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
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
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
E(r,t) = -(4πε0)-1[(q/(c2r''))a⊥(t - r''/c)
r'' = r - r'(t - |r - r'|/c),
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 B = r''/(r''c) × E (SI units), B = r''/r'' × E (Gaussian units).
The energy flux associated with the fields of a point charge is calculated
from the Poynting vector S.
The total power radiated by a point charge moving non-relativistically is
P =∮A S∙dA = ⅔e2a2/c3,
in SI and Gaussian units, with e2 = q2/(4πε0) in SI units. This is the Larmor formula.
Dipole radiation field (non-relativistic)