Point charges, periodic motion

Problem:

A charge q executes simple harmonic motion with amplitude A and angular frequency ω.  Calculate
(a)  the far field components of the electric and magnetic fields, and
(b)  the rate of energy loss.

Solution:

• Concepts:
  The radiation field of a point charge moving non-relativistically, the Larmor formula
• Reasoning:
  Accelerating charges radiate.  At large distances the radiation fields dominate, since they decrease inversely with distance.

• 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).