Retarded potential

The scalar or vector potential electromagnetic of a time varying current or charge distribution.

These are the electromagnetic retarded potentials for an arbitrary source in free space (vacuum). They satisfy the inhomogeneous wave equations for $mathit\; V$ and $mathbf\; A$ in the Lorenz gauge.

:$mathit\; t\_r\; equiv\; mathit\; t\; -\; frac,\; d\; au\text{'}$

Here, $mathbf\; r$ is location, $mathit\; t$ is time, and $mathit\; c$ is the speed of light in a vacuum. $mathit\; t\_r$ is the "retarded time"; the time at which light must be emitted from location r' in order to reach location r at time t. $ho\; (mathbf\; r,\; mathit\; t)$ is the electric charge density, and $mathbf\; J(mathbf\; r,\; mathit\; t)$ is the current density. $epsilon\_0$ is the dielectric constant of free space, and $mu\_0$ is the magnetic permeability of free space. $V\; (mathbf\; r\; ,mathit\; t)$ is the electrical potential, and $mathbf\; A(mathbf\; r\; ,mathit\; t)$ is the vector potential.

The advanced potentials are:

:$mathit\; V\_a\; (mathbf\; r\; ,mathit\; t)\; =\; frac\{1\}\{4piepsilon\_0\}int\; frac\{\; ho\; (mathbf\; r\text{'}\; ,\; mathit\; t\_a)\}\{mathit\; c\}$The subscript "a" stands for advanced, and $mathit\; t\_a$ is the "advanced time".