The one last charge distribution for which we need to know the electric field is a sphere of charge. This problem is analogous to wanting to know the gravitational field of a spherical planet or star. The procedure for calculating the field of a sphere of charge is the same as we used for lines and planes, but the integrations are significantly more difficult. We will skip the details of the calculations and, for now, simply assert the result without proof. In Chapter 24 we’ll use an alternative procedure to find the field of a sphere of charge.

A sphere of charge Q and radius R, be it a uniformly charged sphere or just a spherical shell, has an electric field outside the sphere $(r\ge R)$ that is exactly the same as that of a point charge Q located at the center of the sphere:

This assertion is analogous to our earlier assertion that the gravitational force between stars and planets can be computed as if all the mass is at the center.

FIGURE 23.18 shows the electric field of a sphere of positive charge. The field of a negative sphere would point inward.

FIGURE 23.18 The electric field of a sphere of positive charge.