Question #42185

A horizontal cylindrical container of cross-sectional area $A$ has adiabatic walls and contains one mole of an ideal monatomic gas. A frictionless, adiabatic piston is pushed inwards at a constant speed $v_0$. Integrated into the piston is a conducting rod of length $L$ and electrical resistance $R_e$. The rod slides on two perfectly conducting, parallel rails that run along the inner length of the cylinder and are shorted at the closed end. A non-uniform magnetic field perpendicular to the plane of the rails exists in the region, given by $B(x) = B_0 x^{-1/2}$, where $x$ is the distance from the closed end. The entire electrical resistance $R_e$ is exposed to the gas, so all Joule heating is transferred to the gas. If the initial volume and temperature of the gas are $V_0$ and $T_0$, and the gas is compressed to a volume $V_0/8$, find the final temperature of the gas. (Given: $\frac{B_0^2 L^2 v_0}{R_e} = R T_0$, where $R$ is the universal gas constant).