Classical Electrodynamics · Problem 10.8

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · scattering-good-conductor-sphere-multipole

Problem

Consider the scattering of a plane wave by a nonpermeable sphere of radius

a

and very good, but not perfect, conductivity following the spherical multipole field approach of Section 10.4. Assume that

ka \ll 1

and that the skin depth

\delta < a

. (a) Show from the analysis of Section 8.1 that

\frac{Z_s}{Z_0} = \frac{k\delta}{2}(1-i)

(b) In the long-wavelength limit, show that for

l = 1

the coefficients

\alpha_\pm(l)

and

\beta_\pm(l)

in (10.65) are

\alpha_\pm(1) \approx -\frac{2i}{3}(ka)^3 \frac{\left(1 - \dfrac{\delta}{a}\right) - i\dfrac{\delta}{a}}{\left(1 + \dfrac{\delta}{2a}\right) + i\dfrac{\delta}{2a}}

\beta_\pm(1) \approx \frac{4i}{3}(ka)^3

\delta/a

and lowest order in

ka

. (d) Using (10.61), evaluate the absorption cross section. Show that to first order in

\delta

it is

\sigma_{\text{abs}} \approx 3\pi(k\delta)a^2

. How different is the value if

\delta = a