A sphere of radius R has inhomogeneous charge distribution on its surface. Its surface charge density is

σ = σ0 cos θ,

where σ0 is a positive constant, and θ is the angle from the “north pole” to the point of interest and satisfies 0 ≤ θ ≤ π. Calculate the electric potential φ and the electric field E both inside and outside the sphere. Do not make any approximations. Check that φ is continuous from outside to inside. Check that ∇ · E = 0 strictly inside and ∇ · E = 0 strictly outside.