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Taking the divergence of the Ampère-Maxwell equation and using the Gauss's law $\nabla \cdot E=\rho /{\epsilon }_0$ giveswhich is charge conservation from Maxwell equations ${\rho }_t+\nabla \cdot J=0$.

By Ohm's law, $J=\sigma E$, so $\nabla \cdot J=\sigma \rho /{\epsilon }_0$. Thereforewith charge-relaxation rate $\sigma /{\epsilon }_0$ and time ${\epsilon }_0/\sigma$.

Take the divergence of the Ampère-Maxwell equation. Since the divergence of a curl is zero, Gauss's law givesThuswhich is charge conservation from Maxwell equations. Integrating over a fixed volume and using the divergence theorem yieldsCharge in $V$ is conserved provided no current crosses $\partial V$. For total charge in all space, the corresponding assumption is sufficient decay of $J$ so that the flux at infinity vanishes.

For a localized charge and current distribution satisfying charge conservation from Maxwell equations,