 Solution

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

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

Differentiate

g = ϵ_{0} E \times B

\nabla \cdot E = \nabla \cdot B = 0

. Componentwise vector identities then give

\partial_{t} g_{j} + \partial_{i} σ_{ij} = 0,

where

σ_{ij} = ϵ_{0} (\frac{1}{2} E^{2} δ_{ij} - E_{i} E_{j}) + \frac{1}{μ _{0}} (\frac{1}{2} B^{2} δ_{ij} - B_{i} B_{j}) .

This is the negative of the conventional Maxwell stress tensor; with the sign used here, its divergence is the outward momentum flux.

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