Retarded vector potential ($\mathbf{A}(\mathbf{r},t)=\frac{{\mu }_0}{4\pi }\int \frac{\mathbf{J}({\mathbf{r}}^{\prime },t-|\mathbf{r}-{\mathbf{r}}^{\prime }|/c)}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{d}^3{r}^{\prime }$)

The causal magnetic vector potential generated by a current density in the Lorenz gauge, with no additional incoming homogeneous field. Each source contribution is evaluated at its retarded time. The integral is a convolution with the retarded wave-equation Green function; its validity includes the stated boundary and source conditions.

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.