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\sect{Polarization of two-spheres system inside a tube (Exam2003 Q1)}
Given two balls in a very long, hollow tube, with length ${L}$.
The mass of each ball is ${\mathsf {m}}$, The charge of one ball is
${-q}$ and the charge of the other one is ${+q}$. The ball's radius
is negligible, and the electrostatic attraction between the balls is
also negligible. The balls are rigid and can't pass through each
other. The balls are attached to a drop, whose surface tension
causes it's gravity constant ${\gamma}$ to work on the balls toward
each other. (The force does not depend on the distance between the
balls). The system is in an external electric uniform field
${\bar{\varepsilon}=\varepsilon\hat{x}}$ and in thermic balance in
temperature ${T}$.
\begin {itemize}
\item[(a)]
Write the hamiltonian of the system
${H\left(p_{1},p_{2},x_{1},x_{2}\right)=E_{k}+V\left(x\right)}$ when
${E_{k}}$ is the kinetic energy. Define properly ${V\left(x\right)}$
when ${x=x_{2}-x_{1}}$ and write a diagram of ${V\left(x\right)}$.
\item[(b)]
Calculate the distribution function ${Z\left(\beta,
\varepsilon\right)}$.
\item[(c)]
Find the probability function of $x, \, {\rho\left(x\right)}$ and
the average distance ${\langle x\rangle}$ between the balls. Express
again ${\rho\left(x\right)}$ by ${\langle x\rangle}$.
\item[(d)]
Find the polarization ${p}$ as a function of ${\varepsilon}$.
Use the distribution function.
\item[(e)]
Develop ${p\left (\varepsilon\right)}$ up to first order in
the field: ${P\left(\varepsilon\right)=p_{0}+\chi\varepsilon+O
\left(\varepsilon^{2}\right)}$.
\end {itemize}
This development is valid in a weak field, Define what is a weak
field. Express your answers with ${L, \mathsf{m}, q, \gamma, T,
\varepsilon}$.
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