= Solution
In vacuum, \b[<displacement current> contributes the additional term] $\epsilon_0\partial\mathbf E/\partial t$:
$$
\nabla\times\mathbf B=\mu_0\mathbf J+\mu_0\epsilon_0\frac{\partial\mathbf E}{\partial t}.
$$
Taking the divergence and using <Gauss law> gives $\nabla\cdot\mathbf J+\partial\rho/\partial t=0$, the local conservation of charge. Without <displacement current> the curl equation would incorrectly require a divergence-free current even during charge accumulation.
For a charging <capacitor>, a surface cutting the wire carries conduction current while a surface passing through the gap carries changing <electric flux>. The <displacement current> term makes the circulation independent of the spanning surface. It is not transport of free charge across the vacuum gap. In matter the macroscopic equation is $\nabla\times\mathbf H=\mathbf J_f+\partial\mathbf D/\partial t$, with <electric polarization> included in $\mathbf D$. Together with <Faraday's law of induction>, the vacuum equations give <electromagnetic waves> of <speed> $1/\sqrt{\mu_0\epsilon_0}$.
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