Maxwell's Bright Idea

Bit more of a technical one today, assuming some background in multivariable calculus.

Our starting point is going to be Maxwell's equations for a magnetic field B and an electric field E in the absence of any charges:

We've got these constants mu_0 and epsilon_0 baked into the equations and these are known as vacuum permeability and vacuum permittivity respectively. I'm not entirely sure what that means but the important point is that these are features of the universe and everyone can agree on their value. (Oh look, a gun hanging on the wall).

Taking the time derivative of the first equation and commuting it across the curl gives us:

Then we can use the expression for the time derivative of E from the third equation to give:

Can we say anything about the curl of a curl? We can and because I find index notation therapeutic, I'll even include the details:

But equation 4 tells us that the divergence of B is zero so we can replace our curl of curl of B with just the negative Laplacian, leading to:

Which says that the components of B obey the wave equation! And we can go through the same rigmarole to derive an identical equation for E.

Imagine these oscillations in E and B are lined up so they propagate as plane waves in the same direction, say k. Then what else can we say? E and B take the form

So the waves travel in the direction of k and oscillate in the directions of E_0 and B_0 respectively. Plugging this into equation 1 gives us 

Meaning that the oscillations in B are orthogonal to those in E and also orthogonal to the direction of travel, k!

And this summarises Maxwell's discovery that light is an electromagnetic wave.

What about this funny constant mu_0 epsilon_0? It takes the place of the inverse square of the wave speed in the formula and lo and behold (mu_0 epsilon_0)^-1/2 ~ 3x10^8m/s. But this poses a problem. These are just constants of nature and so the speed of light must be too. But classical Gallilean relativity says that someone travelling along next to a beam of light would perceive it as stationary, despite the fact that they would agree with us about Maxwell's equations and reach the same conclusion as us. This posed a huge issue for physicists and necessesitated the overhaul of Gallilean relativity and the arrival of special relativity. Enter: Einstein.