There is a puzzling aspect of the solution (3.136) for the potential inside a grounded sphere with a uniformly charged wire along a diameter. Very close to the wire (i.e., for ρ=rsinθ≪b), the potential should be that of a uniformly charged wire, namely, Φ=(Q/4πϵ0b)ln(b/ρ)+Φ0. The solution (3.136) does not explicitly have this behavior.
(a) Show by use of the Legendre differential equation (3.10) and some integration by parts, that ln(cscθ) has the appropriate expansion in spherical harmonics to permit the solution (3.136) to be written in the alternative form,
Φ(x)=4πϵ0bQ{ln(rsinθ2b)−1−j=1∑∞2j(2j+1)4j+1(br)2jP2j(cosθ)}
in which the expected behavior near the wire is manifest. Give an interpretation of the constant term Φ0=−Q/4πϵ0b. Note that in this form, for any r/b<1 the Legendre polynomial series is rapidly convergent at all angles.
(b) Show by use of the expansion (3.38) that
21(sinθ/21+cosθ/21)=2j=0∑∞P2j(cosθ)
and that therefore the charge density on the inner surface of the sphere, Eq. (3.137), can be expressed alternatively as
σ(θ)=−4πb2Q{21(sinθ/21+cosθ/21)−j=0∑∞2j+11P2j(cosθ)}
The (integrable) singular behavior at θ=0 and θ=π is now exhibited explicitly. The series provides corrections in ln(1/θ) as θ→0.