Maxwell's equations: differential form
It is sometimes said that the beauty of these equations marks them as one of the great intellectual achievements of our species. You won't see them framed in a museum, or performed at a concert hall‚ but they do occasionally appear printed on the front of tee shirts worn by physics students. And in any case, they don't need a conventional exhibition space, for they are all around you in nearly all phenomena, and they are among the tools with which much of our technology has been created.
Below we write them in a more symmetric and beautiful form: a form that shows more clearly their various symmetries. However, for the moment, let's put them to use.
Maxwell's equations and the speed of light
Let's consider light travelling in a vacuum, ie in a region in which there are no electric charges, so that ρ = 0 and J = 0. Again for the vacuum, Maxwell's equations become:
To get a wave equation, we want second derivatives in both time and space. First, we take the curl of both sides of (1). (Informally, this is like differentiating in space, in the sense described above.) Then substitute in (3), and we have
A theorem in vector calculus is that, for any vector a, 
. Here, ∇.E = 0, so
This is the wave equation in three dimensions. In one dimension, we could write it
Let's look for a solution that is a sinusoidal wave, with speed v and wavelength λ. One such wave has the expression:
Differentiating, we get
and 
Substituting this back into the wave equation, we see that it is a solution, provided that
in other words, the equation has a wave solution and the speed of the waves is 
. When Hertz found this solution, he recognised that the value of this quantity was the speed already known for light, which strongly suggested that light was a wave, and part of the electromagnetic spectrum, along with radio. If you're interested, we show film clips of measurements of the speed of light and of (UHF) radio waves in a multimedia tutorial on The Nature of Light.
In a physics lesson, we're not yet finished: we have to check that, associated with this wave in the electric field, there is another in the magnetic field, and that the two together satisfy the equations in three dimensions. But this page is not a (usual) physics lesson and so, as they say, we leave this as an exercise for the reader.
Other forms of Maxwell's equations
Note that the values of ε0 and μ0 relate our units of mechanics, electricity and magnetism. We could choose our system of units such that ε0 = 1 and μ0 = 1. In fact, many theoretical physicists sometimes use these units. They have the effect of making Maxwell's equations even simpler and more symmetrical. (They have the disadvantage that other units then have uncomfortable sizes. With c=1 in these units, either the unit of length is very long or (more usually) that of time very short.)
Another asymmetry is the appearance of ρ and J in the equations: the electric charge density and electric current density. These arise because individual charges can move independently. In magnetism, a North and South pole always appear together. If there were isolated magnetic monopoles (as Dirac had predicted*), we could define a magnetic charge density ρmag and magnetic current density Jmag. Using grey symbols for the magnetic monopole terms that seem to be zero in our universe, this would give us the following set of Maxwell's equations:
Maxwell's equations: symmetric form (ε0 = 1 and μ0 = 1)
Isn't that lovely? In time (which has only one dimension), there is only a simple derivative. In space (with three) there are different types of derivative, but for the moment just think of ∇ as a derivative in space. See now the symmetries between electric and magnetic quantities, and between time and space. Aaah, wasn't it worth reading this far?
And if you'd like to read a bit more about the practical side of Maxwell's equations and electromagnetism, try The electromagnetic spectrum and the multimedia tutorial on The Nature of Light.
* Dirac and the magnetic monopole - a personal anecdote. When I was a physics student, I remember the thrill of seeing the announcement of a lunchtime lecture to be given by Paul Dirac. Yes, P.A.M. Dirac, one of the greats of quantum mechanics. The Dirac who had predicted the existence of the positron (later found) and the magnetic monopole (later not found). He even has a mathematical function named after him.
On the morning of the lecture, the newspapers carried reports that a team of physicists had reported evidence that appeared to be consistent with a magnetic monopole. What a coincidence - and we should get the great man's reaction. The theatre was packed when he came in. He would not give the lecture whose title had been announced, he said, because he knew what we wanted to hear. (Applause.)
In the decades since he had done the monopole work, I expect that he would have given it little thought: it must have seemed like a lost cause. His notes and papers about it would have been back in Britain. And he would only have had a few hours, at most, to think about it. The theatre was packed and the old man spoke in a quiet voice, but you could have heard a monopole drop as he gave a really clear review of his work, all those years ago. He finished by warning us that he thought that the latest announcement would prove to be a false alarm. He was right. But the lecture remains a treasured memory. Incidentally, with the proceeds from a book made of Dirac's lectures in Australia, the physics school here at UNSW offers a Dirac medal for theoretical physics. Professor Edward Shuryak was the 2004 winner.
