In the Lorentz gauge the potentials A and Φ satisfy the inhomogeneous
∇2Φ - (1/c2)∂2Φ/∂t2 = -ρ/ε0, ∇2A - (1/c2)∂2A/∂t2 = -μ0j.
Φ(r, t) = [1/(4πε0)]∫v' dV' ρ(r', tr)/|r - r'|,
A(r,t) = [μ0/(4π)]∫v' dV' j(r', tr)/|r - r'|.
ρ(r', tr) = ρ(tr(r')) is evaluated at the retarded time tr = t - |r - r'|/c.
For a point charge moving in an arbitrary this yields the
Φ(r, t) = [1/(4πε0)][q/(([1 - β∙n)|r - r'|)]ret,
A(r,t) = [μ0/(4π)][qβ/(([1 - β∙n)|r - r'|)]ret.
Here n|r - r'| 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.
We can find E(r,t) and B(r,t) using E = -∂A/∂t -∇Φ, B = ∇×A.
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
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.
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,
with e2 = q2/(4πε0). This is the Larmor formula.
The radiation field of an oscillating electric dipole with p = p0cos(ωt)
ER(r,t) = -(1/(4πε0c2r''))(d2p⊥(t - r''/c)/dt2).
An electric dipole radiates energy at a rate Prad = <(d2p/dt2)2>/(6πε0c3).
For an oscillating dipole the average total power radiated is <P> = ω4p02/(12πε0c3).
A magnetic dipole radiates energy at a rate Prad = <(d2m/dt2)2>/(6πε0c5).