagate: the displacement current is exactly what makes electromagnetic waves possible.
While a capacitor charges, a current I flows in the wire but stops at the plates. Cap the Amperian loop with a flat surface pierced by the wire and you enclose I; cap it with a bulging surface passing through the gap and you enclose no wire — yet the displacement current epsilon_0 dE/dt through the gap equals exactly the same I, so curl B gives the same answer either way.
J_d = epsilon_0 dE/dt: a changing electric field acts as a current, keeping Ampere's law consistent and making EM waves possible.
The displacement current carries no charge and dissipates no heat — the name is historical, from Maxwell's mechanical model of the ether. It is not optional: without it Ampere's law violates charge conservation and electromagnetic waves cannot exist.