Problem Set 4

Problem Set 4

Due Monday, October 25, in lecture

Problem 1 [10 points]
Consider the same infinitesimally thin charged disk of radius R with uniform surface charge density σ_o that you treated in Problem Set 3, problem 5. Placing the origin of your coordinates at the center of the disk, compute the monopole, dipole, and quadrapole moments of this charge distribution. 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 Problem Set 3.
Problem 2 [10 points]
Given the quadrapole tensor Q in the coordinate system r, derive an expression for the quadrapole tensor Q´ in the coordinate system r´, where r´ = r - d (d is a constant displacement vector). Show that Q = Q´ (i.e. that the quadrapole moment is independent of the choice of orign) only if both the monopole and the dipole moments vanish.
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 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 4 [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)?