4.3 Gravitational potential due to a uniform thin spherical shell

  Gravitation

• At a point outside the shell, i.e. $r>R$ $$V = - \frac{{GM}}{r}$$
• At a point on the surface of the shell, i.e. $r=R$ $$V = - \frac{{GM}}{R}$$
• At a point inside the shell, i.e. $r>R$ $$V = - \frac{{GM}}{R}$$

Proof: At a point outside the shell

The spherical shell behaves like a point mass for a point outside the sphere,

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

Proof: At a point inside the shell

The gravitational field at any point inside the spherical shell is zero. So, the gravitational potential at all point inside the spherical shell is equal to the potential at the surface.

Therefore, $$V = - \frac{{GM}}{R}\quad {\text{for}}\quad r < R$$

Note: The gravitational potential due to a uniform spherical shell varies as,
$$V = - \frac{{GM}}{r}\quad {\text{for}}\quad r \geqslant R$$
$$V = - \frac{{GM}}{R}\quad {\text{for}}\quad r \leqslant R$$
The $V$ vs $r$ can be drawn using the above equations as,