Question #43321

An unconventional capacitor is formed by a solid spherical region of radius $a$ and a concentric thin conducting outer shell of radius $b$ ($b > a$). The inner region $0 \le r \le a$ is filled with a special synthetic material. When a total time-varying charge $Q(t)$ is supplied to the inner region via a thin insulated wire connected exactly to its center ($r=0$), the material instantly distributes this charge such that the volume charge density is $\rho(r, t) = \alpha(t) r$, where $\alpha(t)$ is a time-varying parameter and $r$ is the radial distance from the center. The outer conducting shell carries a charge $-Q(t)$. The wire from the center of the inner region and a wire from the outer shell are connected across an ideal inductor of inductance $L$. Assuming the electric field responds instantaneously to the charge distribution and ignoring any resistance or radiation, what is the angular frequency $\omega$ of the resulting LC oscillations?