And God said, Let there be light: and there was light.
- Genesis 1:3
This post is going to need the standard math disclaimer. Though there’s math that looks horrible, you don’t actually have to know any math to follow the ideas. Give it a try!
Over this week we’ve looked at the four Maxwell equations that describe the entirety of classical electromagnetism. Let’s write them all down:
These four are everything that it’s possible to know about the classical behavior of electromagnetic fields. This notation is a little more modern than what Maxwell originally wrote down in the 1860s, but when he put together this theory one of the first things he did was think along these lines: Faraday’s law says a changing magnetic field produces and electric field. Ampere’s law say a changing electric field produces a magnetic field. It just might be possible for one to produce the other which in turn brings you back to the beginning of the repeating cycle. You don’t need any kind of matter or charge, the mysterious intertwining dance of the fields themselves might be able to propagate itself out forever. Maxwell thought so, and with the machinery of his completed theory he investigated the situation mathematically.
Start with Maxwell’s equations, and let J and rho equal zero. We’re in empty space, so there’s no charge or current floating about. Take the equation for the curl of E, and find the curl of both sides of the equation. Yes, we’re finding the circulation of the circulation. Don’t worry, there’s a reason:
We can apply a vector identity to simplify the left side of the equation, resulting in this:
Some explanation is in order. The triangle is pronounced “del”, and applied to a scalar function it represents the gradient, or in essence the directional rate of change of the field with respect to position. The del squared is called the Laplacian, and roughly represents the rate of change of the gradient. The first term on the left is equal to zero, because the divergence of the electric field is zero by virtue of the fact that there’s no electric charge.
The right hand side simplifies as well. Nothing prevents us for interchanging the order of the space and time derivatives, so let’s do it. Applying all that and canceling the minus signs:
All right, that’s an improvement. We can improve it yet more. What we have on the right is the curl of the magnetic field. And that’s already one of our Maxwell equations at the top of the post. Plug that equation in, remembering that in empty space J = 0, and you’ll get:
To a physicist, I imagine being the first human to see that equation must have been like being struck by lightning. This equation is just the old and venerable wave equation. Maxwell’s equations imply that the electric field can travel along in space as a wave. The same steps lightly modified give the exact same expression for the magnetic field. The two waves are intimately interleaved, creating each other and traveling outward. There’s more. The wave equation says that the speed of the wave is 1 divided by the square root of the constants in front of the time derivative. Plugging in (and feel free to try it yourself!), Maxwell saw that the speed of these electromagnetic waves matched the value of the speed of light.
This was an astonishing revelation. Light must actually be these waves. As Maxwell put it, “We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” He was exactly right, except for the fact that there’s no medium actually vibrating. The fields themselves are all that’s needed.
These equations and this result are some of the greatest triumphs of science. It’s difficult to conceive of much more impressive physics, but I’m an optimist. Physics is not yet out of surprises.