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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 416G \\
    Problem Set \#5, Rev. A, due Friday, 2014-10-17 }
\end{center}

% \vspace{.2 in}

{\bf 1.}~(continuation of HW 4 \#1).  Consider the electric potential $V(\bs
r)$ defined by $\bs E = - \nabla V$.

{\bf \quad a)} Show using one of the boundary theorems that the above
definition is equivalent to $V(\bs r) = -\flow{E} \equiv -\int_{\bs r_0}^{\bs
  r} \bs E \cdot \bs{dl}$.  We call $\bs r_0$ the ground point and $V(\bs
r_0)=0$ the ground potential.  In residential wiring, this point ties to a
physical rod driven deep into the earth.

{\bf \quad b)} Show that (III) implies that $V(\bs r)$ exists, and that (III)
$\Rightarrow$ (V).  Note that the two first-order differential equations (III)
are equivalent to the single second-order differential equation in $V$.

{\bf \quad c)} Using the definition in a), integrate the flow of (I) over an
arbitrary path in $\bs r$ starting at $r_0=\infty$ to show that \mbox{(I)
  $\Rightarrow$ (IV).}  Conversely, calculate the gradient of (IV) to derive
equation (I) from equation (IV).  We say that
$\hrcurs\cdot\bs{dl}/\rcurs\!\!^2=-d(1/\rcurs)$ is a perfect (exact)
differential.

{\bf \quad d)} Calculate the Laplacian in (IV), both directly and from the
divergence of (I) (HW 4), to show that (IV) $\Rightarrow$ (V) and (I)
$\Rightarrow$ (V).

{\bf \quad e)} Show that (IV) $\Leftarrow$ (V) by the decomposition of
$\rho(\bs r)$ into an integral (`forest') over $\dslash q'$ of delta functions
(`poles') and applying $\nabla^{-2} \delta^3(\hrcurs)$ to get a Green's
function (point potential) for each charge.

{\bf \quad f)} Rework the proof of the Helmholtz theorem and use it to show
that (III) $\Rightarrow$ (I).  Note that you need to use both equations of
(III) in the Helmholtz theorem to obtain (I).  The calculation pieces together
parts b), e), and c), involving the tortuous path (III) $\Rightarrow$ (V)
$\Rightarrow$ (IV) $\Rightarrow$ (I).  Use the Helmholtz theorem to prove one
more time that $\bs E(\bs r)$ admits a potential $V(\bs r)$.

{\bf \quad g)} Identify the six formulas used to calculate any field in the
derivative chain $V \stackrel{d}{\to} \bs E \stackrel{\delta}{\to} \rho$ in terms of either of the other two
fields.

\quad Bonus: repeat above for the new formulation (VI): show the four steps of
\mbox{(I) $\Leftrightarrow$ (VI) $\Leftrightarrow$ (V)}.

{\bf 2.}~Sketch the field lines and potentials in the $xy$-plane for: {\bf a)}
a system of three point charges: charge $+q$ at $(0,0)$, charge $-q$ at
$(1,0)$, and charge $-q$ at $(0,1)$; {\bf b)} charge $+q$ at $(a, 0, 0)$ and
charge $-2q$ at $(-a, 0, 0)$.  For b), how many points in space are there
where $\bs E = \bs 0$?  Calcuate their positions, if any.

{\bf 3. a)} Show that the potential due to two parallel line charges of
opposite polarity is $V=\frac{\lambda}{2\pi\epsilon_0} \ln(\frac{s_1}{s_2})$
where $s_1$ is the distance from the field point to the negative line, and
$s_2$ is the distance to the positive line. Hint: use the midpoint of the two
line charges as ground.

{\bf \quad b)} Determine the shape and equation of the equipotential surfaces,
assuming that the negative line charge is along the z-axis at $(x,y)=(-a,0)$
and the positive line charge is at $(+a,0)$.

{\bf 4.}~The Spallation Neutron Source (SNS) at Oak Ridge National Laboratory
accelerates protons to an energy of 1~GeV, to spall (smash) neutrons out of a
mercury target, yielding on average 30 free neutrons per incident proton.  At
the nominal beam current is 1.4~mA, how many neutrons are `produced' per
second?  How much power is dumped into the target?

{\bf 5.}~Calculate the potential energy of a cube of length $a$ on each side
with a point charge $q$ at each of the eight vertices.

Also, Griffiths 3ed[4ed] Ch.~2, \#%
%16 %p75, coaxial cable \qquad
20[20], %p79[80], impossible field \qquad
21[21], %p82[83], potential of sphere \qquad
26[26], %p87[87], conical surface \qquad
%29 %p87,formulation proof \qquad
%31 %p93,like #5 \qquad
32[34], %p95[95], energy of uniform sphere \qquad
34[36], %p96[97], energy two spheres \qquad
39[43], %p106[107], coax capacitors \qquad
46[50], %p107[108], yukawa potential \qquad
bonus:
48[53], %p107[109], Child-Langmuir law space charge \qquad
49[54], %p108[110], beyond Coulomb \qquad
51[56], %p109[111], gravitational potential \qquad
52[57]. %p109[111], elipsoid \qquad
\end{document}
