Consider a thin spherical shell of mass M with an inner radi

Consider a thin spherical shell of mass \"M\" with an inner radius \"a\" and an outer radius \"b\". What is the gravitational potential both inside and outside of the spherical shell? What is the gravitational force on a mass \"m\" at a distance \"r\" from the spherical shell?

Solution

It needs to understood here that for a spherical shell of mass M, the net potential at any point lying inside the shell is zero. While for a point lying outside the shell, the potential can be calculated by considering the shell as a point charge located at its centre and of the same mass as the shell. Plus, we assume that the shell has the mass uniformly spread all across its volume.

That is mass density = 3M / 4(b^3 - a^3)

Hence, mass inside a shell of radius r, where a<r<b is given as

Mr = M (r^3 - a^3)/ (b^3 - a^3)

Therefore, we get: U = -GM / r where is r > b

and U = 0; for r<a

and U = - GM (r^3 - a^3) / r(b^3 - a^3) for a < r < b

Second Part:

The gravitational force on any mass m is given as the gravitational potential multiplied by the mass

That is F = -GMm / r: for r> b

F = 0; for r<a

F = - GM (r^3 - a^3) / r(b^3 - a^3) for a < r < b