Classical Electrodynamics · Problem 2.7

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

Physics — E&M · green-function-half-space-dirichlet

Problem

Consider a potential problem in the half-space defined by z≥0z \geq 0, with Dirichlet boundary conditions on the plane z=0z = 0 (and at infinity). (a) Write down the appropriate Green function G(x,x′)G(\mathbf{x}, \mathbf{x}'). (b) If the potential on the plane z=0z = 0 is specified to be Φ=V\Phi = V inside a circle of radius aa centered at the origin, and Φ=0\Phi = 0 outside that circle, find an integral expression for the potential at the point PP specified in terms of cylindrical coordinates (ρ,ϕ,z)(\rho, \phi, z). (c) Show that, along the axis of the circle (ρ=0\rho = 0), the potential is given by Φ=V(1−za2+z2)\Phi = V\left(1 - \frac{z}{\sqrt{a^2 + z^2}}\right) (d) Show that at large distances (ρ2+z2≫a2\rho^2 + z^2 \gg a^2) the potential can be expanded in a power series in (ρ2+z2)−1(\rho^2 + z^2)^{-1}, and that the leading terms are Φ=Va22z(ρ2+z2)3/2[1−3a24(ρ2+z2)+5(3ρ2a2+a4)8(ρ2+z2)2+⋯ ]\Phi = \frac{Va^2}{2}\frac{z}{(\rho^2 + z^2)^{3/2}}\left[1 - \frac{3a^2}{4(\rho^2 + z^2)} + \frac{5(3\rho^2 a^2 + a^4)}{8(\rho^2 + z^2)^2} + \cdots\right] Verify that the results of parts c and d are consistent with each other in their common range of validity.

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