Smoothed Retarded Potentials and Fields
This note will follow the derivation of the Liénard–Wiechert potentials for a moving charge, which give the entire electromagnetic field generated by that charge including time delays, while concurrently deriving a version for a smoothed charge. The smoothed charge replaces the delta function charge density ρ ∝ δ 3 (x) by a spherically symmetric function f(|x|), so instead of a potential φ ∝ 1/ |x| for a charge at rest, the potential will be the 3D convolution of f with 1/ |x|. The smoothing will be defined in the ‘laboratory frame’ rather than the rest frame of the particle. If a spherically-symmetric distribution is required in a different rest frame, for example the averaged rest frame of a whole particle beam, the particle histories may be Lorentz boosted and the resulting fields Lorentz boosted back. Another motivation for using the lab frame is that if a beam is uniformly accelerated by an electric field, it maintains its lab frame size, rather than shrinking as a rigid object would (if the beam enters an electric field from the side, the particles maintain their time separation and the length actually increases with the beam velocity). Properly defining a charge distribution of fixed size in the changing reference frame of an accelerating particle is somewhat complicated and relates to the concept of Born rigidity.