Classical Electrodynamics · Problem 9.5

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · electric-dipole-potentials-fields

Problem

(a) Show that for harmonic time variation at frequency

\omega

the electric dipole scalar and vector potentials in the Lorenz gauge and the long-wavelength limit are

\begin{aligned} \Phi(\mathbf{x}) &= \frac{e^{ikr}}{4\pi\epsilon_0 r^2}\, \mathbf{n} \cdot \mathbf{p}(1 - ikr) \\ \mathbf{A}(\mathbf{x}) &= -i\frac{\mu_0 \omega}{4\pi} \frac{e^{ikr}}{r}\, \mathbf{p} \qquad [\text{this is (9.16)}] \end{aligned}

where

k = \omega/c

\mathbf{n}

is a unit vector in the radial direction,

\mathbf{p}

is the dipole moment (9.17), and the time dependence

e^{-i\omega t}

is understood. (b) Calculate the electric and magnetic fields from the potentials and show that they are given by (9.18).