Symmetry and Gauss's Law

Application: Charged Conductors in Equilibrium

Imagine that we start with a conductor of an arbitrary shape, with no charge on it. Now we give this conductor a net charge
q--for example, by touching it momentarily with a charged rod.

The charge q will spread throughout the conductor and will eventually settle down to its equilibrium distribution.

Where does the charge q reside in the conductor when it is in equilibrium? We will use Gauss's Law to prove an important property of a charged conductor in equilibrium:

In electrostatic equilibrium, the electric charge on an isolated conductor resides entirely on its surface. No charge is found within the body of the conductor.

Proof using Gauss's Law:

Construct a Gaussian surface just inside the surface of the conductor. By definition, an equilibrium situation is reached when the charges are arranged so that they no longer move within the conductor. That is, there is no electric field within the conductor at equilibrium. (If there were an electric field, it would cause the charges to move.) Therefore, the electric flux about this Gaussian surface. But Gauss's Law says that , so the enclosed charge qenc = 0. One can make the Gaussian surface approach as close to the conductor surface as possible, and the enclosed charge is still zero. Therefore, the only place that the given charge q can reside is on the conductor's surface.

To test your understanding, try out this practice problem.


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