Page:Elementary Text-book of Physics (Anthony, 1897).djvu/83

§ 57] closed surface formed of concentric spherical shells, in each one of which the surface density is uniform.

The force at a point outside a spherical shell of uniform surface density varies inversely with the square of the distance between that point and the centre of the spherical shell. For, describe a sphere concentric with the shell and of radius $$r$$, greater than the radius of the shell. Applying to this sphere the theorem of the flux of force, we have $$\sum Fs = 4\pi M$$, where $$M$$ is the mass of the spherical shell. It is evident, by symmetry, that the force at every point on the sphere to which this theorem is applied must be the same in magnitude and similarly directed along the radius of the sphere.

The flux of force $$\sum Fs$$ therefore equals $$F. 4\pi r^2$$ or $$F = \frac{M}{r^2}\cdot$$ This theorem manifestly holds also for the force at a point outside any mass bounded by a spherical surface, provided that the matter in the sphere is distributed uniformly or in concentric shells, in each one of which the surface density is uniform.