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The Uniform Sphere

Another familiar example of spherical symmetry is the uniformly dense solid sphere of mass (if we are interested in gravity) or the solid sphere of insulating material carrying a uniform charge density  $\rho$  (if we want to do electrostatics). Let's pick the latter, just for variety. If we imagine a spherical "Gaussian surface" concentric with the sphere, with a radius  r  less than the sphere's radius  R, the usual isotropic symmetry argument gives   $\osurfintS \Vec{E} \cdot d\Vec{A} = 4\pi r^2 E$,  where  E  is the (constant) radial electric field strength at radius  r<R. The net charge enclosed within the Gaussian surface is   ${4\over3}\pi r^3 \, \rho$,  giving   $4\pi r^2 E = {1 \over \epsilon_0} {4\over3}\pi r^3 \, \rho$,  or

$$E(r<R) = {\rho \over 3 \epsilon_0} \, r$$ (18.7)

for the electric field inside such a uniform spherical charge density.

A similar linear relationship holds for the gravitational field within a solid sphere of uniform mass density, of course, except in that case the force on a "test mass" is always back toward the centre of the sphere - i.e. a linear restoring force with all that implies . . . .