= Solution
Model the wire as an infinite straight line along $z$. Let $\lambda_0$ be its rest-frame line charge density and $\lambda$ its laboratory line charge density. Length contraction gives $\lambda=\gamma_v\lambda_0$. If the symbol $\rho$ in the source is intended as volume charge density, replace these line densities by their integrals over the cross section; an unspecified wire radius does not determine an interior field.
In the comoving frame the wire is static, so <Gauss's law> and cylindrical symmetry give the exterior fields
$$
\boxed{\mathbf E'(r)=\frac{\lambda_0}{2\pi\epsilon_0r}\hat{\mathbf r},\qquad
\mathbf B'=\mathbf0.}
$$
If the source's $\rho$ means laboratory line density, this electric field is $\rho/(2\pi\epsilon_0\gamma_vr)$; if it means proper line density, it is $\rho/(2\pi\epsilon_0r)$. Stating the density's reference frame is necessary for an unambiguous factor of $\gamma_v$.
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