A conducting sphere of radius a, at potential V0, is surrounded by a thin concentric sph...

A conducting sphere of radius a, at potential V₀, is surrounded by a thin concentric spherical shell of radius b, over which someone has glued a surface charge

where k is a constant and θ is the usual spherical coordinate.

(a) Find the potential in each region: (i) r > b, and (ii) a < r < b.

(b) Find the induced surface charge σ_i (θ) on the conductor.

(c) What is the total charge of this system? Check that your answer is consistent with the behavior of V at large r .