Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 3 / 15D / Solution

With $P^{\mu }=m{U}^{\mu }$, $U^{\mu }=d{x}^{\mu }/d\tau$, and $F^{\mu \nu }$ the field tensor, separating temporal and spatial components yields the Lorentz force and power equations. The invariants are $E\cdot B$ and $E^2-c^2{B}^2$. For perpendicular fields with $E<cB$, boost with $v=E\times B/B^2$ (here $-E\hat{y}/B$); then $E\prime =0$ and $B\prime =\sqrt{B^2-E^2/c^2}\hat{z}$. Perpendicular motion is circular with $\omega =|q|B\prime /(\gamma m)$ and radius $p/(|q|B\prime )$. Transforming back adds the frame velocity, giving the nonrelativistic $E\times B/B^2$ drift, independent of $m,q$; a parallel velocity would additionally produce a helix.

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