a retarded potential

Flip a switch on a distant transmitter and the room does not respond instantly — the news that the current changed spreads outward only at the speed of light. A retarded potential builds this delay directly into the potentials of electromagnetism. It answers the question: if the charges and currents are moving and changing, what are the potentials here and now?

The retarded scalar and vector potentials are the causal solutions of Maxwell's equations in the Lorenz gauge. In SI units, V(r,t) = (1/4 pi epsilon_0) times the integral of rho(r', t_r)/|r - r'| over the source, and A(r,t) = (mu_0/4 pi) times the integral of J(r', t_r)/|r - r'|, where the crucial ingredient is the retarded time t_r = t - |r - r'|/c. Each patch of source contributes not what it is doing now, but what it was doing earlier, by exactly the light-travel time from there to here. Formally these are the solutions of the wave equations box V = -rho/epsilon_0 and box A = -mu_0 J, constructed from the retarded Green's function of the d'Alembertian box.

The wave equation actually admits a second solution, the advanced potential, which uses t_a = t + |r - r'|/c and would let effects precede their causes; we discard it on grounds of causality. The retarded potentials are the launching point for everything that radiates — feed them a single moving point charge and you obtain the Lienard-Wiechert potentials, and from those Larmor's formula and all of antenna theory.