%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \sect{Particle on a ring with electric field} A particle of mass ${\mathsf{m}}$ and charge ${e}$ is free to move on a ring of radius ${R}$. The ring is located in the ${(x,y)}$ plan. The position of the particle on the ring is ${x=R\cos\left(\theta\right)}$ and ${y=R\sin\left(\theta\right)}$. There is an electric field ${\cal E}$ is the $x$ direction. The temperature is ${T}$. \begin {itemize} \item[(1)] Write the Hamiltonian ${H\left(\theta,p\right)}$ of the particle. \item[(2)] Calculate the partition function ${Z\left(\beta,{\cal E}\right)}$. \item[(3)] Write an expression for the probability distribution ${\rho\left(\theta\right)}$. \item[(4)] Calculate the mean position ${\langle x \rangle}$ and ${\langle y \rangle}$. \item[(5)] Write an expression for the probability distribution ${\rho\left(x\right)}$. Attach a schematic plot. \item[(6)] Write an expression for the polarization. Expand it up to first order in ${{\cal E}}$, and determine the susceptibility. \end {itemize} $\frac{1}{2\pi}\int_{0}^{2\pi} \exp\left(z\cos\left(\theta\right)\right)d\theta = I_{0}\left(z\right)$ % $I_{0}'\left(z\right) = I_{1}\left(z\right)$ % $I_{0}\left(z\right) = 1+\left(\frac{1}{4}\right)z^{2}+\left(\frac{1}{64}\right)z^{4+...}$ %%begin{figure} \putgraph{Ex206} %%\caption{}\label{} %%end{figure} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%