Classical Electrodynamics · Problem 2.12

Boundary-Value Problems in Electrostatics: II — John David Jackson

Physics — E&M · poisson-integral-cylinder

Problem

Starting with the series solution (2.71) for the two-dimensional potential problem with the potential specified on the surface of a cylinder of radius

b

, evaluate the coefficients formally, substitute them into the series, and sum it to obtain the potential inside the cylinder in the form of Poisson's integral:

\Phi(\rho, \phi) = \frac{1}{2\pi}\int_0^{2\pi}\Phi(b, \phi')\frac{b^2 - \rho^2}{b^2 + \rho^2 - 2b\rho\cos(\phi' - \phi)}\,d\phi'

What modification is necessary if the potential is desired in the region of space bounded by the cylinder and infinity?

Related theory

Poisson Brackets And Poisson Manifolds

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