Gravitational potential due to a uniform solid sphere

Gravitational potential due to a uniform solid sphere:

1) At a point outside the sphere:

                    Consider a uniform solid sphere having radius  R and  mass  M.  Let  P  be  a point at a distance ‘r' from the centre Of the solid sphere, where the gravitational potential due to the sphere is to be determined.

                      Imagine the solid sphere to be made  up  of  a  large  number  of  thin  concentric spherical shell's of masses m₁, m₂, m₃, ..... etc. Now, the gravitational potential at P due to whole sphere is equal to sum of potentials due to all such shells.

                 .•. V = - [ Gm₁+ Gm+ .....]
                                   r            r

                         = - G ( m₁+ m₂ + .....)
                               r

                 .•. V = - GM
                                r

Where, m₁+m₂ + .... = M = Mass of the sphere.

Intensity  of gravitational field due to solid sphere:

E = - dV   =   - d  ( - GM )
         dr            r        r

.•. E = - GM
               r²

                     Negative sign shows that, the intensity of gravitational field is directed towards the centre O of the sphere.

2) At a point on the surface of the sphere:

At a point on the surface of the sphere, r=R :

V = - GM    and
           R
E = - GM
           R²

3) At a point inside the sphere :

           Consider a uniform point P inside the sphere at a distance‘r' from its centre O , as shown in figure.

Let R = radius of sphere  and
 ϱ = density of the sphere

           Now, with O as a centre and radius OP = r drawn a sphere.

           Then the point P lies on the surface of the solid sphere of the radius ‘r,' and inside the spherical shell of internal radius R and external radius R.

.•. volume of inner solid sphere = 4 π r³
                                                                 3

.•. Mass of inner solid sphere = 4 π r³ϱ
                                                            3

The potential V₁ at P due to inner solid sphere is :

V₁ = - G ( 4πr³ ϱ )  = 4πr² ϱ G     ..... (1)
                     r

               To find the potential V₂ due to the outer spherical shell, draw two concetric spheres of radii ‘x' and ( x + dx ) ,  forming a thin spherical shell of thickness ‘ dx '.

              .•.  Volume of shell = 4πx² dx

              .•. Mass of shell = 4πdx²dx

              .• Potential at P due to this shell

              P  = - G (4πdx²dx )   =   -4πdGx dx
                                  x
              .•. The potential V₂  at P due to thick spherical shell is obtained by integrating above equation between the limits x=r to x=R.

                 V₂ =  -4πϱG ᷊∫  ᷢᷢᷢᷢᷢᷢᷢᷢx dx = -4πϱG [x²/2] ᷊ ᷢᷢᷢᷢᷢᷢᷢᷢ

             .•. V₂ =  -4πϱG [ R² - r²]
                                              2

             .•. V₂ = -2πϱG ( R²-r² )    ......(2)

             .•. Total potential at P is given by :

                  V = V₁+V₂

             .•. V = - 4 π r²ϱ G-2πϱ G(R²-r²)
                          3

             .•. V = - 2πϱ G(2 r² + R² - r²)
                                      3

            .•. V = 2πϱ G(3R²-r²)
                       3

            .•. V =  -G( 4  π R³ϱ ) [3R²-r²/2R³]
                               3

                V = - GM [ 3R³- r²] .......(3) ,
                                      2R³
                Since,  4 πR³ϱ  = M
                             3

           This is an expression for the potential inside a solid sphere, at a distance, from its centre.

            The minimum potential at the centre of the sphere can be obtained by putting r=0.

             V₀= -3GM   ......(4)
                         2R

             The intensity of gravitational field at P inside the solid sphere is:

            E = - dV  =  -  d [-GM ( 3R²-r²)]
                     dr         dr               2R³

            E  = - GM [2r]
                      2R³

       .•. E = - GM   ... .... (5)
                     R³