= Solution
\b[<Integral> conservation.] <Electric charge> cannot be created by ordinary transport. For a fixed <volume (thermodynamics)> $V$, outward current reduces the enclosed charge:
$$
\frac{d}{dt}\int_V\rho_e\,dV=-\oint_{\partial V}\mathbf J\cdot\widehat{\mathbf n}\,dA.
$$
Here $\rho_e$ has units $\mathrm{C\,m^{-3}}$ and $\mathbf J$ has units $\mathrm{A\,m^{-2}}$. Applying the <divergence theorem> to an arbitrary <volume (thermodynamics)> gives the local <continuity equation>
$$
\boxed{\frac{\partial\rho_e}{\partial t}+\nabla\cdot\mathbf J=0.}
$$
It is local <charge conservation>: a net outflow must be accompanied by depletion. In a steady state, $\nabla\cdot\mathbf J=0$. Integrating over a small circuit junction then gives <Kirchhoff's current law>. During a transient the junction can accumulate charge and the incoming and outgoing conduction currents need not yet match.
\b[Consistency with <Maxwell's equations>.] In <vacuum>, take the <divergence> of the <Ampère-Maxwell equation>:
$$
\nabla\times\mathbf B=\mu_0\mathbf J+\mu_0\epsilon_0\partial_t\mathbf E,
\qquad
0=\mu_0\nabla\cdot\mathbf J+\mu_0\epsilon_0\partial_t(\nabla\cdot\mathbf E).
$$
<Gauss's law>, $\nabla\cdot\mathbf E=\rho_e/\epsilon_0$, reproduces the <continuity equation>. The <displacement current> term makes these equations consistent for time-dependent charge distributions, such as a charging <capacitor>. It is not a stream of free charges across the insulating gap. In matter, $\nabla\cdot\mathbf D=\rho_f$ and $\nabla\times\mathbf H=\mathbf J_f+\partial_t\mathbf D$ give the analogous <conservation law> for <free charge>; total charge also includes <bound charge>.
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