Expansion for the potential of a ring of charge

[davesface]

Homework Statement

A total charge q is distributed uniformly along a ring of radius b. The ring is in the x-y plane centered on the origin. The multipole expansion is not valid for r<b. Find an expansion for the potential valid in this region

Homework Equations

The charge density is just $$\lambda=\frac{q}{2\pi b}$$.
The previous problem was about the region r>b, and the book gives the quadrupole-moment tensor components as $$Q_{xx}=Q_{yy}=\frac{b^2}{2}q, Q_{zz}=-b^2q, Q_{xy}=Q_{yz}=Q_{xz}=0$$ and the potential as $$\Phi =\frac{q}{r}+0+\frac{1}{r^5}\frac{b^2q}{4}(x^2+y^2-2z^2)$$.

The Attempt at a Solution

Frankly, I don't understand the problem. Obviously at z=0 we can just integrate $$\Phi=\int_{0}^{2\pi}\lambda d\theta=\frac{q}{b}$$, but that's a pretty trivial introductory level problem. Any thoughts on what exactly this problem is asking me to do?

 

[diazona]

Well, the multipole expansion for large r is basically a Taylor expansion in the parameter b/r. The series only converges if the value of the parameter is less than 1, or equivalently, only if r > b. So I think what this problem wants you to do is come up with a Taylor expansion in a different parameter, one that will be less than 1 when r < b, so that this new expansion will be valid in the region r < b (not just at r = 0).