Switched infinite-wire retarded potential

For a spatially uniform line current $I\Theta (t)$, the causal potential outside the line is $A_z=({\mu }_{0}I/2\pi )\mathrm{arcosh}(ct/r)$ for $ct>r$, and zero before arrival. The finite causal integration length makes it finite at finite time, unlike an absolute magnetostatic potential for an infinite wire.

The induced electric field opposes the increasing axial current, while the magnetic field circles it. Their Poynting vector points radially outward and transports energy into the growing magnetic-field region.

The ideal fields after instantaneous switching behave as $E_z,H_{\varphi }\propto (t-r/c{)}^{-1/2}$ near arrival, and the radial Poynting vector as $(t-r/c{)}^{-1}$. A finite switching rise time smooths the ideal front; the singular solution does not represent a finite-energy real pulse.