Classical Electrodynamics · Problem 2.17

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

Physics — E&M · free-space-green-function-2d

Problem

(a) Construct the free-space Green function G(x,y;x′,y′)G(x, y; x', y') for two-dimensional electrostatics by integrating 1/R1/R with respect to (z′−z)(z' - z) between the limits ±Z\pm Z, where ZZ is taken to be very large. Show that apart from an inessential constant, the Green function can be written alternately as G(x,y;x′,y′)=−ln⁡[(x−x′)2+(y−y′)2]=−ln⁡[ρ2+ρ′2−2ρρ′cos⁡(ϕ−ϕ′)]\begin{aligned} G(x, y; x', y') &= -\ln[(x - x')^2 + (y - y')^2]\\ &= -\ln[\rho^2 + \rho'^2 - 2\rho\rho'\cos(\phi - \phi')] \end{aligned} (b) Show explicitly by separation of variables in polar coordinates that the Green function can be expressed as a Fourier series in the azimuthal coordinate, G=12π∑m=−∞∞eim(ϕ−ϕ′)gm(ρ,ρ′)G = \frac{1}{2\pi}\sum_{m=-\infty}^{\infty}e^{im(\phi - \phi')}g_m(\rho, \rho') where the radial Green functions satisfy 1ρ′∂∂ρ′(ρ′∂gm∂ρ′)−m2ρ′2gm=−4πρδ(ρ−ρ′)\frac{1}{\rho'}\frac{\partial}{\partial\rho'}\left(\rho'\frac{\partial g_m}{\partial\rho'}\right) - \frac{m^2}{\rho'^2}g_m = -\frac{4\pi}{\rho}\delta(\rho - \rho') Note that gm(ρ,ρ′)g_m(\rho, \rho') for fixed ρ\rho is a different linear combination of the solutions of the homogeneous radial equation (2.68) for ρ′<ρ\rho' < \rho and for ρ′>ρ\rho' > \rho, with a discontinuity of slope at ρ′=ρ\rho' = \rho determined by the source delta function. (c) Complete the solution and show that the free-space Green function has the expansion G(ρ,ϕ;ρ′,ϕ′)=−ln⁡(ρ>2)+2∑m=1∞1m(ρ<ρ>)m⋅cos⁡[m(ϕ−ϕ′)]G(\rho, \phi; \rho', \phi') = -\ln(\rho_>^2) + 2\sum_{m=1}^{\infty}\frac{1}{m}\left(\frac{\rho_<}{\rho_>}\right)^m\cdot\cos[m(\phi - \phi')] where ρ<(ρ>)\rho_<(\rho_>) is the smaller (larger) of ρ\rho and ρ′\rho'.

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