Equipotential Surfaces and Conductors

For a conductor in which all charges are at rest, the conductor's surface is equipotential. The electric field is always perpendicular to equipotential surfaces. Therefore, in a conductor with static charges, the electric field just outside the conductor is always perpendicular to the conductor's surface. Any tangential component of the electric field will cause charges to move inside the conductor, which will violate the electrostatic nature of the system. In an electrostatic situation, if a conductor has a cavity with no charges inside it, then there can be no charge anywhere on the surface of the cavity. This means that inside a charged metallic enclosure, with no charges inside the enclosure, one can touch the walls of the enclosure from inside without getting an electrical shock.

Consider two conducting spheres of different radii having different surface charge densities and different amounts of static charges. If a thin conducting wire connects these conductors, the whole system becomes equipotential. The potential of each sphere is the same, and the surface charge density and the electric field are higher on the conductor with a smaller radius of curvature. A practical application of this phenomenon is a lightning rod—a grounded metal rod with a sharp end pointing upward. As positive charge accumulates in the ground due to a negatively charged cloud overhead, the electric field around the sharp point becomes very large. When the field reaches a value of approximately 3.0 × 10⁶ N/C (the dielectric strength of the air), the free ions in the air are accelerated to such high energies that their collisions with air molecules ionize the molecules. The resulting free electrons in the air then flow through the rod to Earth, thereby neutralizing some of the positive charges. This keeps the electric field between the cloud and the ground from becoming large enough to produce a lightning bolt in the region around the rod.