Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2012 / Paper 1 / Section A / 1 / Solution

The shell theorem gives the gravity of a concentric hollow sphere as $\mathbf{g}(r)=-GM(<r)\hat{\mathbf{r}}/r^2$. There is no enclosed mass at the cavity boundary, whereas the entire shell has mass $M=\frac{4\pi \rho }{3}[(2a{)}^3-a^3]=\frac{28\pi }{3}\rho a^3$. ThusBoth nonzero fields point towards the centre. The field is continuous at each boundary; there is no surface mass sheet.