Classical Electrodynamics · Problem 3.22

Multipoles, Electrostatics of Macroscopic Media, Dielectrics — John David Jackson

Physics — E&M · green-functions-two-dimensional-polar

Problem

The geometry of a two-dimensional potential problem is defined in polar coordinates by the surfaces

\phi = 0

\phi = \beta

, and

\rho = a

, as indicated in the sketch. Using separation of variables in polar coordinates, show that the Green function can be written as

G(\rho,\,\phi;\,\rho',\,\phi') = \sum_{m=1}^{\infty} \frac{4}{m} \rho_>^{m\pi/\beta} \left(\frac{1}{\rho_>^{m\pi/\beta}} - \frac{\rho_>^{m\pi/\beta}}{a^{2m\pi/\beta}}\right) \sin\left(\frac{m\pi\phi}{\beta}\right) \sin\left(\frac{m\pi\phi'}{\beta}\right)

Problem 2.25 may be of use.