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
Read more about this topic: Shell Theorem
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