Find the potential and the electric field, both inside and outside
the shell. Use spherical harmonics (just Legendre polynomials, really).
1.
The charge density on a spherical shell of radius a is 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 
state has electron charge
density 
(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 
solve the
problem of a slit of width 2a in a conducting plane having charge
density 
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 (
). Find the charge density on the
conducting plane.
4.
Far below a conducting plane with a slit there is a uniform field 
perpendicular to the plane. Find the field on the y axis and in the
slit 
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).