Radiation produced by moving charges

The radiation field E(r,t) of a point charge moving nonrelativistically is

where

i.e. the vector from the charge to the observer at the retarded time   .

 (SI units),        (Gaussian units).

For the near field it is a better approximation to use the instantaneous Coulomb field than to use the retarded Coulomb field.

The energy flux associated with the fields of a point charge is calculated from the Poynting vector S. The total power radiated is

in SI and Gaussian units, with in SI units. This is the Lamor formula.

The Liénard-Wiechert potentials

The potentials of a point charge moving in an arbitrary way are

Here  is the vector pointing from the point charge to the observer at r.  The potentials of a point charge depend only on the position and the velocity at the retarded time.  The fields E and B depend on the acceleration.

The equations of motion

Neglecting radiation, the equations of motion of a particle of charge q in external fields E and B can be written as

Here  .