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\begin{center}
  {\bf University of Kentucky, Physics 416G \\
    Problem Set \#9, Rev. A, due Monday, 2014-11-24 }
\end{center}

\vspace{.3 in}

{\bf 1.}~~The {\bf Clausius-Mossotti} relation between atomic polarizability
$\alpha$ ($\bs p = \alpha \bs E$ in a linear dielectric) and electric
susceptibility $\chi_e$ ($\bs P = \epsilon_0\chi_e \bs E$ in the same medium).

{\bf \quad a)} The simplest relation can be derived by ignoring the contribution
from individual dipoles to $\bs E$.  Using the definition of $\bs P$, explain
why $\bs P = N \bs p$, where $N$ is the number density of molecules
(number/volume).  Show that in this case $N \alpha = \epsilon_0\chi_e$.

{\bf \quad b)} Correct for the fact a dipole does not feel its own field.  Thus
the definition of polarizability in a dielectric material must be modified to
$\bs p = \alpha \bs E_0$, where $\bs E_0$ is the macroscopic field due to
everything but $\bs p$.  Calculate $\bs E_0$ at the location of $\bs p$ as a
function of the macroscopic field $\bs E$ and susceptibility $\chi_e$ assuming
that $\bs p$ is at the center of a bubble of radius $R$ (the size of the
molecule) inside the dielectric material.  Hint: see example 4.2 and problem
4.16 [both editions].

{\bf \quad c)} Using the modified equation from part a), $N \alpha \bs E_0 =
\epsilon_0\chi_e \bs E$, and part b), derive the relation $N \alpha =
\epsilon_0 \chi_e \frac{3}{3+\chi_e} = 3\epsilon_0
\frac{\epsilon_r-1}{\epsilon_r+2}$.  Note this reduces to the na\"ive relation
as $\chi_e\to0$.

{\bf \quad d)} Use part a) to compare the $\alpha$ and $\chi_e$ for elements which
appear in both tables 4.1 and 4.2.  Calculate the fractional correction term
$\frac{3}{3+\chi_e}$ derived in part c).

{\bf 2.} The {\bf Langevin} formula.  In this problem we calculate the
electric susceptibility $\chi_e$ of a polar material, composed of molecules
with permanent dipole moment $\bs p$.  While the dipole has a tendency to
align along the total field $\bs E$, thermal agitations prevent perfect
alignment.  From statistical mechanics, the number of atoms in state $i$ of
energy $U_i$ is proportional to the Boltzman factor $w=e^{-U_i/kT}$, where $k$
is the Boltzman constant and $T$ the temperature of the system.  Thus colder
systems are more likely to settle into the lowest energy state.  In our case,
the state of a dipole in an electric field is given by its rotational
direction $i \equiv (\theta,\phi)$ with respect to the field $\bs E=E\bh z$.

{\bf \quad a)} Calculate the potential energy $U(\theta)$ of the dipole $\bs p$ in
the field $\bs E$.

{\bf \quad b)} By considering the solid angle of a ring of constant energy
(ie. constant $\theta$), show that the number of states $dn$ of energy between
$U(\theta)$ and $U(\theta+d\theta)$ is proportional to $\sin\theta d\theta$.
The ratio $g=\frac{dn}{d\theta}$ is called the {\it degeneracy of states}, the
{\it density of states}, or the {\it phase space}.

{\bf \quad c)} Calculate the average energy $\langle\, U\rangle \equiv \int dn\,
w(\theta) U(\theta) / \int dn\, w(\theta)$ of a dipole $p$ in the field $E$,
weighted by the Boltzman distribution $w(\theta)$.

{\bf \quad d)} Calculate the weighted average of dipole moment along the field
$\langle p_z\rangle$ of $\langle p\cos\theta\rangle$ as a function of the
electric field $E$ and temperature $T$, also weighted by the Boltzman
distribution.  Explain why $P=N\bs \langle p_z\rangle$ and show that
$P=Np[\coth(pE/kT)-(kT/pE)]$.  Graph this as a function of $E$.

{\bf \quad e)} In the limit where $pE \ll kT$, determine the linear relationship
between $E$ and $P$ to determine the susceptibility $\chi_e$.  Calculate
$\chi_e$ for liquid and vapor water, using $p=3.8\times 10^{-9}~e\,$cm.

Also, Griffiths 3ed[4ed] Ch.~4, \#%
4[4], %p163[170] force of charge on induce dipole
5[5], %p165[171] torque of one dipole on another
6[6], %p165[172] torque on dipole from image
7[7], %p165[172] dipole potential in field
8[8], %p165[172] interaction energy of two dipoles
10[10], %p169[176] sphere with radial polarization
14[14]. %p173[179] neutrality of dielectric

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