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\textbf{Phys 742 -- Assignment 3 -- Due 13 Feb 96}
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\smallskip 

\noindent \textbf{1.} (Similar to Jackson's 2.6; see also 2.14d)

A parallel plate capacitor has a small hemispherical boss of radius $a$ on
one of its plates. Assume that $A$ is the area of each plate, $d$ the
separation between them, with $\sqrt{A}>>d>>a.$ To be definite, assume that
the plates are flat circular disks perpendicular to the $z$ axis and put the
boss at the center of the lower plate. However, the result should hold more
generally, as long as $a$ is much smaller than the distance of the boss from
the closest edge.

\textbf{(a) }Where on the lower plate does the maximum electric field occur,
and how much larger than average is it? Comment on the chances of being
struck by lightning in hilly country.

\textbf{(b) }Show that the presence of the boss does not change the
capacitance, to order $a^2/A$. Do you expect higher order corrections to
increase or decrease $C?$

\medskip\ 

\noindent \textbf{2.} (Similar to Jackson's 2.5)

\textbf{(a) }Solve the problem of an uncharged conducting cylindrical shell
of radius $a$ in a uniform field $E_0$ perpendicular to its axis. You can
use the method of images if you wish. More simply, you can superimpose the
external field and that of an induced line dipole, so that the shell is an
equipotential.

\textbf{(b) }If the cylinder is cut into half by a plane perpendicular to
the field, what force per unit length is required to keep the half-cylinders
from separating?

\textbf{(c) }Same question as in (b), when the charge on the shell is $%
\lambda $ per unit length.

\medskip\ 

\noindent \textbf{3.}

Jackson's 2.4, but you can take parts (b) and (d) together and look at
Estevez and Suen, page 42, for guidance. Be sure to correct the misprints,
however

\medskip\ 

\noindent \textbf{4.}

\textbf{(a)} For the problem of Fig. 2.10, what is the behavior of $\Phi
(x,y)$ near the origin, i.e., for $\rho <<a?$ Sketch equipotentials and
field lines in this (two-dimensional) corner.

\textbf{(b)} For the same geometry as in (a), put $\Phi =0$ on the planes $%
x=0$ and $y=0,$ $\Phi =V$ on the plane $x=a.$ What is now the behavior of $%
\Phi (x,y)$ near the origin? Sketch equipotentials and field lines in this
(two-dimensional) corner.

\textbf{(c)} For the problem of Fig. 2.9 with $a=b=c$ (for a cube), 
what is the behavior of $\Phi
(x,y,z)$ near the origin? What are the charge densities on the three faces
as they meet in this (three-dimensional) corner?

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