%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 ${L, \mathsf{m}, q, \gamma, T, \varepsilon}$. %%begin{figure} \putgraph{Ex207} %%\caption{}\label{} %%end{figure}