Classical Electrodynamics · Problem 10.1

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · scattering-conducting-sphere-long-wavelength

Problem

(a) Show that for arbitrary initial polarization, the scattering cross section of a perfectly conducting sphere of radius

a

, summed over outgoing polarizations, is given in the long-wavelength limit by

\frac{d\sigma}{d\Omega}(\boldsymbol{\epsilon}_0, \mathbf{n}_0, \mathbf{n}) = k^4 a^6 \left[\frac{5}{4} - |\boldsymbol{\epsilon}_0 \cdot \mathbf{n}|^2 - \frac{1}{4}|\mathbf{n} \cdot (\mathbf{n}_0 \times \boldsymbol{\epsilon}_0)|^2 - \mathbf{n}_0 \cdot \mathbf{n}\right]

where

\mathbf{n}_0

and

\mathbf{n}

are the directions of the incident and scattered radiations, respectively, while

\boldsymbol{\epsilon}_0

is the (perhaps complex) unit polarization vector of the incident radiation (

\boldsymbol{\epsilon}_0^* \cdot \boldsymbol{\epsilon}_0 = 1

;

\mathbf{n}_0 \cdot \boldsymbol{\epsilon}_0 = 0

). (b) If the incident radiation is linearly polarized, show that the cross section is

\frac{d\sigma}{d\Omega}(\boldsymbol{\epsilon}_0, \mathbf{n}_0, \mathbf{n}) = k^4 a^6 \left[\frac{5}{8}(1 + \cos^2\theta) - \cos\theta - \frac{3}{8}\sin^2\theta \cos 2\phi\right]

where

\mathbf{n} \cdot \mathbf{n}_0 = \cos\theta

and the azimuthal angle

\phi

is measured from the direction of the linear polarization. (c) What is the ratio of scattered intensities at

\theta = \pi/2

\phi = 0

and

\theta = \pi/2

\phi = \pi/2

? Explain physically in terms of the induced multipoles and their radiation patterns.