= Solution
The <magnetic vector potential> is defined by $\mathbf B=\nabla\times\mathbf A$. The <gauge transformation>
$$
\boxed{\mathbf A\mapsto\mathbf A+\nabla\chi,\qquad\phi\mapsto\phi-\partial_t\chi}
$$
leaves both $\mathbf B$ and $\mathbf E=-\nabla\phi-\partial_t\mathbf A$ unchanged. In <magnetostatics> choose the <Coulomb gauge> $\nabla\cdot\mathbf A=0$. Then <Ampère's law> and the curl-of-curl identity give
$$
\mu_0\mathbf J=\nabla\times(\nabla\times\mathbf A)=\nabla(\nabla\cdot\mathbf A)-\nabla^2\mathbf A,\qquad\boxed{\nabla^2\mathbf A=-\mu_0\mathbf J.}
$$
The free-space <Green function> identity $\nabla^2|\mathbf r-\mathbf r'|^{-1}=-4\pi\delta^{(3)}(\mathbf r-\mathbf r')$ therefore gives
$$
\boxed{\mathbf A(\mathbf r)=\frac{\mu_0}{4\pi}\int\frac{\mathbf J(\mathbf r')}{|\mathbf r-\mathbf r'|}\,d^3r'.}
$$
The <boundary condition> selects the homogeneous solution; the usual decay condition assumes suitably localized sources. An infinitely long static wire needs a reference for its logarithmically divergent potential, although its <magnetic field> is finite away from the wire.
For time-dependent sources use the \b[<Lorenz gauge>], $\nabla\cdot\mathbf A+c^{-2}\partial_t\phi=0$, and the outgoing <retarded potential>
$$
\boxed{\mathbf A(\mathbf r,t)=\frac{\mu_0}{4\pi}\int\frac{\mathbf J(\mathbf r',t-|\mathbf r-\mathbf r'|/c)}{|\mathbf r-\mathbf r'|}\,d^3r'.}
$$
The <scalar potential> has the analogous retarded charge-density integral. The two gauges should not be conflated for a general time-dependent source.<Retarded Potentials by Richard Fitzpatrick>[Equations (73)–(76): Lorenz-gauge retarded scalar and vector potentials]{ref}
For the prescribed uncharged wire, $\rho=0$ and $\phi=0$ is consistent. Translation invariance makes $\mathbf A=A_z(r,t)\hat{\mathbf z}$ with $\nabla\cdot\mathbf A=0$, so this particular solution satisfies both gauge conditions. With $s=z'-z$,
$$
A_z=\frac{\mu_0 I}{4\pi}\int_{-\infty}^{\infty}\frac{\Theta(t-\sqrt{r^2+s^2}/c)}{\sqrt{r^2+s^2}}\,ds.
$$
It is zero for $t<r/c$. For $t>r/c$, set $L=\sqrt{c^2t^2-r^2}$. The causal interval is $-L<s<L$, giving the <switched infinite-wire retarded potential>
$$
\boxed{A_z=\frac{\mu_0I}{2\pi}\operatorname{arsinh}\frac Lr=\frac{\mu_0I}{2\pi}\operatorname{arcosh}\frac{ct}{r}=\frac{\mu_0I}{2\pi}\ln\!\left(\frac{ct+\sqrt{c^2t^2-r^2}}r\right).}
$$
Differentiation gives the fields and <Poynting vector>, for $r>0$, $t>r/c$:
$$
\boxed{\begin{aligned}
\mathbf E&=-\frac{\mu_0Ic}{2\pi\sqrt{c^2t^2-r^2}}\hat{\mathbf z},\\
\mathbf H&=\frac{Ict}{2\pi r\sqrt{c^2t^2-r^2}}\hat{\boldsymbol\phi},\\
\mathbf N=\mathbf E\times\mathbf H&=\frac{\mu_0I^2c^2t}{4\pi^2r(c^2t^2-r^2)}\hat{\mathbf r}.
\end{aligned}}
$$
Here $\mathbf B=\mu_0\mathbf H$ and $B_\phi=-\partial_rA_z$. All three fields vanish before arrival. For positive $I$, $\mathbf E$ opposes the imposed current and the <radial energy flow after switching a line current> points \b[radially outward]: $(-\hat{\mathbf z})\times\hat{\boldsymbol\phi}=\hat{\mathbf r}$.
\Image[/past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2016/paper-1-switched-wire.png]
{title=At fixed radius the causal fields start at t equals r over c and relax toward the magnetostatic limit}
{source=paper-1-switched-wire.py}
For the graph, let $u=ct/r$. Normalized to $E_*=\mu_0Ic/(2\pi r)$, $H_*=I/(2\pi r)$ and $N_*=\mu_0I^2c/(4\pi^2r^2)$, the curves are
$$
E_z/E_*=-\frac1{\sqrt{u^2-1}},\qquad H_\phi/H_*=\frac u{\sqrt{u^2-1}},\qquad N_r/N_*=\frac u{u^2-1}\quad(u>1).
$$
At the <wavefront>, $|E_z|$ and $H_\phi$ diverge as $(u-1)^{-1/2}$ and $N_r$ as $(u-1)^{-1}$. This <abrupt line-current wavefront singularity> reflects the ideal instantaneous turn-on of a line current; a finite rise time smooths the front. At late times,
$$
E_z\sim-\frac{\mu_0I}{2\pi t}\to0,\qquad H_\phi\to\frac I{2\pi r},\qquad N_r\sim\frac{\mu_0I^2}{4\pi^2rt}\to0.
$$
The vector potential grows logarithmically with time in this infinite-wire idealization even as the local <magnetic field> approaches its static limit. The divergent front should not be interpreted as a finite-energy pulse from a real instantaneous switching experiment.
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