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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 416G \\
    Problem Set \#5 (Rev. C), due Wednesday, 2011-10-12 }
\end{center}

\vspace{.2 in}

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

\quad a) Show that (III) implies $V(\bs r)$ is well-defined, and that (III)
$\Rightarrow$ (V).  Note that the two first-order differential equations (III)
are equivalent to the single \mbox{second-order differential equation (V).}
%%% do b) first, since solution of a) requires b) or g)

\quad b) Show that the above definition is equivalent to $V(\bs r) = -\flow{E}
\equiv -\int_{\bs r_0}^{\bs r} \bs E \cdot \bs{dl}$.

\quad c) Using the definition of b), integrate (I) over an arbitrary path in
$\bs r$ to show that \mbox{(I) $\Rightarrow$ (IV).}

\quad d) Evaluate the gradient of (IV) to show that (IV) $\Rightarrow$ (I).
Thus $\bh\scriptr/\scriptr^2$ is a perfect differential.

\quad e) Continue calculating the Laplacian of (IV), both directly and by
taking the divergence of (I), to show that (IV) $\Rightarrow$ (V) and (I)
$\Rightarrow$ (V).

\quad f) Show that (IV) $\Leftarrow$ (V) by the decomposition $\rho(\bs r)$
into an integral over $\dslash q'$ of delta functions and finding $\nabla^{-2}
\delta^3(\bv\scriptr)$ (a Green's function).

\quad g) 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
a), f), and d), involving the tortuous path (III) $\Rightarrow$ (V)
$\Rightarrow$ (IV) $\Rightarrow$ (I).
%%% rederive the fact that E admits a potential V

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

2.~Sketch the field lines and potentials for the system of three point
charges: charge $+q$ at $(0,0)$, charge $-q$ at $(1,0)$, and charge $-q$ at
$(0,1)$, all in the $xy$-plane.

3.~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?

4.~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 corresponding distance to the positive line. If your answer is
0/0, try another method.

\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 $(+a,0)$.

5. A spherical charge distribution inside a sphere of radius $r_0$ is
$\rho(r) = \rho_0(1-r^2/r_0^2)$.

\quad a) Calculate the total charge $Q$.

\quad b) Calculate the electric field $\vec{\bs E}$ both inside and outside
the charge distribution

\quad c) Calculate the electric potential $V$ both inside and outside the
charge distribution.

\quad d) At what radius is the magnitude of electric field the greatest?

\quad e) How does the the potential behave at that radius?

6.~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 chapter 2, problems 
16, % p75, coaxial cable
20, % p79, impossible field
21, % p82, potential of sphere
26, % p26, conical surface
%29 formulation proof
%31 like #5
32, % p95, energy of uniform sphere
34, % p96, energy two spheres
39, % p106, coax capacitors
46, % p107, yukawa potential
bonus: 
48, % p107, Child-Langmuir law space charge
49, % p108, beyond Coulomb
51. % p109, gravitational potential

\end{document}
