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\textbf{Phys 742 -- Assignment 4 -- Due 22 Feb 96}

\noindent \textbf{1.}

The charge density on a spherical shell of radius \textit{a} is $\sigma _0\cos
\theta .$ 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.)


\noindent \textbf{2.}

The Hydrogen atom in the $1p_0$ state has electron charge
density $Az^2e^{-\alpha r}.$

\textbf{(a)} Find the electrostatic potential by using Coulomb's law and the
expansion (3.70).

\textbf{(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.


\noindent \textbf{3.}

\ Using a complex potential proportional to $\sqrt{a^2-z^2},$ solve the
problem of a slit of width \textit{2a} in a conducting plane having charge 
density $ \sigma _0$ far from the slit. 
(Put the slit, in cross section, on the (\textit{-a, a}) segment of the 
\textit{x} axis). Find the field on the \textit{y} axis and in the
slit ($y=0,\ \left| x\right| <a$). Find the charge density on the
conducting plane.

\bigskip\ 

\qquad \qquad ------------------------\qquad ------------------------ $\quad
\rightarrow $ $\,x$

\qquad \qquad \qquad \qquad \qquad $\quad -a$\quad $\;a$


\noindent \textbf{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 \textit{y} axis and in the 
slit $( y=0,\ \left| x\right| <a).$ 
How much electric flux leaks through the slit? (As above, 
put the slit on the (\textit{-a,a}) segment of the \textit{x} axis. 
The applied field is in the \textit{y} direction).

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