Ok, I feel stupid. I tried it another way but forgot to put in a factor of 'r' while doing the integration in polar the first time. Now I have it right. Think of a circle of radius a centered at (a,0). The area of a patch is r*dθdr, so the charge of the patch is σ*r*dθdr. The patch is at a distance r so the contribution to the potential of a wedge is ∫(σ*r)/r dθdr from 0...R which equals σ*R*dθ. The equation for this circle in polar coordinates is R=2*a*cos(θ), so by substitution, the contribution to the potential of a wedge is σ*2*a*cos(θ)*dθ. If you integrate over all the wedges (from -π/2 to π/2), the potential equals 4*π*a. Then part B is trivial to get.

Maybe I didn't clearly explain the problem. It asked to find the amount of energy stored in the electric field created by a uniformly charged disk. This can be said several other ways including "the potential energy stored in a uniformly charged disk" and "the energy required to build up a uniformly charged disk with charges brought in from infinity (where the potential equals zero)." Part A of the question asked for the potential at the edge of a uniformly charged disk (a disk being a cylinder with no thickness) of radius R, and part B asked for the energy required to build the disk (or one of the other equivalent phrases). This leads to believe that the prof wanted us to solve the problem in the way I described (summing up the work done to create rings; the work for one ring being the amount of charge in the ring times the potential on the outside of the disk that it's being added to), which is how I would have done it anyway. To quote my book, "ϕ=∫ (ρ/r)dv" or potential at some point equals the integral over all the charges of the magnitude of the charge times the reciprocal of the distance between the charge and the point you're trying to find the potential at (in the cgs system anyway, in mks there are some constants there too). This is what I was trying to solve. For those tuning in at home, my book claims that the answer to part B is 8*Q^2/(3*π*R) where Q=π*R^3*σ.

Physics Midterm

It was at 6, but I met up and ate dinner with a bunch of kids from the class beforehand. It was interesting meal, with many bad physics jokes and puns. Before I talk about the test itself, let me start off with some statistics. The class average for the midterm last time she taught this course was 28% raw; the average overall grade for the course was 58% (which became a B+). Needless to say, I didn't go in expecting to get much right (she even commented that we'd be lucky to get 50%). The test was 5 questions with 3-5 parts each and we had 2 hours. I was able to get answers for all but one of the questions, and the one I couldn't answer I explained my process and hope to get partial credit for. It asked us to find the potential energy of a uniformly charged disk. My method was to find the potential on the edge of the disk, multiply that by the amount of charge needed to create a thin ring (potential * charge = work or energy), and then integrate over all the rings. I tried to get the potential on the edge by integrating s/d over the surface in polar coordinates (where s is a surface charge density and d is the distance between the patch of area I'm working with and the distance to an arbitrary point on the outside of the disk, (0,R) in polar). However I got something that not even maple could solve because I found d using law of cosigns (d=(R^2+r^2-2*R*r*cos(theta))^.5). I must be missing a trick, feel free to clue me in. The funny thing is, I tried this for a practice problem while studying but gave up (the answer was in the book, but not how to derive it). I feel ok about most of the other problems, so hopefully I did ok.