Classical Electrodynamics · Problem 5.24

Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws — John David Jackson

Physics — E&M · conducting-plane-circular-hole-magnetic-field

Problem

(a) For the perfectly conducting plane of Section 5.13 with the circular hole in it and the asymptotically uniform tangential magnetic field

\mathbf{H}_0

on one side, calculate the added tangential magnetic field

\mathbf{H}^{(1)}

on the side of the plane with

\mathbf{H}_0

. Show that its components for

\rho > a

are

H_x^{(1)} = \frac{2H_0 a^3}{\pi}\frac{xy}{\rho^4\sqrt{\rho^2 - a^2}}

H_y^{(1)} = \frac{2H_0 a^3}{\pi}\frac{y^2}{\rho^4\sqrt{\rho^2 - a^2}} + \frac{H_0}{\pi}\left[\frac{a}{\rho}\sqrt{1 - \frac{a^2}{\rho^2}} - \sin^{-1}\left(\frac{a}{\rho}\right)\right]

(b) Sketch the lines of surface current flow in the neighborhood of the hole on both sides of the plane.