Guass’s theorem for gravitation

Guass’s theorem for gravitation:

              Guass's law states that the total  gravitational  flux  over  a closed  surface,  having  a  unique outward  drawn normal to  it  at every point is - 4πG times the  total mass enclosed by that surface.

                The gravitational field E at a distance ‘r' from a point mass M is given by :

                 E= - GM/r²    .....(1)

                 Then the flux (ϕ) of the gravitational field, through the surface of the sphere of radius ‘r' is given by :

                  ϕ = - GM × 4πr² = - 4πGM

                            r²   

                  As shown in  figure, this flux  ϕ  is  due  to  the  normal component of gravitational field 

                 E = - GM cosϴ

                           r²

                  Let , dϕ = small flux through an  element  of  area  dA.

               .•.dϕ = -GM cosϴ dA  

                              r²

               .•. dϕ= E.dA = n. E dA

               .•. Total gravitational flux enclosed by the closed surface is :

           ϕ = ∫ E. dA = ∫ n^. E dA

       .•. ϕ =  ∫ E1.dA +  ∫ E2.dA + ........

       .•. ϕ = -4πG (M1 + M2 + ......)

       .•. ϕ = -4πGM 

       where , M = sum of all masses enclosed by the surface.

This is Gauss's law in gravitation.