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# 23.9: Charge on a Conductor

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### 23.9: Charge on a Conductor

An interesting property of a conductor in static equilibrium is that extra charges on the conductor end up on its outer surface, regardless of where they originate. Consider a hollow metallic conductor with a uniform surface charge density. Since the conductor itself is in electrostatic equilibrium, there should not be any electric field inside the conductor. Now, assume a Gaussian surface enclosing the hollow portion. Applying Gauss's law, the inner surface of the hollow conductor will not have any charge.

Now, suppose a charge is enclosed inside the hollow conductor. Due to electrostatic equilibrium and Gauss's law, the conductor's inner surface acquires a negative charge. As the net charge inside the conductor is zero, the conductor's surface also acquires a net charge opposite to the cavity. Thus, the charges always reside on the surface of a conductor.

Consider another conductor with two cavities, 1 and 2. Cavity 1 encloses a positive charge, while cavity 2 encloses a negative charge. The polarization of the conductor results in induced negative and positive surface charges, respectively, on the inside surface of cavities 1 and 2, respectively. Similarly, the outside surface of the conductor shows an induced charge equal to the difference between the positive and negative induced charges inside the cavities.

The distribution of charges on the surfaces depends upon the geometry. At electrostatic equilibrium, the charge distribution in a conductor is such that the electric field by the charge distribution in the conductor cancels the electric field of the external charges at all points inside the conductor's body.

In summary, the net charge inside a closed conducting container is always zero. If the closed conducting container encloses a charge and the charge finds a conducting path, it flows to the container's surface. Else, the enclosed charge induces an equal and opposite charge on the inner surface, so the net charge inside is still zero. Any net charge on a conducting object resides on its surface.