Gravitation energy of a spherical shell

[Ananthan9470]

I am asked to find the total gravitational energy of a hollow sphere using the fact that the field energy density is given by $u_g = \frac{-1}{8\pi G}g^2$.

Now, $g = \frac{Gm}{r^2}$ in this case and substituting gives $u_g = \frac{-GM^2}{8 \pi r^4}$. Integrating this over volume will give $U = \iiint \frac{GM^2}{8 \pi r^4} dv = \frac{-GM^2}{2R}$.

Is my solution correct?

 

[TSny]

Yes, that looks correct.