As we march on toward Newton’s birthday, we come to the second of Maxwell’s famous equations, which is Gauss’s Law applied to magnetic fields:
For once, this is pretty much as simple as it looks. The divergence of the magnetic field is zero, full stop. As I said yesterday (albeit using the wrong terminology), the left-hand side of this equation basically means that you look at the magnetic field in the vicinity of some point in space, and ask how many little arrows point toward the point of interest versus how many point away. What this equation tells us is that no matter where you look, there will always be exactly the same number of magnetic field arrows pointing toward you as away from you.
So, why is this important?
The post title ought to be a clue. For the full effect, you need to sort of sing it to the tune of this classic from The Simpsons:
Though that’s also a bit misleading, as what it really says is that you can’t have a monopole in classical electromagnetism.
A “monopole” in physics terms means a source of some sort of field that produces only outward-going arrows or only inward-going arrows. Individual charged particles are monopoles– a positive charge like a proton produces an electric field that points outward everywhere, while a negative charge like an electron produces an electric field that points inward everywhere.
Magnets, on the other hand, are dipoles, which have both inward- and outward-pointing arrows. The magnetic field points out from the north pole of a magnet, and in toward the south pole, and what this equation tells you is that those two things will always appear together. You’ll never have a north pole without a south pole. If you cut a magnet in two, you get two smaller magnets, each with both north and south poles. Even a point-like particle like an electron, which has a tiny intrinsic magnetic field, behaves like it has both north and south poles.
At least, in classical electromagnetism, you can never have a magnetic monopole. When you go to a quantum theory, things get a little more complicated, and it turns out that it’s actually somewhat attractive to have magnetic monopoles. Dirac showed that if you allow the existence of magnetic monopoles, you can explain the quantization of electric charge in a very natural way, so lots of quantum theories include monopoles. A fairly exhaustive treatment can be found in this review on the arxiv.
To date, though, no experiment has produced conclusive proof of the existence of a monopole (other than, you know, the fact that charge seems to be quantized…). One famous experiment led by Blas Cabrera, did produce an apparent detection of a monopole within its first few months of operation, but never saw another.
This means that the equation above remains a really excellent approximation for the state of the universe. It’s also sparked some really interesting research– the apparent lack of magnetic monopoles was one of the problems that led to the development of inflationary cosmology. Dirac’s theory doesn’t require a large number of monopoles to work, just that at least one exists. If you fiddle around a bit, you can arrange for only a relatively small number to have been produced in the Big Bang. If the universe then underwent a brief but dramatic expansion in size– the inflation epoch is supposed to have increased the radius of the universe by a factor of 1026 or thereabouts– then it wouldn’t be surprising that we don’t see any monopoles wandering around– there might be only a handful of them in the entire visible universe, left over from the Big Bang.
It’s sort of amusing in a black comedy kind of way, to imagine that Cabrera’s experiment really did detect a real monopole, which just happened to be the only one in the Milky Way. It’d be simultaneously incredibly momentous, and completely unverifiable.
Anyway, that’s Gauss’s Law for magnetic fields, and the strange places it leads. Come back tomorrow for more about E&M, as we continue our progress through the equations of the season.