Retarded potential and EM wave generation

Retarded potential: The scalar and vector potentials at a point at distance R from a time-varying source, calculated using the value of the source at an earlier time t − R/v, because the effect of the source travels at a finite speed v (= c in free space). The delay R/v is called the retardation.

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

E and H are then found from B = ∇×A and E = −∇V − ∂A/∂t.

Generation of EM waves by a conductor

1. An alternating source drives current along a conductor (e.g. a dipole). Charges move back and forth, so they are continuously accelerated – the basic condition for radiation: l·dI/dt = l·q·dv/dt.
2. In the first quarter cycle, charges collect at the ends and electric field lines form between the two arms; the moving charges (current) produce magnetic field circles around the wire.
3. In the next quarter cycle the current reverses and charges neutralize. The field lines already sent out lose their charges, so they close on themselves into loops and detach.
4. New lines of opposite direction form and push the loops outward. The changing E creates H and the changing H creates E (Maxwell's equations), so the disturbance travels at speed c as an electromagnetic wave.

   source         near field      far field
  (~)--||--  -->   (  ( ) )  -->   ( )  ( )  -->
   dipole         lines form      loops detach

Close to the conductor the reactive (induction) field stores energy; far away the 1/r radiation field carries power away (E ⊥ H ⊥ direction of travel, E/H = 120π Ω).