Sinusoidal electric field and a sphere

[Ido]

A simple question in Electrodynamics is finding the scattering field of a dielectric sphere in constant electric field.
I'm interested in a simple generalization of the above question:
dielectric sphere in sinusoidal electric field.
This problem is much more usefull and even realistic.

And a small matter of intuition:
Can I replace it with a dipole?(like we're doing in the above question)

 

[Claude Bile]

Just off the top of my head, I think the size of the particle relative to the wavelength is of importance, since it determines which scattering 'regime' (for want of a better word) you are in (Rayleigh, Mie etc.).

The theory of Rayleigh scattering approximates scattering points as dipoles, however the theory of Mie scattering (where the particles are larger than those considered for Rayleigh scattering) solve Maxwell's equations with the particular boundary conditions specified by the particle.

Claude.

 

[reilly]

Ido -- Your problem of the scattering of electromagnetic waves from a dielectric sphere is pretty much a standard problem in advanced E&M courses. In class and in most texts the case of scattering from a conducting sphere is worked out. The dielectric sphere case is often assigned as a homework problem. You can find a detailed approach to the conducting sphere in Jackson, and, I'm sure, in other books as well. The homework problem is to take the conducting sphere computations as a guide for the dielectric case.

There are two major issues: finding a spherical coordinate expression for a plane wave -- the so-called Weyl expansion -- which involves bessel functions and spherical harmonics. Then the interior fields and exterior fields must match and obey the boundary conditions at the sphere's surface. This involves quite a lot of tedious calculations.

In the limit of long wavelengths, the dielectric sphere behaves like a dipole, an electric one and a magnetic one. Roughly speaking, a long wavelength radiation field looks like a constant field -- takes a while to propagate through the sphere, and sort of, the problem becomes briefly like the constant field problem. This type of physical reasoning helps guide the more technical stuff, and indicates possible approximations.
The general solution involves all the multipoles.

Good question.
Regards,
Reilly Atkinson