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\sect{Particle on a ring (rotor) with electric field}

Particle with mass ${\mathsf{m}}$ and charge ${e}$ is free to move
along the perimeter of a ring placed on ${x-y}$ plain, (with  a
radius ${R}$). A uniform, electric field
${\bar{\varepsilon}=\hat{x}\varepsilon}$ in the system, and it has a
thermic balance in temperature ${T}$.



\begin {itemize}

\item[(a)]
Write the Hamiltonian ${H\left(p_{\theta},\theta\right)}$ of
the particle.

\item[(b)]
Calculate the distribution function
${Z\left(\beta,\varepsilon\right); \beta^{-1}=k_{B}T}$

\item[(c)]

What is the probability function ${\rho\left(\theta\right)}$ of the
angled coordinate ${\theta}$?

\item[(d)]
Calculate the average place of the particle in a cartesian
coordinates (meaning ${\langle x\rangle , \langle y\rangle)}$

\item[(e)]

What is the probability function ${\rho\left(x\right)}$ of the
coordinate ${x}$? add a schematic drawing.

\item[(f)]

Express the polarization ${P\left(\varepsilon\right)}$. For a weak
${\varepsilon}$ develope ${P\left(\varepsilon\right)}$ up to first
order for - ${\varepsilon}$: ${P\left(\varepsilon\right)=p_{0}+\chi
\varepsilon+0\left(\varepsilon^{2}\right)}$ and find ${\chi}$.

\end {itemize}



Use the next equations:
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\[\frac{1}{2\pi}\int_{0}^{2\pi}e^{\pm z \ln
\theta}d\theta=I_{0}\left(Z\right)\]
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\[I_{0}^{1}\left(Z\right)=I_{1}\left(Z\right)\]
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\[I_{0}\left(Z\right)=1+\frac{1}{4}Z^{2}+\frac{1}{64}Z^{4}+...\]





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