A Uniformly Charged Sphere

An insulating sphere of radius R has a total charge q uniformly distributed throughout its volume.
We want to find the electric field everywhere, that is, at an inside point r < R, and also at an outside point r > R.


For the case where r < R:

We construct a spherical Gaussian surface of radius r and apply Gauss's Law.

We already know that for a spherically symmetric case, . We have to be careful about qenc, though. qenc is the net charge within the Gaussian surface, not the total charge q of the entire sphere. For this reason, we need to multiply the total charge by the ratio of the volume of the Gaussian surface to the volume of the entire sphere:

Therefore,


For the case where r > R:

We construct a spherical Gaussian surface through an outside point r > R, and apply Gauss's Law. In this case,


The graph below is of the magnitude of the electric field due to a uniformly charged sphere, plotted as a function of distance from the center of the sphere.

Now, practice by using a few interactive applets.


Copyright 1998-1999 Rensselaer Polytechnic Institute. All Rights Reserved.