All this week we’re going to be briefly looking at James Clerk Maxwell’s greatest contribution to physics – his theory of electromagnetism. The consequences and applications of the theory fill many volumes, but the conceptual and mathematical foundations of the theory can be expressed as four short and relatively simple equations. These are Maxwell’s equations, and though they’re not as well-known in the pop culture as Einstein’s mass-energy equations our modern world would be impossible without them.
The first equation describes magnetic charge:
In words, you’d say “The divergence of the magnetic field is zero.” The triangle and dot represent divergence, and B represents the magnetic field. (M was already taken by a quantity called magnetization) For this to make any sense we need to know what the heck divergence is. Of course there’s a formal mathematical definition, but we can just take the concept to be a measure of how much field is originating at a point. You’ve seen pictures of magnetic and electric field lines – of course there’s no such thing as literal lines, instead the strength of the field is represented by how closely spaced the lines are and the direction of the field is represented by which way the lines point. Now imagine that you draw a little cube in space. If there’s more lines leaving the cube than entering, that region inside the cube has positive divergence. If there’s not, that region has zero divergence.
From Wikipedia, here’s a drawing of magnetic field lines from current-carrying wires (the circles with the dots and Xs). Notice that no matter where you draw your box, the lines going in are the same in number as the lines going out. They don’t have any particular location they’re diverging from:
Saying that magnetic fields have no divergence is the same as saying there is no such thing as magnetic charge, because charge by definition is a source of field – without divergence, there can be no sources. By “source” I mean a physical location from whence the field lines emerge. Now clearly something has to make the fields! And something does, and we’ll see what it is. But it’s not magnetic charge, and it’s not a source with divergence.
In modern quantum mechanics this is actually a little odd. The great physicist Dirac showed that if there were such thing as magnetic charge, even so much as just one subatomic particle with magnetic charge anywhere in the universe, electric charge would have to be quantized in order for the theory to remain consistent. And as far as we can tell, electric charge is quantized. It comes in discrete chunks like electrons and protons and all the rest. Despite all that, we’ve simply never observed a magnetic charge. They don’t seem to exist.
Maxwell’s equations wouldn’t suddenly become useless if we did discover a magnetic charge. The required alterations are not too hefty, and of course the unmodified theory would still be perfectly valid anywhere without magnetic charges, which seems to be pretty much everywhere.
And that’s the first of the four equations (though it’s usually presented as the second). Three more to go!