Simon Bridge said:
You could also consider what happens to light passing through different dielectrics - here you have E and B fields inducing each other - you can consider how the amplitudes of the waves are affected.
I'm sorry, but I have to insist that this is physically not really correct and didactically misleading. You cannot say that E and B fields are inducing each other; neither in free space, nor in a dielectric.
The electromagnetic field is propagating in free space or through dielectric (i.e., non-conducting) matter as a whole. In a given (inertial) frame of reference it consists of 6 components, none of which induce each other in a causal sense.
Let's take the free Maxwell equations as the most simple example,
\vec{\nabla} \times \vec{E}+\frac{1}{c} \partial_t \vec{B}=0, \quad \vec{\nabla} \cdot \vec{B}=0,
\vec{\nabla} \times \vec{B} - \frac{1}{c} \partial_t \vec{E}=\frac{1}{c} \vec{j}, \quad \vec{\nabla} \cdot \vec{E}=\rho.
The usual way to solve these equations is to use the first two equations (the homogeneous Maxwell equations) to introduce the four-vector potential (or in the 1+3-notation a scalar and a vector potential in the 3D sense of Euclidean configuration space) and then deriving equations of motion for these potentials from the two inhomogeneous equations.
The didactical difficulty with this approach, however, is that you have to deal with gauge invariance, and that the potentials are gauge dependent and thus the causality structure of the entire concept is pretty subtle. Only in the Lorenz gauge the potentials are entirely the retarded solutions. In other gauges, e.g., Coulomb gauge, the potential can contain instantaneous pieces etc.
Of course, the physical em. field is represented by the field-strengths components, \vec{E} and \vec{B}, and they are uniquely defined via apprpriate boundary conditions guaranteeing causality to be retarded solutions.
In classical electrodynamics we can work with the field strengths only. The idea is to separate the electric and magnetic field components, making use of Maxwell's equations. E.g., take the curl of the first inhomogeneous equation:
\vec{\nabla} \times (\vec{\nabla} \times \vec{B}) -\frac{1}{c} \frac{\partial}{\partial t} (\vec{\nabla} \times \vec{E})= \frac{1}{c} \vec{\nabla} \times \vec{j}.
We work in Cartesian coordinates from now on. Then we can write
\vec{\nabla} \times (\vec{\nabla} \times \vec{B}) =\vec{\nabla} (\vec{\nabla} \cdot \vec{B})-\Delta \vec{B}.
The first term vanishes due to the 2nd homogeneous equation. Using also the 1.st homogeneous equation, you find
\frac{1}{c^2} \partial_t^2 \vec{B}-\Delta \vec{B}=\frac{1}{c} \vec{\nabla} \times \vec{j}.
Here, it becomes immediately clear that the source of the magnetic field is the curl of the current density, and since \vec{B} is a physically observable field, it must be causally connected to this source, i.e., the only physically sensible solution is the retarded solution, i.e.,
\vec{B}(t,\vec{x})=\frac{1}{4 \pi c} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x}' \; \left ( \frac{\vec{\nabla}' \times \vec{j}(t',\vec{x}')}{|\vec{x}-\vec{x}'|} \right )_{t'=t-|\vec{x}-\vec{x}'|/c}.
In the same way one can also eliminate the magnetic field components by taking the curl of the first homogeneous Maxwell equation (Faraday's Law)
\vec{\nabla} \times (\vec{\nabla} \times \vec{E})+\frac{1}{c} \frac{\partial}{\partial t} (\vec{\nabla} \times \vec{B})=0.
Again in Cartesian coordinates we have
\vec{\nabla} (\vec{\nabla} \cdot \vec{E})-\Delta \vec{E} + \frac{1}{c} \frac{\partial}{\partial t} (\vec{\nabla} \times \vec{B})=0.
From the 2nd inhomogeneous Maxwell equation (Gauß's Law), used on the first term, and the first inhomogeneous Maxwell equation (Ampere-Maxwell Law) on the third term we get after some rearrangement of terms
\frac{1}{c^2} \partial_t^2 \vec{E}-\Delta \vec{E}=-\vec{\nabla} \rho -1/c^2 \partial_t \vec{j},
which has the retarded solution
\vec{E}(t,\vec{x})=-\frac{1}{4 \pi} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x}' \; \left (\frac{\vec{\nabla}' \rho(t',\vec{x}') + \partial_t' \vec{j}(t',\vec{x}')}{|\vec{x}-\vec{x}'|} \right )_{t'=t-|\vec{x}-\vec{x}'|/c}.
This form of the retarded solutions of the Maxwell equations is known as "Jefimenko's equations" although they have been first derived already mid of the 19th century by Ludwig Lorenz.
Of course, you can rewrite the solutions in some way such as to look as if, say the magnetic field components are somehow "induced" by the electric field components. However, as you'll immediately see, the corresponding equation is not of local causal form as the Jefimenko equations, which in turn only contain the charge and current densities as sources of the em. field. Thus the much more adequate interpretation of Maxwell's equations is that the em. field is caused (or "induced" if you wish to use this word) by the charge and current density.
Of course, the electromagnetic field also propagates in charge-current-free space or in electrically neutral media in form of waves, but also the free Maxwell equations do not admit a rewriting such as some of its components are causally and locally connected with other of its components. The phenomenon of em.-field propagation must be simply seen as a phenomenon of the electromagnetic field as a whole. It follows as solutions of the homogeneous wave equations, which you have to add anyway to the Jefimenko solutions of the inhomogeneous one to fulfill boundary conditions appropriate to your problem. The Jefimenko solutions describe the situation where em. waves are created due to charge-current distributions in some (usually finite) part of space. The boundary condition then of course is that at all times the em. field must vanish at spatial infinity, and that's precisely the case for the Jefimenko equations. That one has to use the retarded and not the advanced (or some superposition of both) is due to the causality constraint that there should no effect before the cause, i.e., at each point in space time the electromagnetic field must be expressible in terms of quantities at this time or earlier times but not at later times. The only solution then is the purely retarded one as given in the Jefimenko solutions.
You can repeat all this, of course, for the case of wave propagation in dielectrics. Only the solutions become somwhat more complicated, because the wave equation is modified due to the dielectric as it must be, because you must describe the non-trivial phenomenon of dispersion (i.e., refraction) of the electromagnetic wave in the medium. For an execellent treatment of the classical dispersion theory for usual dielectrics, see
A. Sommerfeld, Lectures on Theoretical Physics IV (Optics)
