Physics 415: Electromagnetic Theory I
Prof. S. Teitel firstname.lastname@example.org ----- Fall 2003
Problem Set 4
Due Monday, October 27, in lecture
- Problem 1 [10 points]
Consider a charge +q lying at position +s/2 on the z axis, and a charge -q lying at position -s/2 on the z axis.
a) Using Coulomb's law, write down an exact expression for the potential (r) at any point r. Express your answer in spherical coordinates.
b) When s<<r, one can approximate this exact expression by a power series expansion in (s/r).
Derive this expansion directly by doing a Taylor series expansion of each Coulomb term in the exact expression. Carry out the expansion to order (s/r)3.
c) You should be able to express the angular dependence of this expansion in terms of the Legendre polynomials. From this expansion, identify the dipole, quadrupole, and octopole terms.
- Problem 2 [10 points]
Four charges are positioned as follows:
|+3q||at +d along z axis|
|+q||at -d along z axis|
|-2q||at +d along y axis|
|-2q||at -d along y axis|
Compute the potential of this distribution using the multipole expansion up through the quadrupole term. Express your answer in spherical coordinates.
- 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?