Problem Set 4

Due Thursday, October 17, uploaded to Balckboard by 11:59pm (grader Elise)

Problem 1 [10 points]
See Discussion Question 2.4 in Notes 2-4.
Problem 2 [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)
Problem 3 [20 points]
a) Consider a spherical shell of radius R, with uniform surface charge density σ_o, centered on the origin. The shell is spining counterclockwise about the z axis with angular velocity ω. Find the magnetic vector potential A(r), far from the sphere, using the magnetic dipole approximation. Find the magnetic field B within this approximation.
b) Using the method of separation of variables, as applied to the scalar magnetic potential φ_M, find an expression for the exact magnetic field B both inside and outside the spining charged shell of part (a). How does your answer for the field outside compare with that obtained by the magnetic dipole approximation in part (a)?