Maxwell's Equations #1

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!

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The equation explaisn the behavior of magnets when it splits, right? Everytime you break a magnet, there is still a positive and negative pole for both magnets.

Yes and yes. I skirted the "monopole" word because I wasn't sure I could make its meaning clear without going too far afield. But that's exactly right, a magnetic charge is a magnetic monopole. The splitting magnet is just the same; picture pulling the leftmost three wires in the picture further to the left. You still don't have a source with field lines emerging, you just have two smaller copies of the original.

I have to nitpick a little flub about this description of divergence:

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.

If more lines enter the cube than leave, that region has negative divergence. The case of zero divergence only applies if every line that enters the region leaves the region.

One reason I think it makes more pedagogical sense to present the corresponding equation for the electric field first is because it gives you a better intuition for what divergence is. In general, div E is not equal to 0, and the right hand side has a straightforward interpretation (which I presume is the subject of tomorrow's post). Then you follow up with this equation, which as you point out contains the profound statement that no magnetic charges exist (as far as we know).

By Eric Lund (not verified) on 10 Aug 2009 #permalink

I think divB was an interesting place to start, but the difference between divE=0 when a dipole is inside a cube and divE=0 when the cube is empty of charge provides a visual bridge to the magnetic case and your example of the dipole-like field of a short solenoid.

Tiny Nit 1: Saying divB=0 means there is no magnetic charge in your vicinity, not that there are no magnetic charges anywhere in the universe. You can't prove a negative like that, but divB=0 is a fact consistent with our failure to observe one [*]. In a similar way, divE=0 in the vacuum that carries electromagnetic waves.

Small Nit 2: You stumbled into the error of "affirming the consequent". We are not justified in claiming anything about the existence of a monopole from the fact that charge is quantized, because the proof that the existence of a monopole requires the quantization of charge only goes one way. Tempting, yes. Logical, no.

Tiniest Nit 3: We could argue over many beers whether a proton is a discrete chunk of charge because the total of the quarks inside it adds up to "e" within preposterously small experimental uncertainties. The quarks inside it are (at our present level of understanding) the discrete bits of charge, and the ratio of exactly 3 between the smallest one and the electron could be the most intriguing discoveries of my lifetime. Especially if that 3 persists in the number of families.

[*] Maybe. You could write an entire essay about the Valentine's Day Monopole if you looked at all that was written back then rather than the short paragraph on Wiki. However, we still have divB=0 even if that observation was real, because monopoles are exceptionally rare.

E&M is the one subject that still seems rather impenetrable to me even after having an entire course on it. Too much vector calculus and not enough discussing the ideas. I look forward to reading your posts about it.

The Wikipedia drawing shows a single loop made from a single wire, not "wires."


Wow, two days of getting piled on for technicalities in a row. Alas, Eric Lund beat me to my observation :-)

Instead I'll point out that not only is it fairly simple* to include magnetic monopoles (from a theoretical point of view), it's typically a homework assignment given later on in senior-year or reasonably early in 1st-year graduate level electrodynamics classes. Somewhere in Jackson, I forget if it's in Griffiths as such.

*"Simple" meaning in terms of theory. Not necessarily simply effort if your vector calc fu is weak.

I don't mind getting piled on for technicalities. In fact I think it's great! When I aim at an explanation of something technical for a general audience, of necessity some details get washed out. The comments are fantastic expositions and notes for further exploration by the curious or more mathematically inclined. And I count myself as one of those people; more than once comments have fixed errors in my own understanding.

Has anyone actually written down Maxwell's Equations in the case that divB != 0?

I was under the impression that it is much harder than you made it seem. Primarily because of all of the crazy vector potentials that are suddenly required as well as some other details that I'm unable to remember at this hour.

By SimplyHarmonic (not verified) on 11 Aug 2009 #permalink

A magnetic monopole detector is simply a superconducting loop. If a magnetic dipole goes through everything will cancel by induction . If a monopole goes through a quantum of magnetic field remains. It would be no big whoop to launch satellites each with three big orthogonal supercon loops with their Tc above the ambient temperature of space. Add a SQUID for flux counting, and wait.

@SimplyHarmonic: No, it's not that hard. There would be an additional term in the curl E equation which is proportional to the current density due to moving magnetic charges, just as there is a term proportional to the current density due to moving electric charges in the curl B equation. Of course, actually doing useful calculations with the modified Maxwell equations becomes harder, for the reasons you state.

By Eric Lund (not verified) on 12 Aug 2009 #permalink