= Displacement current
{title2=$\mathbf J_D=\epsilon_0\partial_t\mathbf E$}
{wiki}
In vacuum, <displacement current> is the term $\mathbf J_D=\epsilon_0\partial_t\mathbf E$ in the <Ampère-Maxwell equation>. It has the units of <current density>, but need not describe <electric charge> crossing the region. In <Maxwell equations in matter>, the corresponding term is $\partial_t\mathbf D$, with $\mathbf D$ the <electric displacement field>.
Consider a slowly charging <parallel-plate capacitor> with vacuum gap, plate area $A$, and charge $Q(t)$ on its positive plate. Neglect fringing and assume the charging timescale is long compared with the light-travel time across the apparatus. The nearly uniform gap <electric field> has magnitude $E=Q/(\epsilon_0A)$, so the <displacement current> through the gap is
$$
I_D=\epsilon_0 A\frac{dE}{dt}=\frac{dQ}{dt}=I.
$$
For $A=0.010\,\mathrm{m^2}$ and charging <electric current> $I=1.0\,\mathrm{mA}$, this gives $dE/dt\simeq1.13\times10^{10}\,\mathrm{V\,m^{-1}\,s^{-1}}$ and $J_D=0.10\,\mathrm{A/m^2}$. No conducting <electric current> crosses the gap. A spanning surface cutting the wire and another passing between the plates nevertheless give the same magnetic circulation when the full <Ampère-Maxwell equation> is used. https://www.feynmanlectures.caltech.edu/II_18.html#Ch18-S2[Feynman lectures, II.18.2: charging a capacitor]{ref}.
The local consistency check is short. Without <displacement current>, taking the <divergence> of $\nabla\times\mathbf B=\mu_0\mathbf J$ would force $\nabla\cdot\mathbf J=0$, allowing no local accumulation of <electric charge>. With <displacement current>, the <divergence of a curl is zero> and <Gauss's law> give
$$
0=\mu_0\left(\nabla\cdot\mathbf J+\epsilon_0\partial_t\nabla\cdot\mathbf E\right)
=\mu_0\left(\nabla\cdot\mathbf J+\partial_t\rho\right).
$$
This is precisely the <charge continuity equation>. <Electric charge conservation> fixes the required <divergence> of the added term; it does not by itself exclude other divergence-free additions. The specific <Maxwell equations> also yield the <electromagnetic wave equation>, linking this consistency correction to propagation at the <speed of light>.
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