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\textbf{Phys 742 -- Assignment 1 -- Due 30 Jan 96}
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\vspace{0.1in}

\textbf{1.} (Related to Jackson's problem 1.1)

(a) A point charge \textit{q} is located at the center of a conducting spherical
shell having inner radius $R_1$ and outer radius $R_2.$ Find the electric
field everywhere and the surface charge on the inner and outer surfaces of
the shell, when the shell is grounded.

(b) Same questions as in (a), when the shell is insulated and carries no net
charge.

(c) If the point charge \textit{q} is located off-center at $x=R_1/2$, what is 
the electric field for $r>R_1$ when the shell is grounded? Consider separately
the two regions $R_1<r<R_2,\;R_2<r\,$ and compute the field in each region.

(d) Same questions as in (c), when the shell is insulated and carries no net
charge.

\smallskip\ 

\textbf{2.} (Similar to Jackson's problem 1.4)

Each of three charged cylinders of radius \textit{a} and length $>>a,$ one
conducting, one having a uniform charge density within its volume, and one
having a cylindrically symmetric charge density which varies radially as 
$\rho ^n\ (n>-2),$ has a total charge $\lambda $ per unit length. Use
Gauss's theorem to obtain the electric fields both inside and outside each
cylinder. Sketch the behavior of the fields as a function of radius for the
first two cylinders, and for the third with $n=-1.$

\smallskip\ 

\textbf{3.}

(a) Jackson's problem 1.2 gives a possible representation of the Dirac $%
\delta $ function. List (at least) three other representations of $\delta ,$
correctly normalized. You are likely to have encountered already the square
box, the lorentzian and the sinc function representations, for instance. But
you can also be creative.

(b) Using the result of Jackson's problem 1.2, express the three-dimensional 
$\delta $ function in cylindrical coordinates $(\rho ,\phi ,z)$ and in
spherical coordinates $(r,\theta ,\phi ).$

(c) Check that in both cases of part (b) the result is of the form 
\[
\delta (\mathbf{x}-\mathbf{x}^{\prime })=\frac 1J\,\delta (u-u^{\prime
})\delta (v-v^{\prime })\delta (w-w^{\prime }) 
\]
where \textit{J}\/ is a Jacobian. The Jacobian of what, precisely? (Of course,
1/\textit{J} is the inverse Jacobian). Is this equation valid for nonorthogonal
coordinates \textit{(u,v,w)} as well?

\smallskip\ 

\textbf{4.}

Do Jackson's problem 1.5 in reverse. Start with the time-averaged electron
charge density 
\[
\rho (r)=-q\,\frac{\alpha ^3}{2\pi }\,e^{-\alpha r}
\]
and a point charge \textit{q} at the origin, compute the radial electric field
\textit{E} by Gauss's theorem, and then the potential $\Phi(r)$ by integration.
You should get Jackson's $\Phi .$

What is the numerical value of \textit{E} in Volts per centimeter at the
typical atomic distance of $r=$ 1 \AA ngstrom? Supply the value of $\alpha $
for Hydrogen from the store of your knowledge.

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