Equipotential Surfaces and Conductors

Consider a spherical conductor of radius R in which all charges are at rest. The electric field inside the conductor is zero, and it varies inversely with the square of the radial distance outside it.

Now imagine that the electric field has a tangential component outside the conductor's surface.

Such a tangential component would mean that there is also a tangential component of the electric field inside the conductor, causing charges to move in a rectangular loop. This would violate the electrostatic nature of the system.

Therefore no tangential component of the electric field is possible outside the conductor's surface. The electric field can only be perpendicular to the conductor's surface, making it an equipotential surface.

Consider two spherical conductors, with different radii, surface charge densities, and charges, connected by a thin conducting wire.

Here, the complete system is equipotential, and both spheres are at the same potential. Expressing the charge in terms of the surface charge density indicates that the surface charge density and electric field are higher for a smaller radius of curvature.