Applications of Gauss Law

Gauss law can be used to make an important prediction, namely:

An excess charge, placed on an insulated conductor, resides entirely on its outer surface.

This hypothesis was shown to be true by experiment. Consider an insulated conductor (shown dotted) very close inside the actual surface of the conductor. When an excess charge is placed at random on an insulated conductor, it will set up electric fields inside the conductor. These fields act on the charge carriers of the conductor and cause them to move. That is, they set up internal currents. These currents redistribute the excess charge in such a way that the internal electric fields are automatically reduced in magnitude. Finally the electric fields inside the conductor become zero everywhere, the currents automatically stop, and electrostatic conditions prevail. This redistribution of charge takes place in a negligible amount of time.

If E is zero everywhere inside the conductor, it must be zero for every point on the Gaussian surface. Thus
            Ф = Ф E. dS = 0

for the Gaussian surface. According to Gauss’s law, this means that there must be no net charge inside the Gaussian surface. It the excess charge q is not inside this surface, it can only be outside it. That it, the excess charge must be on the actual surface of the conductor.