HRW Ch 24 ex 40 a - e

A point charge of +q is at the center of an electrically neutral, spherically conducting shell with inner radius a and outer radius b. What are the charges on the inner and outer surfaces? What are the fields in each region?

The field inside the conductor has to be zero.

Gauss's Law says that the total charge enclosed has to be zero, so the charge on the inner surface of the conductor is -q. Conservation of charge tells you that the charge on the outer conductor is then +q.

Gauss's Law then says that the field for r<a is given by Coulomb's Law with Q = +q since that is the total charge enclosed in a gaussian surface of radius r.

It also says that outside the the field has the same form since Q = q + (-q) + q = +q.