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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 416G \\
    Problem Set \#4 (Rev. E) due Wednesday, 2011-10-05 }
\end{center}

\vspace{.3 in}

1. In class are discussing five equivalent formulations of electrostatics:
\quad (I)~$\bs E = \int\dslash q'\,\hat{\!\scripty{r}}/4\pi\epsilon_0
\scripty{r}^2$;\\
~(II) $\Phi_E = Q/\epsilon_0$, $\mathcal{E}_E = 0$;
~(III) $\nabla\cdot\bs E = \rho/\epsilon_0$,
$\nabla{\times}\bs E = \bs 0$;
~(IV) $V = \int\dslash q' /4\pi\epsilon_0 \scripty{r}$;
\mbox{~(V) $\nabla^2 V = -\rho/\epsilon_0$.}
In this exercise we explore the connections between the first three and one
new formulation.  We will save the last two %and (I) $\Leftarrow$ (II)
for the next problem set.

\quad a) Show that (I) $\Rightarrow$ (II) by integration of $\bs E$ over an
arbitrary closed surface and path.

\quad b) Show that (I) $\Leftarrow$ (II) by the superposing the fields
obtained by applying Gauss' law on point charge elements $\dslash q'$ at each
point $\bs r'$ in space.

\quad c) Show that (I) $\Rightarrow$ (III) by taking the divergence and curl
of (I).

\quad d) Show that (II) $\Leftrightarrow$ (III).

\quad e) Derive a sixth formulation (VI) by calulating $\nabla^2 \bs E$.
Hint: take derivatives of the two equations in (III).  Thus you are showing
that (III) $\Rightarrow$ (VI).  This completes the symmetry of the
differential vs. integral formulations so that \mbox{(I) $\Leftrightarrow$
  (VI)} corresponds to \mbox{(II) $\Leftrightarrow$ (III)} and \mbox{(IV)
  $\Leftrightarrow$ (V)}.

\quad f) Identify the triangle of six formulas that allow direct calculation
of each of the quantities $\{ \rho,\, \bs E,\, V\}$ from either of the other
two.
                                              

2. Show that the two fields $\bs E_1=r e^{-r^2}\bh r$ and $\bs E_2=
e^{-r^2}\left[ \bh x (x-y) + \bh y (x+y) + \bh z z \right]$ both have the same divergence
$\rho/\epsilon_0 = \nabla\cdot \bs E_1 = \nabla \cdot \bs E_2$.  Using the curl, show which field
is the actual electric field of the charge distribution $\rho$.

\vspace{.166 in}

3. Plot the field lines in the $xy$-plane of a charge $+Q$ at $(a, 0, 0)$ and
a charge $-2Q$ at $(-a, 0, 0)$.  How many points in space are there where $\bs
E = \bs 0$?  Calcuate their positions (if any)?

\vspace{.166 in}

4. A flat sheet of copper carries a constant surface charge density of $\sigma$
on each face. Use Gauss' law to show that the electric field
intensity is $\sigma/\epsilon_0$ just outside the sheet and zero
inside.

\vspace{.166 in}

5. a) Calculate the electric field $\bs E (r)$ for a uniform distribution of
charge $Q$ over a spherical shell of radius $r'$ using
Coulomb's law.  Note the field point $r$ can either be inside or outside the
shell.

\quad b) Calculate the same result using Gauss' law and compare with a).

\quad c) For a spherically symmetric charge distribution $\rho(r) = a e^{-r/r_0}$, calculate the
normalization constant $a$ so that the total charge is $q$ in all of space.

\quad d) Using part a), calculate $\bs E(\bs r)$ for the distribution in c) and compare with Gauss' law.

\vspace{.166 in}

6. Use Gauss' law to calculate the field of two line charges parallel to the
$z$-axis: $+\lambda$ at $(x=a,y=0)$ and $-\lambda$ at $(x=-a,y=0)$.

\vspace{.25 in}

Also, Griffiths chapter 2, problems 4, 5, 9, 10, 18.

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