4.2 Gravitational potential due to a uniform solid sphere

  Gravitation

• At a point outside the sphere, i.e. $r>R$ $$V = - \frac{{GM}}{r}$$
• At a point on the surface of the sphere, i.e. $r=R$ $$V = - \frac{{GM}}{R}$$
• At a point inside the sphere ,i.e. $r<R$ $$V = - \frac{{GM\left( {3{R^2} - {r^2}} \right)}}{{2{R^3}}}$$

Proof: At a point outside the sphere

The solid sphere behaves like a point mass for a point outside the sphere.

So, for the gravitational potential outside the sphere is, $$V = - \frac{{GM}}{r}$$
Similarly, gravitational potential on the surface of the sphere, i.e. $(r=R)$ is, $$V = - \frac{{GM}}{R}$$

Proof: At a point inside the sphere

Consider a sphere of mass $M$ and radius $R$.

Density of sphere is, $$\rho = \frac{M}{V} = \frac{M}{{\left( {\frac{4}{3}\pi {R^3}} \right)}}$$$$\rho = \frac{{3M}}{{4\pi {R^3}}}$$

Mass of sphere of radius $r$ is,
$$M' = \rho V'$$$$M' = \frac{{3M}}{{4\pi {R^3}}} \times \frac{4}{3}\pi {r^3}$$$$M' = M{\left( {\frac{r}{R}} \right)^3} \quad ...(i)$$
Gravitational force at point $P$ is given by, $$F = \frac{{GM'm}}{{{r^2}}} \quad ...(ii)$$
From equation $(i)$ and $(ii)$ we get, $$F = \frac{{GM{r^2}m}}{{{r^2}{R^3}}}$$$$F = \frac{{GMm}}{{{R^3}}}r$$
Work done for displacement $dr$ is, $$dW = - \frac{{GMm}}{{{R^3}}}rdr$$
As we know, $$dV = - \frac{{dW}}{m}$$ So, $$dV = \frac{{GM}}{{{R^3}}}rdr$$
Integrating with proper limits we get,
$$\int\limits_{{V_R}}^{{V_r}} {dV} = \int\limits_R^r {\frac{{GM}}{{{R^3}}}rdr} $$$$\left[ V \right]_{{V_R}}^{{V_r}} = \frac{{GM}}{{{R^3}}}\left[ {\frac{{{r^2}}}{2}} \right]_R^r$$$${V_r} - {V_R} = \frac{{GM}}{{2{R^3}}}\left( {{r^2} - {R^2}} \right)$$$${V_r} - \left( { - \frac{{GM}}{R}} \right) = \frac{{GM}}{{2{R^3}}}\left( {{r^2} - {R^2}} \right)$$$${V_r} = \frac{{GM}}{{2{R^3}}}\left( {{r^2} - 3{R^2}} \right)$$ or $${V_r} = - \frac{{GM}}{{2{R^3}}}\left( {3{R^2} - {r^2}} \right)$$

Note: The gravitational potential due to a uniform solid sphere varies as,
$$V = - \frac{{GM}}{r}\quad {\text{for}}\quad r \geqslant R$$ $$V = - \frac{{GM\left( {3{R^2} - {r^2}} \right)}}{{2{R^3}}}\quad {\text{for}}\quad r \leqslant R$$
The $V$ vs $r$ graph can be drawn using the above equation is,