Phys 742 – Assignment 4 – Due 22 Feb 96 1.

The charge density on a spherical shell of radius textita is σ_{0} cos θ. Find the potential and the electric field, both inside and outside the shell. Use spherical harmonics (just Legendre polynomials, really).

(This problem is related to that of a conducting sphere in a uniform field, and is also good training for the next problem.) 2.

The Hydrogen atom in the 1p_{0} state has electron charge density Az^{2}e^{−αr}.

(a) Find the electrostatic potential by using Coulomb’s law and the expansion (3.70).

(b) Obtain the same result as in (a) by fourier-transforming the charge density, using the convolution theorem, and fourier-transforming back.

For practice, you may want to solve by these methods Jackson’s problem 1.1 ”in reverse” from Assignment 1. Do not forget the nucleus. 3.

Using a complex potential proportional to \sqrt{a^{2} − z^{2}}, solve the problem of a slit of width 2a in a conducting plane having charge density σ_{0} far from the slit. (Put the slit, in cross section, on the (-a, a) segment of the x axis). Find the field on the y axis and in the slit (y = 0, |x| < a). Find the charge density on the conducting plane.

———————— ———————— → x
−a a

4.

Far below a conducting plane with a slit there is a uniform field E_{0} perpendicular to the plane. Find the field on the y axis and in the slit (y = 0, |x| < a). How much electric flux leaks through the slit? (As above, put the slit on the (-a,a) segment of the x axis. The applied field is in the y direction).