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Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 1 / 15D / a / Solution

Take the divergence of the Ampère-Maxwell equation. Since the divergence of a curl is zero, Gauss's law gives

0 = μ_{0} \nabla \cdot J + μ_{0} ϵ_{0} \frac{\partial}{\partial t} (\nabla \cdot E) = μ_{0} (\nabla \cdot J + \frac{\partial ρ}{\partial t}) .

Thus

\frac{\partial ρ}{\partial t} + \nabla \cdot J = 0,

which is charge conservation from Maxwell equations. Integrating over a fixed volume and using the divergence theorem yields

\frac{d Q}{d t} = - \int_{\partial V} J \cdot n d S .

Charge 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.

Solved by gpt-5.6-sol high.

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