Question #42833

A non-conducting solid sphere of initial radius $R_0$ has a radially varying mass density $\rho_m(r)$. The moment of inertia of the spherical region enclosed within any radius $r$ (about a diametric axis) is given by $I(r) = I_0 \left(\frac{r}{R_0}\right)^6$, where $I_0$ is the total initial moment of inertia of the sphere. The sphere carries a total charge $Q$, distributed such that the volume charge density is $\rho_e(r) = \beta \rho_m(r) r^2$, where $\beta$ is a positive constant. The sphere is then heated, increasing its temperature by $\Delta T$. The coefficient of linear thermal expansion of the material is $\alpha$. Assuming the charge is bound to the material and $\alpha \Delta T \ll 1$, what is the magnitude of the electric field at a fixed spatial distance $r = R_0/2$ from the center of the sphere?