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\sect{Charged Bose gas in a divided box}

Consider ${N}$ bosons with mass ${\mathsf{m}}$, positive charge ${e}$, and spin ${0}$. 
The particles are in a box that is divided into two regions: 
zero voltage region of volume $\Omega_0$, and voltage $V$ region of volume $\Omega_v$.  
Assume that the bosons are condensed in the $\Omega_0$ region. 
In items (3,5) assume that the gas in the $\Omega_v$ region 
can be treated using the Boltzmann approximation. 

\begin {itemize}

\item[(1)]
Find the $V=\infty$ condensation temperature $T_c(\infty)$.

\item[(2)]
Find the $V=0$ condensation temperature $T_c(0)$.

\item[(3)]
Assuming an intermediate temperature, 
find the critical voltage $V_c$ \\
below which the bosons are no longer condensed.  

\item[(4)]
Write an exact expression for the energy $E(T,V)$ of the system  

\item[(5)]
Write an expression for the heat capacity $C(T,V)$ of the system.\\
Keep only the leading correction in $V$. 

\end {itemize}


Express the results using the thermal wavelength $\lambda_T$, 
the variables ${\Omega_0, \Omega_v, N, T, eV}$, \\
and the functions ${L_{\alpha}(z)}$ and $\zeta(\alpha)$.      


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