Physics 415: Electromagnetic Theory I
Prof. S. Teitel firstname.lastname@example.org ----- Fall 2002
Problem Set 4
Due Wednesday, October 23, in lecture
- Problem 1 [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 2 [15 points]
Consider a line charge density (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 (r) to leading order only in the multipole expansion, for each of the following three cases:
a) (z) = ocos(z/2a)
b) (z) = osin(z/a)
c) (z) = ocos(z/a)
- Problem 3[10 points]
Consider a uniformly charged disk of radius R in the xy plane at z=0, centered at the origin. Find the monopole moment, the dipole moment vector, and the quadrapole moment tensor. Use these moments to write an approximation for the potential (r) for r far from the disk. Compare your results with what you found in problem 3b of Problem Set 3.
- Problem 4[15 points]
a) Consider a spherical shell of radius R, with uniform surface charge density o, centered on the origin. The shell is spining counterclockwise about the z axis with angular velocity . 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 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?