Question #42938

A closed insulating vertical cylinder of height $H$ is fixed inside an elevator that is accelerating upwards with a constant acceleration $a = g$. The cylinder contains a fluid of total mass $M$ and total charge $Q$. The charge is permanently bound to the fluid particles such that the local charge density is always proportional to the local mass density. Initially, the fluid is in a state where its mass density increases linearly with height $y$ (measured from the bottom of the cylinder): $\rho_m(y) = c y$, where $c$ is a constant. An external non-uniform electric field is maintained inside the cylinder, given by $\vec{E} = E_0 \left(1 - \frac{y}{H}\right) \hat{j}$, where $\hat{j}$ is the unit vector in the upward vertical direction relative to the elevator. Over time, the fluid spontaneously redistributes itself to achieve a uniform mass density throughout the cylinder. If this redistribution process occurs such that the net change in the total potential energy (gravitational + electrostatic) of the fluid in the frame of the elevator is zero, what is the required value of $E_0$?