There is a uniform sphere of mass M and radius R. Find the strength G and the potential `var phi` of the gravitational field of this sphere as a function of the distance r from its centre (with `r lt R` and `r gt R`). Draw the approximate plots of the functions `G(r)` and `varphi(r)`.

Explanation

We have obtained `varphi` and G due to a uniform sphere, at a distance r from it's centre outside it.
`varphi=-(gammamM)/(r)` and `vecG=-(gammaM)/(r^3)vecr` (A)
Accordance with the Eqs. (1) of the solution, potential due to a spherical shell of radius a, at any point, inside it becomes
`varphi=(gammaM)/(a)=Const.` and `G_r=-(deltavarphi)/(deltar)=0` (B)
For a point (say P) which lies inside the uniform solid sphere, the potential `varphi` at that point may be represented as a sum.
`varphi_(i nside)=varphi_1+varphi_2`
where `varphi_1` is the potential of a solid sphere having radius r and `varphi_2` is the potential of the layer of radii r and R. In accordance with equation (A)
`varphi_1=-(gamma)/(r)((M)/((4//3)piR^3)4/3pir^3)=-(gammaM)/(R^3)r^2`
The potential `varphi_2` produced by the layer (thick shell) is the same at all points inside it. The potential `varphi_2` is easiest to calculate, for the point positioned at the layer's centre. Using Eq. (B)
`varphi_2=-gammaunderset(r)overset(R)int(dM)/(r)=-3/2(gammaM)/(R^3)(R^2-r^2)`
where `dM=(M)/((4//3)piR^3)4pir^2dr=((3M)/(R^3))r^2dr`
is the mass of a thin layer between the radii r and `r+dr`.
Thus `varphi_(i nside)=varphi_1+varphi_2=((gammaM)/(2R))(3-(r^2)/(R^2))` (C)
From the Eq. `G_r=(-deltavarphi)/(deltar)`
`G_r=(gammaMr)/(R^3)`
or `vecG=-(gammaM)/(R^3)vecr=-gamma4/3pirhovecr`
(where `rho=(M)/(4/3piR^3)`, is the density of the sphere) (D)
The plots `varphi(r)` and `G(r)` for a uniform sphere of radius R are shown in figure of ansersheet.
Alternate: Like Gauss's theorem of electrostatics, one can derive Gauss's theorem for gravitation in the form `oint vecG*dvecS=-4pigammam_(i nclosed)`. For calculation of `vecG` at a point inside the sphere at a distance r from its centre, let us consider a Gaussian surface of radius r, Then,
`G_r4pir^2=-4pigamma((M)/(R^3))r^3` or, `G_r=-(gammaM)/(R^3)r`
Hence, `vecG=-(gamma*M)/(R^3)vecr=-gamma4/3pirhovecr` (as `rho=(M)/((4//3)piR^3)`)
So, `varphi=underset(r)overset(oo)intG_rdr=underset(r)overset(R)int-(gammaM)/(R^3)rdr+underset(R)overset(oo)int-(gammaM)/(r^2)dr`
Integrating and summing up, we get,
`varphi=-(gammaM)/(2R)(3-(r^2)/(R^2))`
And from Gauss's theorem for outside it:
`G_r4pir^2=-4pigammaM` or `G_r=-(gammaM)/(r^2)`
Thus `varphi(r)=underset(r)overset(oo)intG_rdr=-(gammaM)/(r)`