 Boosted uniform line charge

An infinite uniform line with proper charge per unit length ${\lambda }_0$ moving axially at speed $u$ has Lorentz-contracted density $\lambda =\gamma {\lambda }_0$ and current $I=\lambda u$. Transverse distance $r$ is unchanged by the boost. The rest electric field ${\lambda }_0\hat{\mathbf{r}}/(2\pi {ϵ}_{0}r)$ transforms into $\mathbf{E}=\lambda \hat{\mathbf{r}}/(2\pi {ϵ}_{0}r)$ and $\mathbf{B}=\mathbf{u}\times \mathbf{E}/c^2$. This equals the magnetic field ${\mu }_{0}I\hat{\mathbf{\phi }}/(2\pi r)$ from Ampère's law. The formulas concern a uniform stationary current pattern, not an accelerating finite filament's radiation.