Physics 7701: Problem Set #10

Recent changes to this page:

• 16-Nov-2013 --- original version.
• 18-Nov-2013 --- additional comments for 1-4
• 19-Nov-2013 --- still more comments for 1-4

1. Potential in cylinder.
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.
• We have the full range of φ dependence so we apply the condition for single-valued functions of φ. Don't forget to consider the possibility of a constant term.
• The boundary conditions in z means that the separation constant kappa² will be the opposite sign from what we considered in class, so sinhs and coshs become sines and cosines. Do both contribute?
• You can think of the change in sign of kappa² as meaning that k becomes i*k. This in turn makes the Bessel functions become modified Bessel functions. Look up the properties of modified Bessel functions in a text or online. You will find the asymptotic forms for I(x) and K(x) (which includes what happens at small values of x); which of these functions do you use (or both?) for the interior scalar potential? Why?
• Can you think of limiting cases to check your answer?
2. 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.
3. 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.
4. 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?
5. Free-space Green function in polar coordinates.
Here the only variables are ρ and φ (all functions are independent of z). Some comments:
• Please note the use of Zangwill's conventions. This means that factors of ε₀ and 4π (or 2π in this case) appear in the Green function. This doesn't change anything significant in your solution.
• The idea is to do separation of variables and solve for the angular part in a complete expansion. Remember that we need this part to be single valued and don't forget the possibility of a constant term. Use the completeness relation (the sum that gives a delta function) in the φ functions.
• Don't forget to exploit the idea that the Green function should be symmetric under interchange of the unprimed and primed variables.
• When you solve the one-dimensional equation by integrating across the delta function, make sure you first make any rearrangement needed so that you get the difference of the first derivative of the Green function at the matching point.
• Note that the units seem to be wrong for the ln term (if we associate ρ with a unit of length). You can think of there being a constant there that is equal to one in our units. Where does this enter into your solution for the ρ dependence?
6. 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 G_D 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?