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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 416G \\
    Problem Set \#10, Rev. A, due Monday, 2014-12-08 }
\end{center}

\vspace{.1 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 $\bs P
= P \bh x$.

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

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

{\bf \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 (external plus bound) electric field inside the
rod.  Calculate the values of $\bs E$, $\bs P$, and $\bs D$ inside and outside
the rod.

{\bf \quad d)} Check the boundary conditions $\epsilon_0 E_{2n}- \epsilon_0
E_{1n}=P_n$ and $D_{2n}-D_{1n}=0$ for above.

{\bf \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).
                               
\vspace{.3cm}

{\bf 2.}~~A {\bf capacitive touch-screen}, has strips of the transparent
conductive coating indium tin oxide (ITO) of width $a$ under of a glass plate
of thickness $d$.  The strips are at alternating potentials $\pm V_0$.  Your
finger (a dielectric) is detected by the change of capacitance when it touches
the glass.  We will model a single cell in 2-d (symmetric along the length of
the strip) with period boundary conditions from the middle of one strip to the
middle of the next.  Thus the domain is a semi-infinite rectangle extending
upward from the two half-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_10_fig1}
\hspace{.5in}
\includegraphics[height=2in]{ps_10_fig2}
\vspace{-.1in}

{\bf \quad a)} Solve the boundary value problem for the potential without a
finger ($\epsilon_r^1=\epsilon_r^2=1$).  Note that in this case there is no
discontinuity at the glass and the problem can be solved with a single region
extending from $0<y<\infty$.  Calculate the first two nonzero terms of the
Fourier expansion.

{\bf \quad b)} Solve for the potential when a finger (assume pure water,
$\epsilon_r^2\approx 80$) touches the display and fills the entire region
beyond the glass.  Check that your answer reduces to part a) as
$\epsilon_r^2\to\epsilon_r^1$.

{\bf \quad c)} Plot the equipotentials and field lines in the region with and
without your finger.

{\bf \quad d)} 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
strips.  What glass thickness do you recommend?

{\bf \quad e)} If the voltage on each wire is $\pm V_0 = \pm 2 V$, what
difference in charge must the device be able to measure, for $d=a=3$~mm in a
1~cm long strip?

Also, Griffiths 3ed[4ed] Ch.~4, \#%
16[16], %p177[184] sphere, needle, wafer
20[20], %p185[191] rho embedded in dielectric sphere
21[21], %p185[192] dielectric coaxial cable
24[24], %p190[197] bvp: coated conducting sphere
28[28], %p196[204] dielectric force on oil in coaxial capacitor
33[36], %p198[206] Snell's law for dielectric fields
bonus:
29[29], %p197[205] dipole-dipole force Newton's 3rd law
30[30], %p197[205] dipole force due to wedge potential
32[35], %p198[206] point charge in dielectric sphere
34[37]. %p198[207] point dipole in dielectric sphere

\end{document}
