- #1
cscott
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Homework Statement
Two coaxial cylinders, radii [itex]{a,b}[/itex] where [itex]b>a[/itex]. Find the potential between the two cylinder surfaces.
Boundary conditions:
[tex]V(a,\phi) = 2 \cos \phi[/tex]
[tex]V(b,\phi) = 12 \sin \phi[/tex]
Homework Equations
Solution by separation of variables:
[tex]V(r,\phi) = a_0 + b_0 \ln s + \sum_k \left[ r^k(a_k \cos k\phi + b_k \sin k\phi)+r^{-k}(c_k\cos k\phi + d_k \sin k\phi)\right][/tex]
The Attempt at a Solution
I don't think I can eliminate the [itex]r^{-k}[/itex] term because the origin isn't between the two cylinders.
I think [itex]k=1[/itex] is the only term in the summation that is required for the solution.
[tex]V(r,\phi) = r(a_1 \cos \phi + b_1 \sin \phi)+\frac{1}{r}(c_1\cos \phi + d_1 \sin \phi)[/tex]
I don't see how to have the cosines vanish for [itex]V(b)[/itex] and sines vanish for [itex]V(a)[/itex] because of the common [itex]k[/itex] in both.
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