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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 416G \\
    Problem Set \#10 (Rev. C), due Wednesday, 2011-12-07 }
\end{center}

\vspace{.3 in}

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 $\bs P = \chi_e \epsilon_0 \bs E$, where $\bs E$ is the total
electric field inside the rod, placed in an external electric field $\bs
E_0=E_0\bh x$.
Calculate the values of $\bs E$, $\bs P$, and $\bs D$ inside and outside of
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 constants and $\sigma_f=0$, and 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{-.25in}\hspace{.5in}
\includegraphics[height=1in,trim=0 -.28in 0 .28in]{ps_10_fig1}
\hspace{.5in}
\includegraphics[height=2in]{ps_10_fig2}

\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.

\quad b) Solve for the potential when a finger is present.  Assume the glass
thickness is $d$, and your finger of pure water ($\epsilon_r^2\approx 80$)
fills the whole region beyond the glass.

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

\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
strip.  What glass thickness do you recommend?

\quad e) If the voltage on each wire is $\pm V = \pm 2 V$, 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{.3cm}

Also Griffiths chapter 4, problems 
16, 
18, 
20, 
21, 
22, 
24, 
28, 
29, 
30, 
32, 
33, and
34.

\newpage

3. [bonus] a) Calculate the electric potential due to a point multipole
$Q^{(\ell)}_{\rm int}$ (see HW~8 \#1) placed at the origin, embedded in a
material of dielectric constant $\epsilon_r$.  Hint: put the multipole inside
a spherical bubble of radius $\delta$, solve for the external potential, and
take the limit as $\delta\to0$.  This is similar to the proof of the
Clausius-Mossotti relation in HW~9 \#2.  Note: the solution to the problem
depends on the shape of the for the bubble the multipole is placed in, even in
the limit as $\delta\to0$.

\quad b) Now let the point point multipole $Q^{(\ell)}_{\rm int}$ be embedded
at the center of a sphere of radius $R$ and dielectric constant $\epsilon_r$.
Calculate the electric potential inside and outside the sphere.  Hint: use the
result from part a) to form the boundary condition as $r\to0$.

\vspace{1cm}

4. [bonus] Thermal conductivity $K$ defined by $\bs q = -K \nabla T$, is similar to
electrical conductivity $\sigma$, but for the flow of heat $dQ/dt = \bs q
\cdot \bs{da}$ across a thermal gradient $-\nabla T$, instead of charge
$I=dq/dt=\bs J \cdot \bs{da}$ across and potential gradient $\bs E=-\nabla V$
in the electrical conduction equation $\bs J = -\sigma \nabla V$. 
Note the difference between the symbols: $Q$ (heat energy), \mbox{$\bs q$ (heat
flux), and $q$ (charge).}

\quad a) Show that $K=\frac12 kN v_{\mbox{\tiny rms}} \lambda$, assuming
  classical diffusion, where $k$ is Boltzman's constant, $N$ is the electron
  density, and $\lambda$ the mean free path of electrons.

\quad b) Show that the relation between electrical and thermal conductivity
  is $K / \sigma T = 3k^2/2e^2$.

\vspace{1cm}

5. [bonus] a) Show that the power density in a resistor is $dP/dl = I^2
\rho/A$, where $\rho$ is the resistivity and $A$ the cross sectional area of
the resistor.

\quad b) Show the similar result for a surface resistor: $dP/da = K^2 \rho/d$ where
$d$ is the thickness of the surface resistor, and $dP/d\tau = J^2 \rho$ for
bulk resistance.

\end{document}
