Retarded potential

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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 $V$ and $\mathbf A$ in the Lorenz gauge.

$\mathit t_r \equiv \mathit t - \frac{|\mathbf r - \mathbf r'|}{\mathit c}$

$\mathit V (\mathbf r ,\mathit t) = \frac{1}{4\pi\epsilon_0}\int \frac{\rho (\mathbf r' , \mathit t_r)}{|\mathbf r - \mathbf r'|}\, d\tau'$

$\mathbf A (\mathbf r ,\mathit t) = \frac{\mu_0}{4\pi}\int \frac{\mathbf J (\mathbf r' , \mathit t_r)}{|\mathbf r - \mathbf r'|}\, d\tau'$

Here, $\mathbf r$ is location, $t$ is time, and $c$ is the speed of light in a vacuum. $t r$ is the "retarded time"; the time at which light must be emitted from location r' in order to reaches location r at time t. $\rho (\mathbf r, \mathit t)$ is the electric charge density, and $\mathbf J(\mathbf r, \mathit t)$ is the current density. $ε 0$ is the dielectric constant of free space, and $μ 0$ 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.