### Radiation produced by moving charges

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
E
3 = -(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

where

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).