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Physics 415: Electromagnetic Theory I
Prof. S. Teitel stte@pas.rochester.edu ----- Fall 2002

Problem Set 3

Due Monday, October 14, in lecture

• Problem 1 [10 points total]

  A surface charge density

  () = a sin(5)

  is glued over the surface of an infinite cylinder of radius R ( is the polar angle).

  a) Find the potential inside (r<R) the cylinder. [5 pts]

  b) Find the potential outside (r>R) the cylinder.[5 pts]

• Problem 2 [15 points total]

  The potential at the surface of a sphere of radius R is fixed at

  (R, ) = k cos(3)

  where k is some constant.

  a) Find the potential (r, ) inside (r<R) the sphere. [5 pts]

  b) Find the potential (r, ) outside (r>R) the sphere.[5 pts]

  c) Find the surface charge density () on the surface of the sphere.[5 pts]

  (Assume that there is no charge inside or outside the sphere.)

• Problem 3 [25 points total!]

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

  (r) = da´

  (r´) |r - r´|

  do the integral to find the exact value of the potential 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 azimuthal symmetry, we can express the solution for the potential (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. (Hint: read Jackson section 3.3) [15 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 _o, as we know from general principles that it must be. [5 pts]