Shell Theorem - Derivation Using Gauss's Law

Derivation Using Gauss's Law

The shell theorem is an immediate consequence of Gauss's law for gravity saying that

where M is the mass of the part of the spherically symmetric mass distribution that is inside the sphere with radius r and

is the surface integral of the gravitational field g over any closed surface inside which the total mass is M, the unit vector being the outward normal to the surface

The gravitational field of a spherically symmetric mass distribution like a mass point, a spherical shell or a homogenous sphere must also be spherically symmetric. If is a unit vector in the direction from the point of symmetry to another point the gravitational field at this other point must therefore be

where g(r) only depends on the distance r to the point of symmetry

Selecting the closed surface as a sphere with radius r with center at the point of symmetry the outward normal to a point on the surface, is precisely the direction pointing away from the point of symmetry of the mass distribution.

One therefore has that

and

as the area of the sphere is 4πr2.

From Gauss's law it then follows that

i.e. that