Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S29 / 3 / a / Solution

For a thin uniform spherical shell, $dC=(R\sin \theta {)}^{2}dm$ and angular averaging gives $⟨{\sin }^2\theta ⟩=2/3$. Thus $\boxed{C=2M{R}^2/3}$. The printed formula requires a negligibly thin outer shell. For a finite-thickness hollow sphere of uniform density between $r_0$ and $R$,For example $r_0=R/2$ gives $C/(M{R}^2)=31/70$, not $2/3$. The thin-shell limit $r_0\to R$ restores the printed result. A uniform outer layer of unspecified thickness does not by itself establish it.