Retarded time and retarded potentials

Retarded time is the earlier time at which a signal must leave the source to reach the observation point at the present time t:

t' = t − R/v

where R = distance from source to point and v = speed of propagation (c in free space). The field at P now depends on what the source was doing at t'.

Retarded potentials

For static sources:

V = (1/4πε) ∫ ρ / R dv,     A = (μ/4π) ∫ J / R dv

For time-varying sources, ρ and J are taken at the retarded time:

V(r, t) = (1/4πε) ∫ [ρ(r', t − R/v)] / R  dv'
A(r, t) = (μ/4π)  ∫ [J(r', t − R/v)] / R  dv'

These are the retarded scalar potential and retarded vector potential. They satisfy the wave equations:

∇²V − με ∂²V/∂t² = −ρ/ε
∇²A − με ∂²A/∂t² = −μJ

with the Lorenz gauge ∇·A = −με ∂V/∂t.

Phasor (time-harmonic) form, with e^(jωt) and k = ω/v = 2π/λ:

V = (1/4πε) ∫ ρ e^(−jkR) / R  dv'
A = (μ/4π)  ∫ J e^(−jkR) / R  dv'

The factor e^(−jkR) is the phase delay of retardation.

For a line current I (wire antenna): A = (μ/4π) ∫ I e^(−jkR)/R dl.

Fields are then found from

B = ∇ × A
E = −∇V − ∂A/∂t   (= −∇V − jωA)

Importance: retarded potentials explain why time-varying currents produce fields that travel outward with finite speed and a 1/r part – the radiation field. All antenna fields (dipole, loop, arrays) are derived from them.