CP CALC Gausss Law for Gravitation. The gravitational

Chapter 22, Problem 22.59

(choose chapter or problem)

Gauss’s Law for Gravitation. The gravitational force between two point masses separated by a distance r is proportional to \(1 / r^{2}\), just like the electric force between two point charges. Because of this similarity between gravitational and electric interactions, there is also a Gauss's law for gravitation. (a) Let \(\vec{g}\) be the acceleration due to gravity caused by a point mass m at the origin, so that \(\vec{g}=-\left(G m / r^{2}\right) \hat{r}\). Consider a spherical Gaussian surface with radius r centered on this point mass, and show that the flux of \(\vec{g}\) through this surface is given by

         \(\oint \vec{g} \cdot d \vec{A}=-4 \pi G m\)

(b) By following the same logical steps used in Section 22.3 to obtain Gauss's law for the electric field, show that the flux of \(\vec{g}\) through any closed surface is given by

         \(\oint \vec{g} \cdot d \vec{A}=-4 \pi G M_{\mathrm{encl}}\)

where \(M_{\mathrm{encl}}\) is the total mass enclosed within the closed surface