Question #42679

A perfectly rigid, sealed parallel-plate capacitor has a plate area $A$ (in $\text{m}^2$) and a plate separation $d$ (in $\text{m}$). The space between the plates is completely filled with an incompressible solid dielectric of dimensionless dielectric constant $K$. The capacitor is placed horizontally at the bottom of a vessel whose cross-sectional area increases exponentially with height. The vessel is initially filled with a non-conducting liquid of density $\rho$ (in $\text{kg/m}^3$) such that the top of the capacitor is at a depth $h$ (in $\text{m}$) below the liquid's free surface. The atmospheric pressure is $P_0$ (in $\text{Pa}$). The capacitor is connected to an ideal battery that maintains a constant potential difference $V$ (in $\text{V}$) across its plates. If the liquid is slowly drained from the vessel until it is completely empty, what is the net electrical work done by the battery during this draining process? (Assume acceleration due to gravity is $g$ in $\text{m/s}^2$ and ignore fringing edge effects of the electric field).