Physics 834: Problem Set #9

Here are some hints, suggestions, and comments on the assignment. Remember to keep track of the amount of time you spend doing the (entire) assignment and record this number on your problem solution.

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  1. Electrostatic potential in hemisphere. This is a boundary value problem: solve Laplace's equation with the given boundary conditions. This problem has a lot in common with the example worked out on pages 142-144 of the Lecture 15 notes (we'll review this example on Monday). That problem had a full sphere rather than a hemisphere, but the "trick" here is to solve a full sphere problem that will give you the hemisphere solution as a byproduct. (Remember that to solve a differential equation you just need a solution that satisfies the equation and the boundary conditions.) So how could you choose the potential on the top and bottom halves of a sphere so that the top half is V0 and the slice through the middle is zero? Then look for a solution as in the example. To solve the necessary integrals, you can use Mathematica but try also using a recursion relation that let's you do the integrals directly (because they are total derivatives). NOTE: It is perfectly ok to solve the hemisphere directly by imposing the boundary condition on the flat part as opposed to using the "trick" described above.
  2. Expansion and application.
    1. The first part is quite straightforward, once you realize what r< and r> let you do. They solve the problem of knowing which of r and r' is smaller, so you know how to expand the square root in the generating function. Then it's pretty much just substitution.
    2. The application requires you to know that the magnetic vector potential for a wire with current density J(x) is μ0/(4π) times the integral of J(x) times equation (3). So first you have to find J(x) for a circular loop of radius a that carries current I. This should involve a couple of δ functions. Then substitute this and the expansion for 1/|x-x'|. You can use the delta functions for two integrals, but you'll have to do the φ integral. Simplify your final result using results such as Equation (8.53) in Lea.
  3. Heating (or cooling) a sphere. Another boundary value problem. Do a full separation of variables on this problem, taking into account that we (apparently) have spherical symmetry. (So what is the dependence on θ and φ?) When finding solutions to the separated equations, don't forget the case where the separation constant is zero (you'll need this to match the boundary conditions!). Your final answer should be in the form of a sum over functions in r times functions in t, with all coefficients determined.
  4. Current in a conducting sheet.
    1. In steady state, how does the charge density change with time? (Or does it?).
    2. If we have a circular copper plate, what three dimensional coordinate system is most appropriate? You should be able to follow section 8.4 in Lea. What do you expect for the z dependence of j or Φ?
    3. Given the general expansion from part (b), use the boundary conditions to determine the coefficients. Try to do the plots in Mathematica using ContourPlot.
  5. Harmonic oscillator.
    1. Play the game of multiplying by a positive function p(x) and requiring the new equation have the standard Sturm-Liouville form, as in the last problem set.
    2. The weight function should come out as a by-product of solving part (a). Review the orthonality proof to verify that the surface terms vanish in this case.
    3. Revisit our series solution methods. When you get a recurrence relation for the coefficients of your Taylor expansion, see what it reduces to for large n (if n is the index of your expansion). You should be able to sum the series based on that simplified form, which tells you the asymptotic form of the series. Does it go to zero for large x? If not, we need to require the series to terminate, which determines the possible values of the eigenvalue λ.
    4. Just plug into your solution and normalize according to the guideline in the problem.

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