(a) Evaluate the exterior spherical multipole moments for a shell of radius R which carries a surface charge density σ(θ, φ) = σ₀ sin θ cos φ.

(b) Write φ(r < R,θ,ϕ) in the form φ(x, y, z, r).

(c) Evaluate the interior spherical multipole moments for the shell of part (a).

(d) Write φ(r < R,θ,ϕ) in the form φ(x, y, z, r).

(e) Check the matching conditions for φ and E at r = R.

(f) Extract the dipole moment p of the shell from your answer to part (b).