Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 1 / 37B / b / Solution

Differentiate $\mathbf{g}={ϵ}_0\mathbf{E}\times \mathbf{B}$, use the vacuum Maxwell equations, and use $\nabla \cdot \mathbf{E}=\nabla \cdot \mathbf{B}=0$. Componentwise vector identities then givewhereThis is the negative of the conventional Maxwell stress tensor; with the sign used here, its divergence is the outward momentum flux.