From Jackson, problem 10.3:
A solid uniform sphere of radius $R$ and conductivity $\sigma$ acts as a scatterer of a plane-wave beam of unpolarized radiation of frequency $\omega$, with $\omega R /c \ll 1$. The conductivity is large enough that the skin depth $\delta$ is small compared to $R$. (a) Justify and use a magnetostatic scalar potential to determine the magnetic field around the sphere, assuming the conductivity is infinite. (Remember that $\omega \neq 0$.)
We'd like to show $\nabla \times {\bf B} = \nabla \cdot {\bf B}=0.$
I have two questions. First, what does it mean for a plane wave to have a definite frequency and be unpolarized? For example, are there many sources out of phase, all at a given frequency, radiating with varying amplitudes and in all directions? If this is true, then is it possible for the superposition of all these waves to vary faster than the original frequency due to interference?
Assuming that the above question is resolved and that we can make the long-wavelength approximation so the magnetic field is roughly constant over the sphere, why does
$$\nabla \times {\bf B} =0\neq \frac{1}{c^2}\frac{\partial {\bf E}}{\partial t}$$ (assuming that the electric field is just like an oscillating dipole and has a term free from powers of $\omega$)?
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