 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-1/37b/d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

J_{k} = \int ϵ_{k ℓ j} x_{ℓ} g_{j} d^{3} x

. Local momentum conservation and integration by parts, with the boundary term vanishing for a localized field, give

\dot{J}_{k} = - \int ϵ_{k ℓ j} x_{ℓ} \partial_{i} σ_{ij} d^{3} x = \int ϵ_{kij} σ_{ij} d^{3} x = 0,

because

σ_{ij}

is symmetric. Hence the total electromagnetic angular momentum is independent of time.

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