Chapter 17.4, Problem 48E

To determine

(a) The potential that depends only on the distance $r$ from $P$ to the center of the sphere. Therefore, it suffices to compute $V(P)$ for a point $P=(0,0,r)$ on the $z-axis$(with $r\ne R$) using symmetry.

(b) Proof of $V(0,0,r)$ is equal to $-\frac{mG}{4\pi }{\displaystyle {\int }_0^{\pi }{\displaystyle {\int }_0^{2\pi }\frac{\sin \varphi d\theta d\varphi }{\sqrt{R^2+r^2-2Rr\cos \varphi }}}}$ using spherical co-ordinates.

(c) The proof of $V(0,0,r)=-\frac{mG}{2Rr}\left(\left|R+r\right|-\left|R-r\right|\right)$ using the substitution $u=R^2+r^2-2Rr\cos \varphi$.

(d) The verification of equation $V(P)=-\frac{Gm}{r}$ and $V(P)=-\frac{Gm}{R}$ for $r>R$ and $R>r$ respectively.