Solid Sphere - Field

A solid sphere of radius a has a charge Q distributed throughout its volume. Find the electric field at all points within and outside the sphere.

Solution:

First we find the volumetric charge density...

Next, we consider points inside the sphere (r < a), so we take a Gaussian surface of radius r less than a. The field is the flux divided by the area of the Gaussian surface:

Notice that, as r approaches zero, so does the field.

At r = a, the Gaussian surface encloses all of the charge Q and there is no need to use a charge density:

We could have obtained this from the first relationship, since it was valid for r = a as well as r < a.

For values of r > a, the same charge Q is enclosed by the Gaussian surface or radius r, so

This is essentially Serway's Example 24.5.