Physics 7701: Problem Set #7

Recent changes to this page:

• 16-Oct-2013 --- original version.

1. Vector operations in cylindrical and spherical coordinates. Here the idea is to apply the formulas on the back cover of Jackson (see handouts).
   • For the divergences and curls, this requires identifying A1, A2, and A3 in each case (they should be very simple and mostly zero for unit vectors), then simply plugging into the formulas. That is all you need to do in your written solution.
   • For the gradient and Laplacians, note that the functions only depend on ρ and r, so derivatives with respect to angles vanish. Assume first that ρ and r are greater than zero but then comment on what you expect to happen when they equal zero.
   • Use units as a check of your results.
2. Vector calculus theorems.
   1. Practice with Stokes's theorem. Assume a direction for going around C in the x-y plane. If you go counterclockwise, what is nhat? What is an easy surface to choose?
      • It may be simplest to evaluate the surface integral in polar coordinates.
   2. Practice with the Divergence theorem.
      • What is the easiest coordinate system for a volume integral over a hemisphere? You are welcome (and encouraged!) to use Mathematica to evaluate the resulting integral, but write down the command used. Check the Mathematica examples on the 7701 Mathematica web page.
3. Charge distributions with delta functions.
   1. Is there theta or phi dependence? The "shell" is of zero thickness. Be sure to check that the charge density integrates to Q.
   2. You could consider a fixed length L of the cylinder, which then has total charge lambda*L. Then do as in part (a).
   3. For each variable, decide if it has a theta function, delta function, or if it doesn't depend at all. Use that the charge density integrates to Q to determine any overall constant. Check units!
   4. This is the only somewhat tricky one. Be careful of how many R's and how many r's appear in the charge density. It could integrate to Q but still be incorrect!
4. Gauss's theorem applied to charged spheres.
   • For all of these, use the spherical symmetry and a standard Gaussian surface.
   • How do you expect the electric fields to compare for r > a?
   • Are the electric fields always continuous at the surface?
5. Electrostatic potential of neutral hydrogen atom.
   • Remember that you are given the electrostatic potential.
   • Be careful of taking the Laplacian of 1/r: delta functions lurk!
   • What are the parts of a neutral hydrogen atom? How do you expect their contributions to look?
6. 3-D wave equation.
   1. This part should be a direct generalization of examples we've considered or those in Lea in which we take the Fourier transform of an equation. Don't go to a particular coordinate system yet (i.e., keep x and k as vectors. The transform S is given in terms of the transform F (which is left as a function until parts c and d).
   2. Assume α > 0. Just find where the poles are! (What kind of equation do you need to solve?)
   3. First find F and then set up the inverse Fourier transform. You might find it simplest to do the ω integral first (but it is not essential). This is the same in parts c and d, and uses the result from part b. Consider t<0 and t>0 separately (one of them is very easy!). For the k integrals, you'll want to use spherical coordinates, as in problem 5 from the last problem set. Use the same trick of choosing the k_z axis so that the angle between k and x is the polar angle. Do the r integral last and you are encouraged to use Mathematica (but include a printout if you do).
   4. This is quite similar to the last part, right up to the r integral. But this one is easy, if you remember about Fourier transforms and delta functions. Also remember when getting your answer that t>0.
7. More practice with the vector calculus theorems.
   1. Practice with Stokes's theorem. Assume a direction for going around C in the x-y plane. If you go counterclockwise, what is nhat? What is an easy surface to choose?
      • This one should be easy!
   2. Practice with the Divergence theorem.
      • Pick a convenient (natural) coordinate system for the volume integral.