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\begin{center}
  {\bf University of Kentucky, Physics 416G \\
    Problem Set \#7, Rev. A, due Friday, 2014-11-07 }
\end{center}

\vspace{.3 in}

{\bf 1.~Sturm-Liouville equation}.  In class we saw that the Fourier series
$\sin(kx)$ and $\cos(kx)$, and Legendre polynomials $P_\ell(\cos\theta)$ are
all orthogonal.  This holds in general for a large class of ODEs, including
those from separation of variables of the Laplacian.  We will prove this
result using the same technique used for symmetric matrices (see the
\href{http://www.pa.uky.edu/~crawford/phy416_fa14/notes_01_1b.pdf}{class
  notes} on eigenvectors).

Let $y_1(x)$ and $y_2(x)$ be two solutions ({\it eigenfunctions}) of the
differential equation $L[y_i]=\lambda_i \, y_i$ with real eigenvalues
$\lambda_i$, where $L$ is a {\it linear differential operator} of the form
\begin{equation}
  L[y(x)] \equiv \frac{1}{w(x)}\left[-\frac{d}{dx} p(x) \frac{d}{dx}
 + q(x)\right] y(x)% = \lambda \, y(x)
\end{equation}
acting on the function $y(x)$ with boundary conditions $y(a)=y(b)=0$.  We will
show that these eigenfunctions are {\it orthogonal},
\begin{equation}
\langle y_1|y_2\rangle \equiv \int_a^b w dx \, y_1 y_2 = 0 \where[if]
\lambda_1 \neq \lambda_2.
\end{equation}
Note that $L$ is just a shorthand for the above derivatives, but is analogous
to a matrix operator, and the {\it inner product} $\langle y_1|y_2\rangle$ is also a
shorthand for the above integral, which is analogous to the vector dot
product.  $y_1$, $y_2$, $p$, $q$, and $w$ are all functions of $x$, and
$\lambda_1$ and $\lambda_2$ are real numbers.

{\bf \quad a)} Show by integration of parts that $\int_a^b y_1 \frac{d}{dx}
y_2 dx = -\int_a^b y_2 \frac{d}{dx} y_1 dx$.

{\bf \quad b)} Extend part a) to show that $\int_a^b y_1
\frac{d}{dx}p\frac{d}{dx} y_2 dx = \int_a^b y_2 \frac{d}{dx}p\frac{d}{dx} y_1
dx$.  Note that each derivative operator acts on everything to the right.

{\bf \quad c)} Use part b) to show that $\int_a^b w dx \, y_1 L [y_2] \, -
\int_a^b w dx \, y_2 L [y_1] =0$.  We say that the operator $L$ is {\it
  self-adjoint} with respect to the weight $w dx$.  It is the analog of a {\it
  symmetric} matrix.

{\bf \quad d)} Replace each operator in part c) with its eigenvalues, using
$L[y_i]=\lambda_i\,y_i$ and factor out the inner product $\langle
y_1|y_2\rangle$, to conclude that $y_i$ and $y_2$ are orthogonal if $\lambda_1
\neq \lambda_2$.

\quad Bonus: give other examples of self-adjoint operators with orthogonal
eigenfunctions, specifying the integration interval and weight function
associated with each.

{\bf 2.}~~A {\bf spherical capacitor} of radius $R$ is made of two
hemispherical conductors separated by a negligible gap.  The top hemisphere is
held at potential $+V_0$ and the bottom hemisphere at $-V_0$.  Evaluate the
first two non-zero terms in each case.

{\bf \quad a)} Calculate the potential everywhere inside and outside the
sphere.  Note that both regions must be solved indpendently, because of the
``external'' boundary condition $V=\pm V_0$ at $r=R$.

{\bf \quad b)} Calculate the surface charge density on both the inside and
outside of the sphere, and integrate the total charge on either hemisphere to
determine the capacitance.

{\bf \quad c)} Draw the equipotentials and field lines, demonstrating the
dominant features of the lowest non-zero multipole, both inside and outside
the sphere.

Also, Griffiths 3ed[4ed] Ch.~3, \#%
%12[13], %p136[140], inf slot, two metal strips
%13[14], %p136[140], inf slot, charge density 
%14[15], %p136[140], rect pipe
15[16], %p136[141], cubic box
23[24], %p145[150], sep cyl coords
24[25]. %p145[150], cyl pipe

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