The scalar and vector potentials in the Coulomb gauge are where (a) The scalar potential C

The scalar and vector potentials in the Coulomb gauge are

where

(a) The scalar potential ϕ_C(r, t) is not retarded because it depends on ρ(r' , t). The vector potential A_C(r, t) looks retarded, because it depends on j_⊥ (r , t − |r − r '|/c), but it is not. Use the properties of the transverse current density, j_⊥, to explain why at least some contributions to AC are not retarded.
(b) Show by direct calculation that A_C(r, t) nevertheless produces a properly retarded magnetic field, B(r, t).
(c) Exploit the Amp`ere-Maxwell equation to derive a retarded expression for E(r, t).

Transcribed Image Text:

Oc(r, t) = = 1 160 fd³r. 4π €0 d³p P(r', t) |rr| and Ac(r, t) = Ho 4π ‚ j₁ (r, t — \r— r'\/c) = r-r d³r'