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\centerline{\bf GAUSS'S LAW}
\medskip

Each group will be given one of the charge distributions given below:
($\alpha$ and $k$ are constants with appropriate dimensions.)
\smallskip

\begin{enumerate}
\item A positively charged (dielectric) spherical shell of inner radius $a$
and outer radius $b$ with a spherically symmetric internal charge density
$\rho(\vec r)=\alpha\, r^3$.\\

\item A positively charged (dielectric) spherical shell of inner radius $a$
and outer radius $b$ with a spherically symmetric internal charge density
$\rho(\vec r)=\alpha\, e^{(kr)^3}$.\\

\item A positively charged (dielectric) spherical shell of inner radius $a$
and outer radius $b$ with a spherically symmetric internal charge density
$\rho(\vec r)=\alpha\, {1\over r^2}\, e^{kr}$.\\

\item An infinite positively charged (dielectric) cylindrical shell of inner radius $a$
and outer radius $b$ with a cylindrically symmetric internal charge density
$\rho(\vec r)=\alpha\, r^3$.\\

\item An infinite positively charged (dielectric) cylindrical shell of inner radius $a$
and outer radius $b$ with a cylindrically symmetric internal charge density
$\rho(\vec r)=\alpha\, e^{(kr)^2}$.\\

\item An infinite positively charged (dielectric) cylindrical shell of inner radius $a$
and outer radius $b$ with a cylindrically symmetric internal charge density
$\rho(\vec r)=\alpha\, {1\over r}\, e^{kr}$.

\end{enumerate}

\bigskip

For your group's case, answer each of the following questions:
\begin{enumerate}
\item  Use Gauss's Law and symmetry arguments to find the electric field at
each of the three radii below:
\begin{enumerate}[(i)]
\item  $r_1>b$
\item  $a<r_2<b$
\item $r_3<a$
\end{enumerate}

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\item  What dimensions do $\alpha$ and $k$ have?
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\item For $\alpha=1$, $k=1$, sketch the magnitude of the electric field as a
function of $r$.


\end{enumerate}

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