Problem

Consider a spherically symmetric density distribution. Using the Poisson equation, obtain the potential ($\phi$) and the gravitational acceleration ($g$) for a density distribution shown below.

$$\rho\left\{\begin{array}{ll} =\rho_0 &{\rm for} r<R \\ =0 & {\rm for} r\ge R \end{array} \right.$$

Hint: The Poisson equation (2.11) for the spherically symmetric system is

$$\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2\frac{\partial \phi}{\partial r}\right)=4\pi G \rho.$$