 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ib/paper-2/6h/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

In SI units, the sourced Maxwell equations are

\nabla \cdot E = ρ / ϵ_{0}, \nabla \cdot B = 0, \nabla \times E = - \partial_{t} B, \nabla \times B = μ_{0} J + μ_{0} ϵ_{0} \partial_{t} E .

Take the divergence of the last equation. The divergence of a curl vanishes; using the first equation gives the necessary local conservation law

\partial_{t} ρ + \nabla \cdot J = 0.

For sources supported in a fixed region, integrate over a containing surface where the current vanishes. The divergence theorem gives

\frac{d}{d t} \int_{V} ρ d^{3} x = - \int_{\partial V} J \cdot n d S = 0.

Thus the total charge is constant. For each component of the electric dipole moment, integration by parts gives

\frac{d}{d t} \int_{V} x_{i} ρ d^{3} x = - \int_{V} x_{i} \partial_{j} J_{j} d^{3} x = - \int_{\partial V} x_{i} J \cdot n d S + \int_{V} J_{i} d^{3} x .

The surface term again vanishes, so

\frac{d}{d t} \int_{V} x ρ d^{3} x = \int_{V} J d^{3} x .

Smooth compact support suffices for these manipulations; equivalently one may integrate the charge continuity equation over all space and use the fixed support to identify the integrals with those over

V

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