Classical Electrodynamics · Problem 5.17

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

Physics — E&M · current-distribution-near-permeable-slab-image

Problem

A current distribution J(x)\mathbf{J}(\mathbf{x}) exists in a medium of unit relative permeability adjacent to a semi-infinite slab of material having relative permeability μr\mu_r and filling the half-space, z<0z < 0. (a) Show that for z>0z > 0 the magnetic induction can be calculated by replacing the medium of permeability μr\mu_r by an image current distribution, J∗\mathbf{J}^*, with components, (μr−1μr+1)Jx(x,y,−z),(μr−1μr+1)Jy(x,y,−z),−(μr−1μr+1)Jz(x,y,−z)\left(\frac{\mu_r - 1}{\mu_r + 1}\right)J_x(x, y, -z), \qquad \left(\frac{\mu_r - 1}{\mu_r + 1}\right)J_y(x, y, -z), \qquad -\left(\frac{\mu_r - 1}{\mu_r + 1}\right)J_z(x, y, -z) (b) Show that for z<0z < 0 the magnetic induction appears to be due to a current distribution [2μr/(μr+1)]J[2\mu_r/(\mu_r + 1)]\mathbf{J} in a medium of unit relative permeability.

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