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 10^{26} 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.

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Some grand unified theories explain quantization of electric charge in a different way, by having electromagnetism be part of a more complex force whose charges are forced by the group-theoretical mathematics to be quantized. But these GUTs are, themselves, the same ones that have topological magnetic monopoles! I've always thought that was a beautiful result.

(I recall somebody, probably John Baez, posting a concise proof that involved differential topology, which I actually understood for about an hour.)

If a magnet were large enough, and we stand at one of its poles, we might perceive it as a monopole...

Only if you ignore the field going the other way in the interior of the magnet.

Despite the presumed equivalence of E and B (or is "B" just our funny way of talking about velocity-dependent aspects of E?), monopoles have a problem with the A field (vector potential) presuming it is "real": the A field can't be reasonably distributed around the MM. Instead, a IMHO tacky "string" or tail of tangled lines of A has to trail it, or connect it to another monopole. Well, if E and B are equivalent then there's an A analog, the curl of which gives E: creating the same problem. Yuck. But it would be fun to find a monopole. It's odd, that one interaction that seemed to be one (forget details), and never again.

That problem with the "string" is actually crucial to Dirac's charge quantization argument. If I'm recalling it correctly, you can move the tail anywhere you want with a gauge transformation, so it's possible to have a well-behaved A field in any local neighborhood. But if there's an electrically charged object hanging around in the vicinity of the monopole, the transformation can only be done consistently if electric charge is quantized.

Interestingly, there's a formulation for some phenomena in magnetized plasma physics in which an net magnetic field acts exactly as a monopole due to the interaction of plasma with magnetic field. And I mean, it's freaky - the math is quite literally the same as for electrostatic fields, at which point you invoke Feynman's quip about the same differential equations having the same solutions.

Unfortunately that's about limit of what I remember from that particular class years ago...

Matt, true that the monopole issue "explained" quantized electric charge - but only the ratio of electric to magnetic, not the actual value (ie, the FSC being around 1/137 is still mysterious.) Still, the "tail" is a creepy thing if imagined as physically real, and again: what about then needing an A-field analog for E?

Neil @4, the reason that magnetic monopoles are a problem for the vector potential A is that the divergence of the curl of any vector is identically zero (this is why one can equate B to the curl of some vector A), so one would have to add a source term to B in addition to the curl of A. This need not be a problem for E because it has an electrostatic source term (the gradient of a scalar potential phi) and a dynamic source term (dA/dt). Both terms are needed because the curl of any gradient is also identically zero, so the dA/dt term is needed to represent the curl of E. Of course, you will also need to modify Faraday's law by adding a term representing the magnetic charge current similar to the electric current term in Ampère's law.

No, I haven't worked out what form the additional source term for B would take.

Eric, I understand basically where you're coming from, however: true equivalence, means by logical necessity that whatever there is for B, must be for E and vice versa. If monopoles existed, then the left-over part of EM based on electric charges and no magnetic charges would be false: there would be both charges, both fields and epi-fields (the vector potentials), a magnetic field analog of phi, there would be an analog of dA/dt to make magnetic field when monopoles were accelerated (you must have overlooked that), etc.

Neil: Remember that you also have some freedom due to gauge transformations to simplify the problem (this is true even without magnetic monopoles). In the case we're considering, you claim in effect that there must be, in addition to the scalar potential for E and the vector potential for B, a scalar potential for B and a vector potential for E. So far so reasonable, and I agree that there must be a scalar potential for B, but it is not obvious to me that you cannot choose a gauge in which one of the vector potentials is identically zero. The argument is that you can separate the vectors E and B into curl-free and divergence-free parts. But because of Faraday's Law and Ampère's Law, these pieces are not independent of each other; my intuition is that you will be able to express one of the four potentials in terms of the other three, and therefore eliminate it at least by substitution if not by gauge transformation.

You probably cannot do this without forcing your remaining potentials to be explicitly time-dependent. But the divergence-free part of E already has explicit time dependence, so this is no big deal.