Assume an observer is located at the origin. The electric field produced by a point charge q which moves in an arbitrary way at the location of the observer is
E(t) = -(q/(4πε0))[(r''/r''3) + (r''/c)(d(r''/r''3)/dt) + (1/c2)(d2(r''/r'')dt2)].
Here r'' is the position of the charge at the retarded time (t -r''/c); r'' points from the observer to the charge. [Note r''/r'' is the unit vector.}
E = E1 + E2 + E3.
E1 = Ec(t - r''/c) = retarded Coulomb field. E2 = (r''/c)(dE1/dt).
E(t) = E1(t - r''/c) + (r''/c)(dE1(t - r''/c)/dt) + ...
The retardation is removed to first order. For the near field it is a better approximation to use the instantaneous Coulomb field than to use the retarded Coulomb field.
E3 is the radiation field. For a point
charge moving non-relativistically we have
E3 = -(q/(4πε0c2r''))a⊥(t-r''/c).
If the observer is not located at the origin but at position r then the radiation field E(r,t) of a point charge moving non-relativistically is
i.e. the vector from the charge to the observer at the retarded time t -|r - r'|/c, and r' is the position of the charge at the retarded time.
For the radiation field we have
(SI units), (Gaussian units).
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 e2 = q2/(4πε0) in SI units. This is the Larmor formula.
The radiation field of an oscillating electric dipole with p = p0cos(ωt) is
ER(r,t) = -(1/(4πε0c2r''))(d2p⊥(t - r''/c)/dt2),
and the average total power radiated is <P> = ω4p02/(12πε0c3).
The potentials of a point charge moving in an arbitrary way are
|(SI units)||(Gaussian Units)|
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.