Every now and then, I run across a question in class that I genuinely don’t know how to answer. If I’m lucky, this happens when I’m prepping a class, rather than when a student asks it live. Like today, when I noticed the above discussion question in my slides (reproduced at the bottom as well for those reading via RSS).
The discussion question itself is perfectly straightforward– the current in the wire creates a magnetic field, the moving electron interacts with that, and getting the direction of the force is a straightforward right-hand-rule problem. I’ll pose this to the class and ask them to work out their answers on small white boards, then “poll” the class by having them hold up the boards.
The return question, though, that I don’t entirely expect somebody to ask, but realized I don’t know the answer to, is what happens to the wire? Most of the time, when I do this kind of one-object-interacting-with-another question, I like to look at both action and reaction forces. But here, I don’t think that works– the magnetic field of the electron is zero straight ahead of it, and in opposite directions on either side. That would produce a zero net force (though possibly a very slight torque).
But this is a problem because then we have a situation where one object is changing its momentum (the electron is going to curve in some direction due to the field), but no corresponding change in momentum of something else. Which isn’t consistent with points we’ve spent a lot of time banging on in the intro courses.
I think the answer here is something to do with the momentum of the electromagnetic field itself, but I’m not sure there’s an easy way to explain it. It’s been so long since I took the sort of class where you would calculate that sort of thing that I don’t even know where to start, let along how I would explain it in terms that first-year students would follow.
This isn’t entirely academic– I have one student in my class with a gift for asking really difficult questions, who had me explaining the self-energy of the electron in QED the other day. I can probably wave hands convincingly enough to get through if it comes up, but it’s a nice reminder that teaching intro E&M umpteen times doesn’t actually mean you understand everything in it perfectly.