Write down Maxwell's equations in the presence of a charge density $\rho$ and current density $\mathbf{J}$. Show that it is necessary that $\rho, \mathbf{J}$ satisfy a conservation equation.

If $\rho, \mathbf{J}$ are zero outside a fixed region $V$ show that the total charge inside $V$ is a constant and also that

$$\frac{\mathrm{d}}{\mathrm{d} t} \int_{V} \mathbf{x} \rho \mathrm{d}^{3} x=\int_{V} \mathbf{J ~}^{3} x .$$