= Solution
The magnetostatic <Ampère's law>, in vacuum, is
$$
\boxed{\oint_{\partial S}\mathbf B\cdot d\boldsymbol\ell=\mu_0I_{\rm enc},\qquad\nabla\times\mathbf B=\mu_0\mathbf J}.
$$
The path direction and surface normal are related by the right-hand rule. Taking the <divergence> of the differential law gives $\nabla\cdot\mathbf J=0$, since the <divergence> of a <curl> vanishes. However, <charge conservation> requires the <continuity equation> $\nabla\cdot\mathbf J=-\partial\rho/\partial t$, which need not vanish during charging.
Using <Gauss's law>, $\nabla\cdot\mathbf E=\rho/\epsilon_0$, the combination $\mathbf J+\epsilon_0\partial\mathbf E/\partial t$ has zero <divergence>. Replacing $\mathbf J$ by this conserved total current density gives the <Ampère-Maxwell equation>:
$$
\boxed{\nabla\times\mathbf B=\mu_0\mathbf J+\mu_0\epsilon_0\frac{\partial\mathbf E}{\partial t}}.
$$
Its integral form is $\oint\mathbf B\cdot d\boldsymbol\ell=\mu_0[I_{\rm enc}+\epsilon_0\,d\Phi_E/dt]$. The extra term is the <displacement current>. In a charging parallel-plate <capacitor>, a surface cutting the wire encloses a conduction current $dQ/dt$, while a surface through the vacuum gap encloses no conduction current but has $\epsilon_0d\Phi_E/dt=dQ/dt$. This modification makes the magnetic circulation independent of which spanning surface is chosen. The vacuum <displacement current> represents a changing <electric field>, not literal transport of charge across the gap.
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