Classical Electrodynamics · Problem 2.3

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

Physics — E&M · method-of-images-intersecting-planes

Problem

A straight-line charge with constant linear charge density λ\lambda is located perpendicular to the xx-yy plane in the first quadrant at (x0,y0)(x_0, y_0). The intersecting planes x=0x = 0, y≥0y \geq 0 and y=0y = 0, x≥0x \geq 0 are conducting boundary surfaces held at zero potential. Consider the potential, fields, and surface charges in the first quadrant. (a) The well-known potential for an isolated line charge at (x0,y0)(x_0, y_0) is Φ(x,y)=(λ/4πϵ0)ln⁡(R2/r2)\Phi(x, y) = (\lambda/4\pi\epsilon_0)\ln(R^2/r^2), where r2=(x−x0)2+(y−y0)2r^2 = (x - x_0)^2 + (y - y_0)^2 and RR is a constant. Determine the expression for the potential of the line charge in the presence of the intersecting planes. Verify explicitly that the potential and the tangential electric field vanish on the boundary surfaces. (b) Determine the surface charge density σ\sigma on the plane y=0y = 0, x≥0x \geq 0. Plot σ/λ\sigma/\lambda versus xx for (x0=2,y0=1)(x_0 = 2, y_0 = 1), (x0=1,y0=1)(x_0 = 1, y_0 = 1), and (x0=1,y0=2)(x_0 = 1, y_0 = 2). (c) Show that the total charge (per unit length in zz) on the plane y=0y = 0, x≥0x \geq 0 is Qx=−2πλtan⁡−1 ⁣(x0y0)Q_x = -\frac{2}{\pi}\lambda \tan^{-1}\!\left(\frac{x_0}{y_0}\right) What is the total charge on the plane x=0x = 0? (d) Show that far from the origin [ρ≫ρ0\rho \gg \rho_0, where ρ=(x2+y2)\rho = \sqrt{(x^2 + y^2)} and ρ0=(x02+y02)\rho_0 = \sqrt{(x_0^2 + y_0^2)}] the leading term in the potential is Φ→Φasym=4λπϵ0(x0y0)(xy)ρ4\Phi \to \Phi_{\rm asym} = \frac{4\lambda}{\pi\epsilon_0}\frac{(x_0 y_0)(xy)}{\rho^4} Interpret.

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