Shell Theorem - Outside The Shell

Outside The Shell

A solid, spherically symmetric body can be modelled as an infinite number of concentric, infinitesimally thin spherical shells. If one of these shells can be treated as a point mass, then a system of shells (i.e. the sphere) can also be treated as a point mass. Consider one such shell:

Note: dθ in the diagram refers to the small angle, not the arclength. The arclength is R dθ.

Applying Newton's Universal Law of Gravitation, the sum of the forces due to mass elements in the shaded band is

However, since there is partial cancellation due to the vector nature of the force, the leftover component (in the direction pointing toward m) is given by

The total force on m, then, is simply the sum of the force exerted by all the bands. By shrinking the width of each band, and increasing the number of bands, the sum becomes an integral expression:

Since G and m are constants, they may be taken out of the integral:

To evaluate this integral one must first express dM as a function of dθ

The total surface of a spherical shell is

while the surface of the thin slice between θ and θ + dθ is

If the mass of the shell is M one therefore has that

and

By the law of cosines,

These two relations link the 3 parameters θ, s and φ that appear in the integral together. When θ increases from 0 to π radians φ varies from the initial value 0 to a maximal value to finally return to zero for θ = π.

s on the other hand increases from the initial value r − R to the final value r + R when θ increases from 0 to π radians.

This is illustrated in the following animation

To find a primitive function to the integrand one has to make s the independent integration variable instead of θ

Performing an implicit differentiation of the second of the "cosine law" expressions above yields

and one gets that

where the new integration variable s increases from r − R to r + R.

Inserting the expression for cos(φ) using the first of the "cosine law" expressions above one finally gets that

A primitive function to the integrand is

and inserting the bounds r − R, r + R for the integration variable s in this primitive function one gets that

saying that the gravitational force is the same as that of a point mass in the centre of the shell with the same mass.