### Radiation produced by moving charges

In the Lorentz gauge the potentials A and Φ satisfy the inhomogeneous wave equation.
2Φ - (1/c2)∂2Φ/∂t2 = -ρ/ε0,    ∇2A - (1/c2)∂2A/∂t2 = -μ0j.
Solutions are
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 Lienard-Wiechert potentials
Φ(r,t) = [1/(4πε0)][q/((1 - βn)|r - r'|)]ret,
A(r,t) = [μ0c/(4π)][qβ/([1 - βn)|r - r'|)]ret.
Here r' is the position of the point charge at the retarded time,  n|r - r'| is the vector pointing from the position of the point charge at the retarded time 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 also 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.

E
3 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
E(r,t) = -(4πε0)-1[(q/(c2r'')]a(t - r''/c),
where
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.