Potentials for a Hertzian Oscillating Dipole

1 Problem Give expressions for the potentials of a Hertzian (point) oscillating dipole in various gauges. The form of Maxwell's equations for the fields E and B permit these fields to be related to potentials V and A according to, E = −∇V − 1 c ∂A ∂t , B ∇ × A, (1) in Gaussian units, where c is the speed of light in vacuum, and the potentials obeys the wave equations, ∇ 2 V + 1 c ∂ ∂t ∇ · A = −4ππ, ∇ 2 A − 1 c 2 ∂ 2 A ∂t 2 = − 4π c J + ∇ ∇ · A + 1 c ∂V ∂t , (2) in terms of source charge and current densities and J. Using the Lorenz-gauge condition [1], ∇ · A (L) = 1 c ∂V (L) ∂t , (3) the potentials obey the wave equations ∇ 2 V (L) − 1 c 2 ∂ 2 V (L) ∂t 2 = −4ππ, ∇ 2 A (L) − 1 c 2 ∂ 2 A (L) ∂t 2 = − 4π c J. (4) The solutions to these wave equations are the famous retarded potentials of Lorenz [1] and Riemann [2]. As deduced in eq. (16) of [3], a formal expression for the vector potential in the any other gauge is given in terms of the Lorenz-gauge potentials, and the scalar potential in the other gauge, as A(r, t) = A (L) + ∇χ = A (L) (r, t) + c∇ t −∞ {V (L) (r, t) − V (r, t)} dt = A (L) (r, −∞) − c t −∞ {E(r, t) + ∇V (r, t)} dt , (5)