Twirling, Twirling, Twirling…

Oh, yes, the midterm. Well, apparently the students don’t hate me as a result of it. (Actually I have not seen any of them since before the midterm (I was away during the actual midterm itself), so I’m not entirely sure about that…)

I stayed up until 2:00am or so on Wednesday writing and typesetting the thing, and in the end I think I set a relatively straightforward exam. Furthermore, after I finished writing it, I realized that a good chunk of the computation I’d prepared for them had already been done in a previous midterm. I’d completely forgotten. You can see a (bit blurry, sorry – in a cafe trying not too look too conspicuous taking photos) snap of my notebook with the computations that I did in preparation for the midterm in the little photo to the right.

Since Elliot asked (thanks!), I’ll say a bit more about what they had to do. I decided to go for the model of picking a specific physical situation that they come back to for the entire midterm, having them work out several of its properties using lots of the techniques and concepts that they’ve been learning so far. It almost worked, but the last bit I cut out since it was making the exam too long for them to solve in and hour and forty minutes. The system was simply a spherical shell with electric charge spread uniformly on it. When stationary the thing just has the usual simple radial electric field, but if you set it spinning, it has a nice dipole magnetic field as well. These are the fields I give them at the beginning of the exam (they’ve seen this setup before in a previous course) and then as a warmup they are to check that the boundary condition on the fields (which jump at the shell in going from inside to outside) are consistent with there being a spinning sphere of charge. After that they set out to compute the energy in the fields, the flow of energy in the fields (computing the Poynting vector) and then study the momentum in the fields. Then they compute the total angular momentum in the fields too, and think about that. Then I get them to notice that the expressions for the magnetic energy and the angular momentum factorize nicely in a way they have not seen in class, ([tex] U_M=\frac12 I_{M} \omega^2[/tex] and [tex] {\vec L}=I_M {\vec\omega}[/tex]) implying the existence of something ([tex]I_M[/tex]) that looks a lot like a moment of inertia for electromagnetic fields. The next thing they do is imagine slicing the sphere along the equator with a sharp sword. Then they get to ask the question of whether it will fly apart or not. To do that, they need to compute the stress tensor of the system, finding a nice surface over which to integrate the beast in order to evaluate the forces on the pieces. They find (I hope) that the magnetic force is attractive, and the electric bit is repulsive…. so whether it flies apart depends upon how fast it is spinning. But then I ask them to compute the ratio of the two forces to see what the condition on the spin is, and they find (I hope) that since [tex]\omega[/tex] is the ratio of the speed of light to the radius of the sphere (times a factor of [tex]\sqrt{2}[/tex]), the setup is more likely to fly apart, since you’d either need a really huge (and unphysical) sphere, or unphysical spin speeds [update: Better put, as someone in class said later on, the speed at the equator is always superluminal, so… bad].

The fun thing I was going to do next was have the thing radiate (by jiggling the charge distribution a bit), and then examine all of the above again to see transport of energy and momentum and so forth… But there’s no way they’d have had time…

So there you have it. This is the kind of fun detective work I put my students to from time to time.

-cvj