Paper 3, Section II, C

Given a spherically symmetric mass distribution with density $\rho$, explain how to obtain the gravitational field $\mathbf{g}=-\nabla \phi$, where the potential $\phi$ satisfies Poisson's equation

$\nabla^{2} \phi=4 \pi G \rho$

The remarkable planet Geometria has radius 1 and is composed of an infinite number of stratified spherical shells $S_{n}$ labelled by integers $n \geqslant 1$. The shell $S_{n}$ has uniform density $2^{n-1} \rho_{0}$, where $\rho_{0}$ is a constant, and occupies the volume between radius $2^{-n+1}$ and $2^{-n}$.

Obtain a closed form expression for the mass of Geometria.

Obtain a closed form expression for the gravitational field $\mathbf{g}$ due to Geometria at a distance $r=2^{-N}$ from its centre of mass, for each positive integer $N \geqslant 1$. What is the potential $\phi(r)$ due to Geometria for $r>1$ ?

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