Charge conservation

Remarks

Take the divergence of the Ampère–Maxwell law $\nabla \times B = \mu_0 J + \mu_0 \varepsilon_0 \partial_t E$. The left side vanishes because the divergence of a curl is zero, and Gauss's law $\nabla \cdot E = \rho / \varepsilon_0$ turns the right side into $\mu_0 (\nabla \cdot J + \partial_t \rho)$. The integral form follows from the divergence theorem. The $C^2$ hypothesis lets the divergence pass through the time derivative and the mixed partial derivatives commute.

Physically, conservation of electric charge is an experimental law that holds independently of Maxwell's theory. The proposition shows that Maxwell's equations are consistent only with sources that conserve charge. Without the displacement term $\mu_0 \varepsilon_0 \partial_t E$, the same computation would force $\nabla \cdot J = 0$, which fails for a charging capacitor.

Example: a charge $q$ moving with constant velocity $v$ has $\rho = q \, \delta(x - vt)$ and $J = q v \, \delta(x - vt)$, and indeed $\partial_t \rho = -v \cdot \nabla \rho = -\nabla \cdot J$ in the sense of distributions.

Going deeper

For a student

The continuity equation is the local form of conservation, and it has the same shape as conservation of mass in fluid mechanics. In the covariant formulation it reads $\partial_\mu J^\mu = 0$ for the four-current $J^\mu = (c\rho, J)$, and it follows from the antisymmetry of the electromagnetic tensor. By Noether's theorem the conservation of charge also corresponds to the gauge symmetry of electromagnetism.