Why does the speed of light within a solid depend on frequency? Different frequencies of light travel at different speeds through solids, which along with Snell's law allows for rainbows. Has this phenomenon of variable speeds been predicted through derivations? What does it tell us about the interactions that occur when light travels through a solid?
 A: The index of refraction is found to be  a function of the frequency in an analysis of radiation scattering in a medium: for example in the book of Panofsky and Philips  "classical electricity and magnetism" chapter 22,  radiation scattering and dispersion ,on  paragraph 7. The book seems to be available for a free download here.
Here is a link with Feynman lectures on light to to get the extended framework on light.
A: You can find this effect in Maxwell's Equations combined with basic models of atomic polarization. 
Consider an electron somewhat loose in an atom exposed to an electric field with wave length much larger than the atomic spacing. A decent equation of motion is:
$m\frac{\partial^2\vec{x}}{\partial t^2}=-k\vec{x}-b\dot{\vec{x}}-q\vec{E_0}e^{-i\omega t}$
Where the first term is a binding term roughly representing atomic attraction of the electron as simple harmonic motion, the second term is a breakign term, and the third term on the right is a time varying electric field. 
One finds $\vec{x}=\frac{\vec{E_0}e^{iwt}}{m\omega^2-iwb+k}$. 
Only take the real part in further analysis. 
Solving this equation for $\vec{x}$ yields a formula for $\vec{p}$, the dipole moment of the atom. There is a density of atoms in the solid. Taking this into account we get $\vec{P}$, the Polarization Vector. 
The solid has a Polarization Vector directly proportional to the applied Electric field. The term of proportionality is the Electric Susceptibility. N otice it is Frequency Dependent. 
The permitivitty is the susceptibility  plus 1 times the permitivity of free space. 
So we have a frequency dependent permitivity which changes Maxwell's equations. 
The resulting solution yields a wave equation with frequency dependent speeds. 
$\nabla^2\vec{E}-(\mu_0\epsilon)\frac{\partial^2\vec{E}}{\partial t^2}=0$
The coefficient of the second time derivative is the reciprocal of the speed squared. Not it is dependenton the frequency dependent permittivity. 
