In Ex. 3.9, we derived the exact potential for a spherical shell of radius R, which carr...

In Ex. 3.9, we derived the exact potential for a spherical shell of radius R, which carries a surface charge σ = k cos θ.

(a) Calculate the dipole moment of this charge distribution.

(b) Find the approximate potential, at points far from the sphere, and compare the exact answer (Eq. 3.87). What can you conclude about the higher multipoles?

Reference equation 3.87

Reference example 3.9

A specified charge density σ₀(θ) is glued over the surface of a spherical shell of radius R. Find the resulting potential inside and outside the sphere.