Ampère's circuital law describes magnetic fields produced by steady currents, but changing electric fields also contribute to magnetic circulation. Maxwell's correction adds this effect:

$$\oint_C\mathbf B\cdot d\boldsymbol\ell =\mu_0 I_{\mathrm{enc}}+\mu_0\varepsilon_0\frac{d\Phi_E}{dt}.$$

The second term is associated with the changing electric flux through the surface bounded by $C$.

Why the correction is needed

Consider a charging capacitor. Conduction current flows in the wires, but no charge crosses the dielectric gap between the plates. Nevertheless, the electric field and electric flux in that gap change with time.

Without Maxwell's term, applying Ampère's law to different surfaces spanning the same loop would give contradictory results.

Displacement current

The quantity

$$I_D=\varepsilon_0\frac{d\Phi_E}{dt}$$

is called the displacement current. It has units of current but need not represent charges physically crossing the surface.

Coupled fields

Faraday's law says a changing magnetic field produces circulating electric field. The Ampère–Maxwell law says a changing electric field contributes to circulating magnetic field.

This symmetry between time-varying electric and magnetic fields is the key step that allows electromagnetic waves to exist even in empty space.