Past exam of the optics and electrodynamics course of the University of Cambridge / 2012 / Paper 1 / 4 / b / Solution

A material segment of the line with rest length $d{x}^{\prime }$ has laboratory length $dx=d{x}^{\prime }/\gamma$, while its charge $dQ$ is unchanged. ThusEquivalently the line-integrated four-current transforms from $(c{\lambda }^{\prime },0)$ to $(\gamma c{\lambda }^{\prime },\gamma u{\lambda }^{\prime })$. Gauss's and Ampère's laws in $S$ now give $E_r=\lambda /(2\pi {ϵ}_{0}r)$ and $B_{\phi }={\mu }_{0}I/(2\pi r)$, exactly the fields found in part (a). Charge density is frame-dependent even though the charge of the same set of particles is invariant.