Classical Electrodynamics · Problem 10.6

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · scattering-amplitude-optical-theorem-multipole

Problem

(a) Show that for the scattered wave (10.57) the normalized scattering amplitude (10.138) is

\mathbf{f} = \frac{1}{ik}\sqrt{\frac{\pi}{2}}\sum_l \sqrt{2l+1}\left[\alpha_\pm(l)\mathbf{X}_{l,\pm 1} \pm i\beta_\pm(l)\mathbf{n} \times \mathbf{X}_{l,\pm 1}\right]

where the polarization vector of the incident wave is

(\boldsymbol{\epsilon}_1 \pm i\boldsymbol{\epsilon}_2)/\sqrt{2}

. (b) Deduce an expression for the total cross section of

\sigma_t

from the optical theorem (10.139) and the above expression for

\mathbf{f}