So far we have introduced notions of the electromagnetic (EM) wave via the Maxwell equations but I have left unanswered a key question: what happens when this EM wave interacts with matter? We should expect the EM field to have a strong effect on the sub-atomic particles, particularly the negatively-charged electrons that surround the positively charged nucleus according to the
classical model of the atom. This interaction is accounted for in the material relations of Maxwell equations that relate the electric and magnetic fields to the electric and magnetic displacements, respectively. In the simplest case of isotropic media, the electric displacement is proportional to the electric field via a scalar quantity known as the
permittivity; it is this quantity that incorporates physical properties of matter into Maxwell's theory. Let us only concern ourselves for the time being with
dielectric materials which are for the most part simply insulators and ask the question: what happens to these dielectrics in the presence of an electric field? The answer is charge separation: the applied field acts to separate the electrons from the nucleus (beyond their equilibrium positions) inducing what is commonly referred to as a dipole moment. A measure of the extent of this charge separation is a quantity termed
atomic polarizability. In order to solve for the atomic polarizability, one employs a classical electron model where individual electrons are bound elastically to the nucleus and are driven using a harmonic forcing function with known frequency, this amounts to solving a second order
ordinary differential equation which has a
Lorentzian solution. An important part of this theory is to suppose that each atom has a resonant frequency of electron oscillation about the nucleus whose calculation need not concern us at this point, the salient point to be made is that for most materials in the presence of
visible light, this resonant frequency is much larger than the light frequency and allows us to simplify our result showing that the atomic polarizability (and hence the permittivity) is independent of frequency. We deal with the case of frequency-dependent permittivity, the topic of dispersion next.