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\begin{document}
\begin{center}
  {\bf Arizona State University, Physics 311 \\
    Problem Set \#7, Rev. A, due Wednesday, 2014-03-19 }
\end{center}

\vspace{.3 in}

{\bf 1.}~~An infinite {\bf cylindrical rod} of radius $R$ centered in the
$xy$-plane and extending along the $z$-axis has a uniform polarization $\bs P
= P \bh x$.

\quad a) Calculate the bound surface charge density $\sigma_b$.

\quad b) Calculate $\bs E_b$ (due to the bound charge) inside and outside the
\mbox{cylinder as a function of $\bs P$.}

\quad c) Suppose the rod is placed in an external electric field $\bs
E_0=E_0\bh x$, and that $\bs P = \chi_e \epsilon_0 \bs E$, where $\bs E=\bs
E_0 + \bs E_b$ is the total electric field inside the rod.  Calculate the
values of $\bs E$, $\bs P$, and $\bs D$ inside and outside the rod.

\quad d) Show that $\epsilon_0 E_{2n}- \epsilon_0 E_{1n}=P_n$ and that
$D_{2n}-D_{1n}=0$ for your solution.

\quad e) Solve the same problem (part c) as a boundary value problem with
dielectric constant $\epsilon_r = 1 + \chi_e$ inside the rod and $\sigma_f=0$
on the surface (only bound surface charge).  Show that the two answers are
consistent.
                               
\vspace{.3cm}

2.~~In a {\bf capacitive touch-screen}, your dielectric finger modifies the
capacitance between two strips of the transparent conductor indium tin oxide
(ITO).  We will model a single cell (from the middle of one strip to the
middle of the next) of this display as an semi-infinite rectangle extending
upward from the strips.  Use the boundary conditions $\partial V/\partial x =
0$ on the sides at $x=\pm a$ to ensure the correct periodic symmetry.
                                      
\vspace{-.3in}\hspace{.5in}
\includegraphics[height=1in,trim=0 -.28in 0 .28in]{ps_07_fig1}
\hspace{.5in}
\includegraphics[height=2in]{ps_07_fig2}
\vspace{-.1in}

\quad a) Solve a boundary value problem for the potential when a finger is
touching the glass.  Assume the glass thickness is $d$, and your finger of
pure water ($\epsilon_r^2\approx 80$) fills the whole region beyond the glass.
Calculate the first two nonzero terms of the Fourier expansion.  Set
$\epsilon_r=1$ to determine the solution with a finger touching the glass.

\quad b) Plot the equipotentials and field lines in the region with and
without your finger.

\quad c) Calculate the capacitance/length with and without your finger, as a
function of $d/a$.  Don't forget fields on both the top and bottom of the
strip.  What glass thickness do you recommend?

\quad d) If the voltage on each wire is $\pm V_0 = \pm 2 V$, what difference
in charge must the device be able to measure?  ($d=a=3$~mm, 1~cm long) If the
strips were joined into an $LRC$ circuit, what frequency shift $\Delta\omega$
would the device need to discriminate?

\vspace{.25 in}

Also, Griffiths 3ed[4ed] Ch.~4, \#%
8[8],	%p165, p.p energy
16[16],	%p177, sphere,needle,wafer
24[24],	%p190, dielectric shell
29[29],	%p197  p-p force
33[36].	%ppsnell-type law

\end{document}
