Problem Set 4

Problem Set 4

Due Wednesday, October 23, in lecture

Problem 1 [30 points]
a) Consider an infinitesmally thin charged disk of radius R and uniform surface charge density σ_o, that is lying in the xy plane centered on the orign. Using the Coulomb formula for the potential,

φ(r) = ∫ da´ σ(r´)
|r - r´|
do the integral to find the exact value of the potential φ for points along the z axis, both above and below the disk (make sure you get this part correct before proceeding with the following parts!). [5 pts]
b) Since the above problem of the charged disk has rotational symmetry about the z axis, we can express the solution for the potential φ(r) in terms of a Legendre polynomial series using the method of separation of variables. In this case, the "boundary condition" that determines the unknown coefficients of this series is the requirement that the potential agree with the known exact values along the z axis, as computed in part (a). Find the coefficients of this expansion up to order ℓ=5, for |r|>R and |r|<R, both above and below the disk. [10 pts]
c) Using your results of part (b), compute the electric field E just above and just below the disk, and explicitly show that the discontinuity in E is given by the surface charge density σ_o, as we know from general principles that it must be. [5 pts]
d) Now, keeping the origin at the center of the disk, compute the monopole moment, the dipole moment vector, and the quadrapole moment tensor of the charged disk. 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 part (b). [10 pts]
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 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)?