Classical Electrodynamics · Problem 2.18

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

Physics — E&M · green-function-interior-dirichlet-cylinder

Problem

(a) By finding appropriate solutions of the radial equation in part b of Problem 2.17, find the Green function for the interior Dirichlet problem of a cylinder of radius bb [gm(ρ,ρ′=b)=0g_m(\rho, \rho' = b) = 0. See (1.40)]. First find the series expansion akin to the free-space Green function of Problem 2.17. Then show that it can be written in closed form as G=ln⁡[ρ2ρ′2+b4−2ρρ′b2cos⁡(ϕ−ϕ′)b2(ρ2+ρ′2−2ρρ′cos⁡(ϕ−ϕ′))]G = \ln\left[\frac{\rho^2\rho'^2 + b^4 - 2\rho\rho' b^2\cos(\phi - \phi')}{b^2(\rho^2 + \rho'^2 - 2\rho\rho'\cos(\phi - \phi'))}\right] or G=ln⁡[(b2−ρ2)(b2−ρ′2)+b2∣ρ−ρ′∣2b2∣ρ−ρ′∣2]G = \ln\left[\frac{(b^2 - \rho^2)(b^2 - \rho'^2) + b^2|\boldsymbol{\rho} - \boldsymbol{\rho}'|^2}{b^2|\boldsymbol{\rho} - \boldsymbol{\rho}'|^2}\right] (b) Show that the solution of the Laplace equation with the potential given as Φ(b,ϕ)\Phi(b, \phi) on the cylinder can be expressed as Poisson's integral of Problem 2.12. (c) What changes are necessary for the Green function for the exterior problem (b<ρ<∞b < \rho < \infty), for both the Fourier expansion and the closed form? [Note that the exterior Green function is not rigorously correct because it does not vanish for ρ\rho or ρ′→∞\rho' \to \infty. For situations in which the potential falls off fast enough as ρ→∞\rho \to \infty, no mistake is made in its use.]

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