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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 416G \\
    Problem Set \#8, Rev. A, due Friday, 2014-11-14 }
\end{center}

\vspace{.3 in}

{\bf 1.}~~We will use a spherical boundary value problem to develop a more
systematic derivation of the {\bf multipole expansion} than that presented in
the text.  Consider the potential $V(r,\theta)$ both inside and outside a
spherical shell of radius $r'$ due to a ring of total charge $q$ on the shell
at polar angle $\theta=\theta'$.  In the limit $\theta'\to 0$, it is a point
charge at the top of the sphere.

{\bf \quad a)} Write down a formula for $\sigma(\theta)$ of the ring charge
using a delta function $\delta(\theta-\theta')$.  Hint: make sure that $\int
\sigma\, da = q$ if the region includes the ring charge, and 0 otherwise.  In
the same way, express the surface charge as a function of $x=\cos\theta$.
Confirm the the change-of-variables formula for delta functions:
$\delta(\theta-\theta')d\theta = \delta(x-x')dx$ where $x'=\cos\theta'$.

{\bf \quad b)} Solve for $V(r,\theta)$ both inside and outside the shell due
to a uniform ring of charge $q$ on the shell at the angle $\theta=\theta'$.
Compare your answer for a point charge at the north pole ($\theta'=0$) with
$V(r)= q/4\pi\epsilon_0 \rcurs$ to derive the addition formula of Griffiths
Eq. 3.94.

{\bf \quad c)} Substitute $q\to \int \dslash q'$ to obtain the general
multipole expansion for an azimuthally symmetric charge distribution with
multipoles $Q^{(\ell)}_{\rm ext,int}$ discussed in class:
\begin{eqnarray}
  \label{ext}
  V_{\rm ext}(r,\theta) = \frac{1}{4\pi\epsilon_0} \sum_{\ell=0}^{\infty} 
  Q^{(\ell)}_{\rm int} \frac{P_\ell(\cos\theta)}{r^{\ell+1}}
  &\quad\mbox{where}\quad&
  Q^{(\ell)}_{\rm int} = \int \dslash q' r'^\ell P_\ell(\cos\theta') \\
  \label{int}
  V_{\rm int}(r,\theta) = \frac{1}{4\pi\epsilon_0} \sum_{\ell=0}^{\infty} 
  Q^{(\ell)}_{\rm ext} \, r^\ell P_\ell(\cos\theta)
  &\quad\mbox{where}\quad&
  Q^{(\ell)}_{\rm ext} = \int  \dslash q' \frac{P_\ell(\cos\theta')}{r'^{\ell+1}}
\end{eqnarray}
Note that the external multipole potential is only valid outside the entire
charge distribution, while the internal multipole potential is only valid
completely inside the charge distribution.

{\bf \quad d)} Identify monopole, dipole, and quadrupole terms, and compare
with Griffiths.  Sketch equipotentials and field lines of both the internal
and external dipole.  Match each multipole with the corresponding coefficient
$A_\ell$ or $B_\ell$ in the general solution from separation of variables.

{\bf 2.}~Calculate the (external) {\bf quadrupole tensor} of two dipoles $\pm
\bs p$ separated by a displacement $\bs d$ (pointing from $-\bs p$ to $+\bs
p$).  Consider two cases: a) $\bs p = p \hat{\bs x}$ and $\bs d = d \hat{\bs
  y}$; and b) $\bs p = p \hat{\bs x}$ and $\bs d = d \hat{\bs x}$.  The other
cases follow the same pattern.

\quad Hint: construct each dipole $\bs p$ with two opposite charges separated
appropriately, and calculate the quadrupole moment of all four point charges.

\quad Bonus: a) calculate the general formula for $\bs Q(\bs p, \bs d)$; b)
compare and contrast the quadrupole tensor $\bs Q$ with the moment of inertia
tensor $\bs{\mathcal I}$; c) compare with the hydrogen atom 3d orbitals.

Also, Griffiths 3ed[4ed] Ch.~3, \#%
% 17[18], %p144[149], sphere V0=const
% 19[20], %p144[149], sphere V0=k cos(3 theta)
% 22[23], %p144[150], sphere charge from V(theta)
33[36], %p155[160], field of pure dipole
%More problems on Chapter 3:
38[44], %p156[162], line charge z=-a..a
41[47], %p156[163], average field inside a sphere
%42?[extra credit],
bonus:
45[52], %p158[165], quadrupole expansion
49[56]. %p159[166], dipole pendulum

\end{document}
