In Ex. 3.9, we obtained the potential of a spherical shell with surface charge σ(θ) = k...

In Ex. 3.9, we obtained the potential of a spherical shell with surface charge σ(θ) = k cos θ. In Prob. 3.30, you found that the field is pure dipole outside; it’s uniform inside (Eq. 3.86). Show that the limit R → 0 reproduces the delta function term in Eq. 3.106.

Reference prob 3.30

Reference equation 3.86

Reference equation 3.106

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.