A Conducting Spherical Shell

Consider a conducting spherical shell with a net negative charge. At equilibrium the excess charge is uniformly distributed over the outside edge of the shell.

None appears on the inside edge because that would mean field lines would either pass through the shell or would converge at the center. Field is excluded from the bulk of the shell, and field lines would not converge at the center unless there was a charge there.

If the outer radius of the shell is R, the field is zero for r < R, and the field looks like that of a point charge (kQ/r2) for r > R.

Concentric Spheres

What happens with two concentric spheres, or with a point charge at the center of a spherical shell? The net field outside any of the objects is simply the vector sum of the fields from the different objects. The field inside any conductors is zero - charge on the conductor shifts to ensure this.

In the example above, a positive point charge Q is placed at the center of a conducting shell that has a net charge of -2Q. What is the field as a function of r, if the shell has inner radius R1 and outer radius R2?

For r < R1 the field comes only from the point charge. The charge on the sphere does not produce a field in this region, so E = kQ/r2 directed out from the center.

For R1 < r < R2 the field is zero because that's inside the conductor. For this to be true there must be a net -Q on the inside surface of the shell to stop all the field lines coming from the +Q charge at the center.

For r > R2 the field once again looks like the field from a point charge equal to the total charge in the system, which is +Q - 2Q = -Q. Therefore the field here is also given by E = kQ/r2, but now is directed in toward the center.