= Solution
Boost the comoving electric field perpendicular to itself along the wire. The transverse <electric field> gains a factor $\gamma_v$, while a circulating <magnetic field> appears. Equivalently apply <Gauss's law> and <Ampère's law> with laboratory line charge $\lambda$ and current $I=\lambda v$:
$$
\boxed{\mathbf E(r)=\frac{\lambda}{2\pi\epsilon_0r}\hat{\mathbf r}
=\gamma_v\mathbf E',\qquad
\mathbf B(r)=\frac{\mu_0\lambda v}{2\pi r}\hat{\boldsymbol\varphi}
=\frac1{c^2}\mathbf v\times\mathbf E.}
$$
These are exterior fields of the ideal straight wire. For laboratory line density $\rho$, substitute $\lambda=\rho$; for proper line density $\rho$, substitute $\lambda=\gamma_v\rho$. The radial distance is transverse to the boost and unchanged between the two frames.
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