A charge q executes simple harmonic motion with amplitude A and angular
(a) the far field components of the electric and magnetic fields, and
(b) the rate of energy loss.
Details of the calculation:
(a) The radiation field E(r,t) of a point charge moving non-relativistically is (in SI units)
E(r,t) = -(q/(4πε0c2r''))a⊥(t - r''/c), r'' = r - r'(t - |r - r'|/c).
r' ≈ 0, r'' ≈ r for a charge oscillating about the origin.
a⊥(t - r/c) = -a(t - r/c)sinθ (θ/θ), z = A cos(ωt), a = -ω2A cos(ωt).
E(r,t) = -[q/(4πε0c2r)]ω2A cos(ω(t - r/c))sinθ(θ/θ).
B(r,t) = (r/(rc))×E(r,t) = -[q/(4πε0c3r)]ω2A cos(ω(t - r/c))sinθ (φ/φ).
(b) S(r,t) = (1/μ0)E(r,t)×B(r,t) = [q2/(4πε0)2][1/(c5r2μ0)]ω4A2 cos2(ω(t - r/c))sin2θ (r/r).
Now let us integrate over a spherical surface of radius r.
P = ∫S∙dA = (2πr2)[q2/(4π)2][1/(c3r2ε0)]ω4A2 cos2(ω(t - r/c))∫sin3θdθ
= [q2/(6πc3ε0)]ω4A2 cos2(ω(t - r/c)).
<P> = [q2ω4A2/(12πc3ε0)] = rate of energy loss.
This is also what we get from the Larmor formula,
P =∮A S∙dA = ⅔e2a2/c2, e2 = q2/(4πε0).