Question #42537

A vertical, perfectly thermally insulated cylinder of cross-sectional area $A$ contains a monatomic ideal gas. The cylinder is closed by a frictionless, electrically conducting but thermally insulating piston of mass $m$. The piston is constrained to move along a frictionless helical groove of pitch $p$ (vertical displacement $p$ per revolution). A uniform vertical magnetic field $B$ exists throughout the region. The center and the rim of the piston are connected via zero-resistance sliding contacts to a heating coil of resistance $R$ and negligible heat capacity located entirely inside the gas. Initially, the system is in equilibrium with the piston at height $h_0$ and external atmospheric pressure $P_0$. Let $P_1 = P_0 + \frac{mg}{A}$. A block of mass $M$ is suddenly placed on the piston. The piston descends, and the system eventually comes to rest at a new equilibrium height $h_f$. Let $P_2 = P_0 + \frac{(m+M)g}{A}$. Which of the following expressions gives $h_f$? (Ignore the induced magnetic field).