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Pengwuino
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Jackson 3.12: An infinite, thin, plane sheet of conducting material has a circular hole of radius a cut in it. A thin, flat disc of the same material and slightly smaller radius lies in the plane, filling the hole, but separted from the sheet by a very narrow insulating ring. The disc is maintained at a fixed potential V, while the infinite hseet is kept at zero potential.
a) Using appropriate cylindrical coordinates, find an intergral expression involving Bessel functions for the potential at any point above the plane.
Attempt!
So considering there is no\phi dependence, I know the potential \Phi = Z(z)R(\rho)\Phi(\phi) reduces down to 2 equations: Z(z) = Asinh(kz) + Bcosh(kz) and R(\rho) = CJ(k\rho) + DN(k\rho).
By the fact that V is finite at \rho = 0, D = 0.
So the solution looks like
\sum\limits_{m = 0}^\infty {\sum\limits_{n = 1}^\infty {(A_{mn} \sinh (k_{mn} z) + B_{mn} (\cosh (k_{mn} z))J_m (k_{mn} \rho )} } where k_{mn} = \frac{x_{mn}}{a} where x_{mn} is the n-th root for the m-th Bessel function.
At z=0, I have
V_0 = \sum\limits_{m=0}^\infty{\sum\limits_{n=1}^\infty{B_{mn}J_m(k_{mn}\rho)}. At this point, i integrate both sides by <br /> \int_0^a {\rho J_m (k_{mn'} \rho )d\rho } and get:
<br /> \begin{array}{l}<br /> \int_0^a {V_0 J_m (k_{mn'} \rho )\rho d\rho = } \int\limits_0^a {\sum\limits_{m = 0}^\infty {\sum\limits_{n = 1}^\infty {B_{mn} J_m (k_{mn} \rho )J_m (k_{mn'} \rho )\rho d\rho } } } \\ <br /> \int_0^a {V_0 J_m (k_{mn'} \rho )\rho d\rho = } \frac{1}{2}\sum\limits_{m = 0}^\infty {\sum\limits_{n = 1}^\infty {B_{mn} |J_{m + 1} (k_{mn} )|^2 \delta _{nn'} } } \\ <br /> \end{array}
Now I'm quite new to Bessel functions so at this point, I'm not sure what to do. It seems like I still have one series on the right and a number on the left. I ran the left side through mathematica and it said it was simply V_0 but I don't know enough about Bessel functions to tell if that's right or not. Any tips or problems to be pointed out would be greatly appreciated :)
a) Using appropriate cylindrical coordinates, find an intergral expression involving Bessel functions for the potential at any point above the plane.
Attempt!
So considering there is no\phi dependence, I know the potential \Phi = Z(z)R(\rho)\Phi(\phi) reduces down to 2 equations: Z(z) = Asinh(kz) + Bcosh(kz) and R(\rho) = CJ(k\rho) + DN(k\rho).
By the fact that V is finite at \rho = 0, D = 0.
So the solution looks like
\sum\limits_{m = 0}^\infty {\sum\limits_{n = 1}^\infty {(A_{mn} \sinh (k_{mn} z) + B_{mn} (\cosh (k_{mn} z))J_m (k_{mn} \rho )} } where k_{mn} = \frac{x_{mn}}{a} where x_{mn} is the n-th root for the m-th Bessel function.
At z=0, I have
V_0 = \sum\limits_{m=0}^\infty{\sum\limits_{n=1}^\infty{B_{mn}J_m(k_{mn}\rho)}. At this point, i integrate both sides by <br /> \int_0^a {\rho J_m (k_{mn'} \rho )d\rho } and get:
<br /> \begin{array}{l}<br /> \int_0^a {V_0 J_m (k_{mn'} \rho )\rho d\rho = } \int\limits_0^a {\sum\limits_{m = 0}^\infty {\sum\limits_{n = 1}^\infty {B_{mn} J_m (k_{mn} \rho )J_m (k_{mn'} \rho )\rho d\rho } } } \\ <br /> \int_0^a {V_0 J_m (k_{mn'} \rho )\rho d\rho = } \frac{1}{2}\sum\limits_{m = 0}^\infty {\sum\limits_{n = 1}^\infty {B_{mn} |J_{m + 1} (k_{mn} )|^2 \delta _{nn'} } } \\ <br /> \end{array}
Now I'm quite new to Bessel functions so at this point, I'm not sure what to do. It seems like I still have one series on the right and a number on the left. I ran the left side through mathematica and it said it was simply V_0 but I don't know enough about Bessel functions to tell if that's right or not. Any tips or problems to be pointed out would be greatly appreciated :)
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