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\begin{document}
\begin{center}
  {\bf Arizona State University, Physics 311 \\
    Problem Set \#4, Rev. A, due Wednesday, 2014-02-12 }
\end{center}

\vspace{.2 in}

{\bf 1. a)} An amount of charge $Q$ is uniformly distributed over a spherical
shell of radius $r'$.  Integrate Coulomb's law over this charge distribution
to calculate the electric field $\bs E (r)$ everywhere (both inside and
outside the shell).

{\bf \quad b)} Compare part b) with the field of the same spherical charge
distribution, this time calculated using Gauss' law.

{\bf \quad c)} Given a spherically symmetric charge distribution $\rho(r) =
\rho_0(3-2r^2/r_0^2) e^{-r^2/r_0^2}$, where $\rho_0$ and $r_0$ are constants,
plot $\rho(r)$ and show that the total charge everywhere is zero.  Bonus: what
is the total positive charge?

{\bf \quad d)} Explain why the charge in a shell of radius $r'$ and thickness
$dr'$ is $\dslash q = \rho(r') 4\pi r'^2 dr'$.  Calculate the electric field
of the charge distribution in part c) by substituting $\dslash q$ for $Q$ in
part a) and integrating over $r'$.

{\bf \quad e)} Calculate the divergence of the electric field $\bs E(r)$ of
part d) and compare your \mbox{answer with c).}

{\bf \quad f)} Show that the impostor field $\bs E'(r)= \rho_0/\epsilon_0
\left[ \bh x (x-y) + \bh y (x+y) + \bh z z \right]e^{-r^2/r_0^2}$ has the same
divergence, but calculate $\bs\nabla\times\bs E'$ to show that is cannot be
the electric field for $\rho(r)$.

\vspace{.25 in}

Also, Griffiths 3ed[same in 4ed] Ch.~2, \#%
4,  %p64, square loop\qquad
5,  %p64, circular loop\qquad
6,  %p64, circular disk\qquad
%7,  %p64, spherical shell\qquad %see #5 above
9,  %p69, rho(E)\qquad
%10, %p69, flux cube\qquad
16, %p75, coaxial cable \qquad
18. %p75, overlapping spheres \qquad

\end{document}
