the Liénard-Wiechert potentials
/ lee-ay-NAR VEE-khairt /
Take the retarded potentials and ask the sharpest possible question: what are the potentials of a single point charge moving along some arbitrary trajectory? The answer is the Liénard-Wiechert potentials, and they carry a surprising twist — a factor with no analogue in electrostatics that ultimately makes moving charges radiate.
For a point charge q whose retarded position and velocity are read off at t_r, the potentials are V(r,t) = (1/4 pi epsilon_0) qc / (Rc - R·v) and A(r,t) = (v/c^2) V, where R points from the charge's retarded position to the field point and R = |R|. The denominator can be written R(1 - n·beta), with n the unit vector toward the field point and beta = v/c; the factor 1/(1 - n·beta) is a geometric 'beaming' or bunching factor, largest when the charge moves toward you. Differentiating these potentials gives the exact fields of a moving charge, which split cleanly into a velocity field falling as 1/R^2 (the generalized, boosted Coulomb field carried along with the charge) and an acceleration field falling only as 1/R (the radiation field).
That 1/R acceleration term is the whole origin of radiation: only a term dying as slowly as 1/R lets energy — which flows as field-squared, about 1/R^2 — survive out to infinity. Set the acceleration to zero and the radiation field vanishes, leaving just the boosted Coulomb field of a uniformly moving charge. The beaming factor, raised to high powers for a fast charge, is what concentrates synchrotron radiation into a narrow forward cone.
For a slowly moving charge (v << c) the bunching factor 1 - n·beta is nearly 1, and the Liénard-Wiechert scalar potential reduces to the familiar V = q/(4 pi epsilon_0 R) evaluated at the retarded time — ordinary Coulomb, just delayed.
A point charge's potential = Coulomb's law, retarded, times the beaming factor 1/(1 - n·beta).
The fields depend on the charge's retarded position, velocity AND acceleration; only the acceleration part radiates. A common error is to drop the (1 - n·beta) factor — it is precisely what beams a relativistic charge's radiation forward and cannot be neglected when v approaches c.