Problem Set 5

Due Friday, October 20, uploaded to Balckboard by noon

Problem 1 [10 points]
See Discussion Question 2.4 in Notes 2-4.
Problem 2 [10 points]
Consider an infinitesmally thin charged disk of radius R and uniform surface charge density σ_o, that is lying in the xy plane centered on the orign, just as in problem 2 of Problem Set 4. In that problem you solved for the potential φ(r) by expanding in a Legendre polynomial series. Here you will solve for φ(r) far from the disk by using the multipole expansion.
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) of problem 2 on Problem Set 4.
Problem 3 [10 points]
Consider a line charge density λ(z) that is localized on the z axis from z=−a to z=+a. By considering the monopole, dipole, and quadrapole moments of the charge distribution, find an approximation for the potential φ(r) to leading order only (i.e. the first non-vanishing term) in the multipole expansion, for each of the following three cases:
a) λ(z) = λ_ocos(πz/2a)
b) λ(z) = λ_osin(πz/a)
c) λ(z) = λ_ocos(πz/a)