Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 3 / 15D / c / Solution

Take the wire along the $x$-axis and let $r$ be perpendicular distance from it. Gauss's law and cylindrical symmetry giveFor an observer boosted parallel to the wire,The moving observer sees length contraction and hence line density ${\lambda }^{\prime }=\gamma \lambda$, together with current $I^{\prime }=-\gamma \lambda v$. The transformed magnetic field is exactlywith the direction prescribed by the current. This is Ampère's law for the current seen in the moving frame.