Classical Electrodynamics · Problem 10.5

Radiating Systems, Multipole Fields and Radiation — John David Jackson

Physics — E&M · scattering-dielectric-sphere-magnetic-dipole

Problem

The scattering by the dielectric sphere of Problem 10.4 was treated as purely electric dipole scattering. This is adequate unless it happens that the real dielectric constant ϵ/ϵ0\epsilon/\epsilon_0 is very large. In these circumstances a magnetic dipole contribution, even though higher order in kRkR, may be important. (a) Show that the changing magnetic flux of the incident wave induces an azimuthal current flow in the sphere and produces a magnetic dipole moment, m=i4πσZ0kμ0(kR)2R330Binc\mathbf{m} = \frac{i4\pi\sigma Z_0}{k\mu_0}(kR)^2\frac{R^3}{30}\mathbf{B}_{\text{inc}} (b) Show that application of the optical theorem to the coherent sum of the electric and magnetic dipole contributions leads to a total cross section, σt=12πR2(RZ0σ)[1(ϵr+2)2+(Z0σ/k)2+190(kR)2]\sigma_t = 12\pi R^2(RZ_0\sigma)\left[\frac{1}{(\epsilon_r+2)^2 + (Z_0\sigma/k)^2} + \frac{1}{90}(kR)^2\right] (Compare Landau and Lifshitz, Electrodynamics of Continuous Media, p. 323).

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