Insert $\rho=\epsilon_0\nabla\cdot E$ and $J=\mu_0^{-1}\nabla\times B-\epsilon_0E_t$ into the <Lorentz force> and use <vector> identities plus <Faraday's law>. Rearrangement gives $\partial_tg_i+\partial_j\sigma_{ij}=-f_i$. Here $g=\epsilon_0E\times B$ is field <momentum> density and $\sigma$ is the <Maxwell stress tensor>, whose surface traction transports <momentum>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 36D / a / Solution

 Solution (source code)