Physics 7701: Problem Set #10

Here are some hints, suggestions, and comments on the assignment.

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  1. Potential in cylinder.
  2. In general, we follow the expansion method in class for a finite cylinder. The difference here is that the boundary conditions are, in a sense, reversed: the end caps are grounded and we specify the potential on the cylindrical surface.
  3. Cylinder again.
    1. The first part is just to carry out the integrals to find the coefficients given the particular potential in equation (1).
    2. Use the asymptotic (small ρ) limits of the modified Bessel function. [An open question is whether this is really justified, given that the sums go up to infinity so the argument of these Bessel functions will eventually not be small in the sum for fixed b/L. That the sum converges is not in question because it is the ratio that appears.] You will need to do sums, which you can do using Mathematica, but you will still have some manipulations to do to make it equal to equation (2). In class we'll discuss how to do sums like this "by hand" (you can apply the power series for arctan twice!). A helpful identity is tan(A+B) = (tan A + tan B)/(1 - tan A tan B), remembering that tan(tan-1(x))=x.
  4. Concentric spheres.
    1. Use the general expansion for a φ-independent scalar potential. Remember that restrictions on how the potential behaves at the origin won't be relevant here. You'll determine the coefficients from the boundary conditions on the hemispheres.
    2. The large b limit is a bit unexpected, because the potential is not zero at infinity because of the boundary conditions.
  5. Spherical surfaces. Note that we have a spherical surface of charge here and not a conductor. So we want to find a potential given a charge distribution, which suggests using a Green function.
    1. What is the charge density written with appropriate theta and delta functions? What is the relevant Green function (again remembering that we are not counting the sphere as a conductor) to integrate over to get the scalar potential. You can also write an expansion for the potential on the inside of the sphere. Equate at a simple choice for theta and find the coefficients. You'll need to use recursion relations to get the form requested.
    2. Same procedure as for inside the sphere.
    3. Use the standard formula for the electric field given the potential.
    4. What happens to α in this limit?
  6. Free-space Green function in polar coordinates.
  7. Here the only variables are ρ and φ (all functions are independent of z). Some comments:
  8. Green function inequalities.
    1. What would the Green function be in the absence of boundary conditions on the surface of the volume V? What are those boundary conditions for a Dirichlet Green function? Could the opposite inequality to the one stated in equation (8) be possible if these boundary conditions are to hold? Consider the physical interpretation of the second term in equation (7).
    2. Earnshaw's theorem says that the scalar potential in a charge-free region takes both its minimum and maximum value on the boundary of that region. Suppose GD were less than zero; what would that mean about the contribution of the second term in equation (7)? If we considered just this term, what would it imply for the scalar potential in V?

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Physics 7701: Assignment #10 hints.
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