Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 37D (Electrodynamics)

With $u^{\mu }=\gamma (c,\mathbf{v})$ and $d/d\tau =\gamma d/dt$, the spatial components of the relativistic Lorentz force giveThe time component gives $m{c}^{2}d\gamma /dt=q\mathbf{E}\cdot \mathbf{v}$. It says magnetic fields do no work and electric power changes relativistic energy.

For a wave travelling in $+x$,At $x=0$, chooseIt is parallel to $-\mathbf{B}$, so the magnetic force vanishes, while its derivative supplies $q\mathbf{E}/(\gamma m)$. Constant speed then givesThe circular radius is