laplace's equation in two dimensions_clyindrical coordinates


by haidara
Tags: coordinates, equation, laplace
haidara
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#1
Nov22-10, 02:44 AM
P: 1
1. The problem statement, all variables and given/known data
an infinitely long thin conducting cylindrical shell(radius R) of surface charge density

[tex]\sigma[/tex]=[tex]\sigma_{1}[/tex]sin(2[tex]\Phi[/tex])+[tex]\sigma_{2}[/tex]cos([tex]\Phi[/tex]).
what are the four boundary conditions for this problem?
using the four boundary conditions and the identification of the coefficients of sin(n[tex]\Phi[/tex])and cos(n[tex]\Phi[/tex])find the expression of the potential inside and outside the cylindrical shell.
2. Relevant equations
the general solution of laplace's equation in this case can be written:
V(r,[tex]\Phi[/tex])=[tex]\sum^{n=1}_{\infty}[/tex][[tex]A_{n}[/tex]cos(n[tex]\Phi[/tex])+
[tex]B_{n}[/tex]sin(n[tex]\Phi[/tex]))r[tex]^{n}[/tex]+(C[tex]_{}n[/tex]cos(n[tex]\Phi[/tex])+D[tex]_{}n[/tex]sin(n[tex]\Phi[/tex])r[tex]^{-n}[/tex]]+A[tex]_{}0[/tex]ln(r)+C[tex]_{}0[/tex].
take C[tex]_{0}[/tex]=0 inside the cylinder
3. The attempt at a solution
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