Displacement current

Remarks

With it the Ampère's circuital law becomes the Ampère–Maxwell law $\nabla \times H = J_f + J_D$ of Maxwell's equations, where $J_f$ is the free current density. Taking the divergence gives $\nabla \cdot J_f + \partial_t (\nabla \cdot D) = 0$, and Gauss's law $\nabla \cdot D = \rho_f$ turns this into the continuity equation for charge. So the displacement term is exactly what makes the law consistent with conservation of charge.

Example: a capacitor with parallel plates of area $a$ in vacuum carries charges $\pm Q(t)$. Neglecting edge effects, the field between the plates has size $Q / (\varepsilon_0 a)$, so the displacement current through a surface between the plates is $a \, \varepsilon_0 \, \frac{d}{dt} \frac{Q}{\varepsilon_0 a} = \frac{dQ}{dt}$. This equals the conduction current $I$ in the wires.

Non-example: in electrostatics and magnetostatics the fields do not depend on time, so $J_D = 0$ and Ampère's original law holds.

Despite its name, the displacement current is not a flow of charge in vacuum.