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\begin{document}
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\exnumber{3030}
\heading{Charged Bose gas in box with potential difference}
\auname{Gilad Granit}
{\bf The problem:}
\Dn
Consider ${N}$ bozons with mass ${\mathsf{m}}$, positive charge
${e}$ and spin ${0}$. The particles are in a tank in thermic
equilibrium, and temperature ${T}$. The tank has two zones ${A}$
and ${B}$, The volume of each zone is ${L^{3}}$.A battery creates
potential difference ${V}$ between the zones. The potential in
every zone is homogenous.
\begin{itemize}
\item[1)]
Find the condition on ${N}$, so if ${V=0}$ then there's no
condensation, but if ${V=\infty}$ then there's condensation.
\item[2)]
assume that the particles in zone ${A}$ are in a
condensation state and the particles in zone ${B}$ can be described
in the Boltzman proximity frame.
\begin {itemize}
\item[(a)]
What is the number of the particles in zone ${B}$.
What is the condition for ${V}$, so that Boltzman proximity will be
valid
\item[(b)]
What is the number of the particles in zone ${A}$. How many
of them are in condensation state?
\item[(c)]
Show that the condensation in zone ${A}$ as long as
${V_{c}0}$ ,starting from Eq (8)
\[
N - \frac{L^3}{\lambda_{T}^{3}} \left[ \zeta \left( \frac{3}{2} \right)+ e^{- \beta \left( eV \right)} \right]>0
\]
\[
\frac{N\lambda_{T}^{3}}{L^3}> \zeta \left( \frac{3}{2} \right) + e^{- \beta \left( eV \right)}
\]
\begin{align}
eV_c = -T \ln\left[ \frac{N\lambda_{T}^{3}}{L^3} -\zeta \left( \frac{3}{2} \right) \right]
\end{align}
as can see we found a critical voltage, below it there will be no condensate state for the Bosons, that is ${V_c < V}$ is our condition.
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\end{itemize}
\end{enumerate}
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\end{document}