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finalnothing

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## Homework Statement

As per Griffiths 3.21, I am given the on axis potential a distance r from a uniformly charged disk of radius R as a function of [tex]\sigma[/tex]. Using this and the general solution for laplace's equation in spherical coordinates with azimuthal symmetry, calculate the first three terms in the general solution. Assume [tex]r>R[/tex].

## Homework Equations

[tex]V(r,\theta)=\sum^\infty_{l=0}{(A_lr^l +\frac{B_l}{r^{l+1}})P_l(\cos\theta)}[/tex]

[tex]V(r,0)=\frac{\sigma}{2\epsilon_0}(\sqrt{r^2+R^2}-r)[/tex]

## The Attempt at a Solution

As I know that V goes to 0 at infinity, all the A terms must be 0. Applying the boundary condition V(r, 0) and nothing that [tex]P_l(1)=1[/tex] leaves me with

[tex]V(r,0)=\sum^\infty_{l=0}{\frac{B_l}{r^{l+1}}}[/tex]

Now, my major idea was to rewrite V(r, 0) in terms of [tex]u=\frac{R}{r}[/tex] and then perform a taylor series expansion in terms of u. This will generate successive terms of the form [tex]\frac{C_l}{r^{l+1}}[/tex], then I simply assign my B variables equal to the C variables. However, I hit a moderate snag that I was not able to reason out. What point should I expand my taylor series about? Should I expand it about u=0 (r >> R) or about u=1 (r=R)? Or should I take the average of the two expansions? Each expansion does give different C values, when the problem expressed suggests this should not be the case. Am I simply going in the wrong direction or does this problem expect an approximate answer? Any help would be much appreciated.