Classical Electrodynamics · Problem 9.7

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · instantaneous-radiated-power-dipole-quadrupole

Problem

(a) By means of Fourier superposition of different frequencies or equivalent means, show for a real electric dipole

\mathbf{p}(t)

that the instantaneous radiated power per unit solid angle at a distance

r

from the dipole in a direction

\mathbf{n}

\frac{dP(t)}{d\Omega} = \frac{Z_0}{16\pi^2 c^2}\left|\left[\mathbf{n} \times \frac{d^2\mathbf{p}}{dt^2}(t')\right] \times \mathbf{n}\right|^2

where

t' = t - r/c

is the retarded time. For a magnetic dipole

\mathbf{m}(t)

, substitute

(1/c)\ddot{\mathbf{m}} \times \mathbf{n}

for

(\mathbf{n} \times \ddot{\mathbf{p}}) \times \mathbf{n}

. (b) Show similarly for a real quadrupole tensor

Q_{\alpha\beta}(t)

given by (9.41) with a real charge density

\rho(\mathbf{x},\, t)

that the instantaneous radiated power per unit solid angle is

\frac{dP(t)}{d\Omega} = \frac{Z_0}{576\pi^2 c^4}\left|\left[\mathbf{n} \times \frac{d^3 \mathbf{Q}}{dt'^3}(\mathbf{n},\, t')\right] \times \mathbf{n}\right|^2

where

\mathbf{Q}(\mathbf{n},\, t)

is defined by (9.43).