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\begin{center}
  {\bf Arizona State University, Physics 311 \\
    Problem Set \#6, Rev. A, due Wednesday, 2014-03-05 }
\end{center}

\vspace{.3 in}

{\bf 1. Sagging tent potential.}  Solve Laplace's equation for the
potential $V(x,y)$ in the region $-a{<}x{<}a$ and $-b{<}y{<}b$ with boundary
conditions $V(x,\pm b) = 0$ and $V(\pm a, y) = V_0 (1-|y/b|)$.  Sketch the
solution.

{\bf 2.} The purpose of this exercise is to systematically develop the {\bf
  multipole expansion} in cylindrical coorindates using a boundary value
problem.  We will solve Laplace's equation for the potential $V(r,\theta)$
inside and outside a spherical shell of radius $r'$, due to a uniformly
charged ring of total charge $q$ at the polar angle $\theta'$ on the shell.
In the limit $\theta'\to 0$, it becomes a point charge at the top of the
sphere.

{\bf \quad a)} Write down a formula for the surface charge density
$\sigma(\theta)$ of the ring charge distribution using a delta function of
$\theta$. Normalize it so that the total charge is $\int \sigma\, da = q$.

{\bf \quad b)} Repeat part a) in terms of the variable $x=\cos\theta$.
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 c)} Solve $V(r,\theta)$ both inside and outside the shell due to
the above ring charge distribution.

{\bf \quad d)} Calculate the potential in the limit $\theta'\to 0$, noting
that $P_\ell(1)=1$ always.  Compare your answer with $V(r)= q/4\pi\epsilon_0
\rcurs$ to derive the addition formula [Griffiths Eq. 3.94].

{\bf \quad e)} Substitute $q\to \int \dslash q'$ and factor out the resulting
integral over the charge distribution to obtain the following multipole
expansions for an azimuthally symmetric charge distribution:
\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; otherwise the potential will have a
mixture with both contributions.

{\bf \quad f)} Compare this result with Griffiths Eqs. 3.95, 3.97, and
3.98--99.  Identify the terms corresponding to the monopole, dipole, and
quadrupole terms.  What is potential inside an external monopole (uniform
shell of charge)?  What is the correspondence between the internal/external
multipoles and the coefficients $A_\ell$ and $B_\ell$ of the general solution
to Laplace's equation by separation of variables in cylindrical coordinates?

\vspace{.25 in}

Also, Griffiths 3ed[4ed] Ch.~3, \#%
%12, %p136, inf slot, two metal strips
13[14], %p136, inf slot, charge density
%14, %p136, rect pipe
%15[16], %p136, cubic box
22[23], %p150, sph capacitor
%23[24], %p145, sep cyl coords
24[25], %p145, cyl pipe
% 41, %p156, average field inside a sphere
45[52]a,b. %p158, quadrupole expansion

\end{document}
