A point charge with a charge of -3Q is placed at the center of a conducting spherical shell. The shell has an inner radius a, an outer radius b, and a net charge of +2Q.

What is the electric field as a function of r?

For r > b, E = kQ/r2 and points toward the center

For a < r < b, E = 0

For r < a, E = 3kQ/r2, directed toward the center

What is the electric potential as a function of r?

To find the total potential we can simply use superposition. The net potential is the potential from the different pieces, the point charge, the +3Q charge on the inner surface of the conductor, and the -Q on the outer surface of the conductor.

Region Point Charge Conductor
Inner surface
Outer surface
Net Potential
r > b -3kQ/r +3kQ/r -kQ/r -kQ/r
a < r < b -3kQ/r +3kQ/r -kQ/b -kQ/b
r < a -3kQ/r +3kQ/a -kQ/b -3kQ/r + 3kQ/a - kQ/b