Consider a thin spherical shell of uniform density of mass M and radius R:

Say the shell has acquired a mass m and further a mass dM is to be added

$dW=VdM=-\frac{GMdM}{R}or,W={\displaystyle {\int }_0^M\frac{GMdM}{R}}$

$=\frac{G{M}^2}{2R}=$ Self energy =U

Say F is now the attractive force per unit area. if the shell expands from R to R+dR then work done by attractive force is$-F\times 4\pi R^{2}dR$  since this is the work done by gravitational field, this may be equal to reduction in gravitational potential energy.

or, $-\pi R^{2}FdR=dU$ or

$F=-\frac{1}{4\pi R^2}\frac{dU}{dR}=\frac{G{M}^2}{8\pi R_{}^4}$

Now total force experienced by one hemisphere$=F\times \pi R^2$

(Vector addition adds only components perpendicular to the equatorial plane)

$\Rightarrow$ Net gravitational force $F_G=\frac{G{M}^2}{8R^2}$