Question #42131

A thick uncharged conducting spherical shell has an inner radius $R_1$ and outer radius $R_2$. The spherical cavity ($0 \le r < R_1$) is completely filled with a uniformly charged compressible fluid of total charge $Q$ which always fills the cavity. The shell is made of a material whose coefficient of linear thermal expansion varies radially as $\alpha(r) = \frac{c}{r^2}$ for $R_1 \le r \le R_2$, where $c$ is a positive constant. The outer surface of the shell ($r = R_2$) is mechanically fixed such that it cannot move. The system is heated, increasing its temperature at a constant rate $\frac{dT}{dt} = \beta$. Assuming the radial displacement $u(r)$ of the shell material is purely kinematic and governed by $\frac{\partial u}{\partial r} = \alpha(r) \Delta T$, find the rate of change of the electric field magnitude $\frac{\partial E}{\partial t}$ at a fixed distance $r$ ($r < R_1$) from the center at $t=0$.