= Solution
Take the <divergence> of the <Ampère-Maxwell equation>, $\nabla\times\mathbf B=\mu_0\mathbf J+\mu_0\epsilon_0\partial_t\mathbf E$. Since the <divergence of a curl is zero> vanishes, and <Gauss's law> gives $\nabla\cdot\mathbf E=\rho/\epsilon_0$, the result is
$$
\boxed{\partial_t\rho+\nabla\cdot\mathbf J=0.}
$$
This is the <charge continuity equation>.
At an internal point of a homogeneous <Ohmic conductor>, <Ohm's law> gives $\mathbf J=\sigma\mathbf E$. Since $\sigma$ is constant, the <charge continuity equation> becomes $\partial_t\rho=-(\sigma/\epsilon_0)\rho$. Therefore
$$
\boxed{\rho(\mathbf x,t)=\rho(\mathbf x,0)e^{-t/\tau},\qquad \tau=\epsilon_0/\sigma.}
$$
This is <charge relaxation>. Here $\rho$ is the total charge appearing in the vacuum form of <Gauss's law>; in a homogeneous dielectric description of free charge the corresponding constant permittivity replaces $\epsilon_0$.
For a finite isolated body, charge travels to the surface. The bulk formula applies at internal points, not across the interface where <electrical conductivity> jumps and <surface charge density> accumulates. Integrating the <charge continuity equation> over the body, including its surface charge, gives constant total <electric charge> because no <electric current> crosses the external boundary. Bulk decay and <conservation of electric charge> are therefore compatible.
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