Problem Set 4

Due Wednesday, October 26, in lecture

Skip problem 2, we have not gotten far enough for you to do it now. It will appear again on Problem Set 5!

Problem 1 [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)?
Problem 2 [20 points]
A spherical dielectric shell, with inner radius a, outer radius b, and dielectric constant ε, is placed in a uniform external electric field E_o. Find the electric field outside the shell (r>b), inside the shell (r<a), and in the dielectric (a<r<b). What is the field inside the shell in the limit that ε gets infinitely large?
Problem 3 [10 points]
Consider a "dielectric" material formed by a regular cubic lattice of small conducting spheres. The radius of the spheres is R and the spacing between the spheres is d. Assume d>>R.
Find the linear dielectric constant ε for this material in the presence of an applied field E_o. You may assume that each sphere is only influenced by the externally applied field and not by the other spheres.