Retarded potential

A retarded potential evaluates each source at the event on the source's past light cone. This accounts for propagation at the speed of light and enforces causality.

For a source point $y(s)$ and field event $(x,t)$, the retarded time solves

For $R(s)=|x-y(s)|$ and $|v|<c$, the function $F(s)=s+R(s)/c$ hasIt is strictly increasing, so the retarded-time equation has at most one solution.

The potentials of a point charge $q$ arewith every source quantity evaluated at the retarded time.