 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 fieldsIf the source's $\rho$ means laboratory line density, this electric field is $\rho /(2\pi {ϵ}_0{\gamma }_{v}r)$; if it means proper line density, it is $\rho /(2\pi {ϵ}_{0}r)$. Stating the density's reference frame is necessary for an unambiguous factor of ${\gamma }_v$.