Problem 1 [10 points]
Two concentric spherical shells of radii R1 and R2, with R1<R2, are fixed with the following values of the electrostatic potential:
φ(R1, θ) = φ1cosθ and
φ(R2, θ) = φ2
where φ1 and φ2 are constants and θ is the usual polar spherical angle. Find the electrostatic potential for:
a) r<R1, inside the inner shell
b) r>R2, outside the outer shell
c) R1<r<R2, between the two shells
d) Find the surface charge density σ(θ) on each of the two shells.
Problem 2 [20 points]
a) Consider an infinitesmally thin charged disk of radius R and uniform surface charge density σo, that is lying in the xy plane centered on the orign. Using the Coulomb formula for the potential,
|
φ(r) = ∫ da´ |
σ(r´)
|r - r´| |
do the integral to find the exact value of the potential φ for points along the z axis, both above and below the disk (make sure you get this part correct before proceeding with the following parts!).
b) Since the above problem of the charged disk has rotational symmetry about the z axis, we can express the solution for the potential φ(r) in terms of a Legendre polynomial series using the method of separation of variables. In this case, the "boundary condition" that determines the unknown coefficients of this series is the requirement that the potential agree with the known exact values along the z axis, as computed in part (a). Find the coefficients of this expansion up to order ℓ=5, for |r|>R and |r|<R, both above and below the disk.
c) Using your results of part (b), compute the electric field E just above and just below the disk, and explicitly show that the discontinuity in E is given by the surface charge density σo, as we know from general principles that it must be.
d) Now, keeping the origin at the center of the disk, compute the monopole moment, the dipole moment vector, and the quadrapole moment tensor of the charged disk. Use these moments to write an approximation for the potenial φ(r) far from the disk (i.e. r >> R). Compare with your results from part (b).