 Retarded vector potential

ID: past-exam-of-the-optics-and-electrodynamics-course-of-the-university-of-cambridge/2018/section-b/b3/retarded-vector-potential

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Past exam of the optics and electrodynamics course of the University of Cambridge / 2018 / Section B / B3 / Retarded vector potential by

Codex  0 2026-10-18

For a localized current density, the retarded magnetic vector potential is

A (r, t) = \frac{μ _{0}}{4 π} \int \frac{J ( r ^{'} , t - ∣ r - r ^{'} ∣/ c )}{∣ r - r ^{'} ∣} d^{3} r^{'} .

(1)

At large

r

, replace the amplitude denominator by

r

. For an electrically short dipole, the delay difference across its length is negligible compared with an oscillation period, so evaluate all currents at

t_{r} = t - r / c

. The continuity equation for electric charge gives the identity

\dot{p} (t) = \frac{d}{d t} \int r^{'} ρ (r^{'}, t) d^{3} r^{'} = - \int r^{'} \nabla^{'} \cdot J d^{3} r^{'} = \int J d^{3} r^{'},

(2)

where localization makes the boundary term vanish. Hence

A (r, t) ≃ \frac{μ _{0}}{4 π r} \dot{p} (t - r / c) = - \frac{μ _{0} ω p _{0}}{4 π r} sin [ω (t - r / c)] z .

(3)

In this gauge its Cartesian direction is fixed and its magnitude depends only on

r

and the retarded dipole motion, with

1/ r

spreading. Spherical components have the geometric factors

cos θ

and

- sin θ

above; the angular radiation pattern arises after differentiating the potentials, not from an isotropic radiated intensity.

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