See that equation there? It’s the unsung hero of Maxwell’s equations. It says in English, “This is where I would tell you how magnetic fields are generated from magnetic charges, but there aren’t any such thing as magnetic charges.”
In its integral form as above, it’s usually the second of Maxwell’s equations that my students see in their intro E&M class, after Gauss’ Law and before Ampere’s Law. It doesn’t get a lot of press. In fact it doesn’t really even have a name, unlike the other four equations. In a way that’s understandable. Rarely is a problem solved by suddenly remembering that there are no magnetic charges. But it is a fact, and it’s an important one, and it’s one that I make sure my students know.
The professor I’m teaching for this semester takes a different tack - one that’s legitimate, but not one I’m entirely comfortable with. He begins the subject of magnetism by talking about permanent magnets. Obvious enough, but his mathematical treatment of them involves a fictitious magnetic charge on the pole ends of the magnet. Which is to say, he treats the dipole field of a bar magnet as literally produced by “north charge” and “south charge” distributed on the surfaces of the ends of the magnets.
It works, and it’s mathematically justifiable as a calculation device. You want to figure out how magnetic forces from a permanent magnet scale with distance, and this method will serve you well. Everyone is happy. The problem is that it’s wrong. Magnetic monopoles: there ain’t no such animal. This is probably why our textbook treats magnetism from the other way around. It spends a chapter talking about magnetic fields and what they do to moving charged particles. Then it spends another chapter on what makes magnetic fields, which focuses almost exclusively on current-carrying wires. Finally it spends a brief few pages with a purely qualitative discussion of paramagnetism, diamagnetism, and ferromagnetism. That’s the way I’d prefer to do it. If you’re going to include the artificial monopole approximation, do it only after the basic concepts are straight. Certainly don’t lead off with it.
Oh well. I guess this is why we have recitation instructors, so the professors’ lectures can be translated into something that doesn’t seem quite so arcane to new students.
3 responses so far ↓
1 Uncle Al // Jul 11, 2008 at 10:12 am
Magnetic monopole: Thick-walled hollow ball radially ferromagnetized. One pole as the outer surface, the other as the inner. Applying Gauss’ law thereafter is hate language.
If the universe contains even one magnetic monopole Maxwell’s equations are symmetric. The whole planet should be strewn and orbited with orthogonal triplets of large area superconducting loops SQUIDly awaiting the first induced persistent quantum of flux.
Contemporary physics theory is sodden with symmetries. Reality arises from symmetry breakings. Discoveries await at the interfaces (pdf). Somebody should look.
2 Carl Brannen // Jul 11, 2008 at 12:07 pm
When I was a physics grad student, I mostly kept out of teaching by using my skills in electronics. But they did require all students to teach for one quarter so I ended up running a recitation or homework section one quarter.
The subject was relativity. There was this one question that got kind of sticky. Two spacecraft leave the earth going in opposite directions, each at speed 0.8c; what speed are they separating at in the rest frame of the earth?
The answer, of course, is 1.6c, but the students weren’t having any of that. They claimed that the professor told them that a speed couldn’t be larger than c. I explained that in this case, there is no change of reference frame and so no need to transform velocity, etc. They said, no, the professor had been very explicit in this. I told them that tenured professors don’t make mistakes and surely they had misunderstood what he had said.
3 CCPhysicist // Jul 11, 2008 at 12:59 pm
How often does that professor teach this subject? That is a really awful approach to magnetism and violates rule 1 (don’t make the class harder than it needs to be for the target audience by ignoring the book as well as pedagogical studies).
Uncle Al is wrong. Gauss’ Law of Magnetism is perfectly symmetric. It has the “magnetic charge enclosed” on the right side. It is zero because we have yet to find one, provided we ignore the one observation that Blas Cabrera published.
I introduce Gauss’ Law for Magnetism in conjunction with Ampere’s Law, since it says that there is no radial component to B. I find it helpful to have a “point infinite current” equation for B that can be compared to the “point charge” equation for E, and do a compare and contrast for them - since they differ in just about every way imaginable. I do Biot Savart last because our students do not have Calc III as a prerequisite for Physics 2.
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