Finding potential inside the cavity of a Conductor

[xcgirl]

Homework Statement

A conductor, maintained at a potential V, contains a spherical cavity of radius R.
A point charge q is placed at a distance a ( a < R)from the center of the cavity.
Find the potential of the electric field in the cavity .

Homework Equations

I was considering using Poisson's Equation because we recently did that in class. Also, we are learning about the method of images right now. I am really unsure of where to start with this problem. I did another problem successfully that had 2 point charges above an infinite conductor and I simply replicated the opposite of those charges the same distance below the conductor and used the method of images to find the force on one of the charges. But, this problem is asking for potential, not force and the sphere cavity thing makes it weird for me. I don't even know where to start. Any hints would be helpful :\

The Attempt at a Solution

 

[Bessel]

This is just a simple boundary value problem (of the Dirichlet type) where one seeks a function obeying Poisson's equation in the spherical cavity and taking the constant value V on its boundary. If you can find such a function, you're done. This problem is designed to make it especially easy and you can check that V(r) = q(1/r-1/a)/4πε+V is the solution, where r is the distance from the point charge q.