Question

At time t=0 a uniform surface current K = Kj is suddenly switched on in the entire x-y plane. K is a constant and everything is set in an infinite uniformiy magnetized linear isotropic homogenous material with permeability and magnetic susceptibility xm (which is not a dielectric, i.e. ? = €9). You may assume that the current appears everywhere on the plane at the same instant (which is not realistic since it implies instantaneous transfer of current to infinity). Now at t=0 magnetic field lines appear parallel to the plane both above and below it and start propagating out to infinity. After a very long period of time the field is constant and uniform everywhere and can be calculated using the ordinary static case Ampere's law. a) Calculate the magnitude and direction of the H, B and M field after an infinite amount of time. b) As the H field propagates outward, there are clearly regions where the field has not appeared yet. Use Ampere's law again (without Maxwell's correction) to argue that a time changing D must exist. This problem can in fact be used as another proof of Maxwell's correction of Ampere's law.

Answer 1

(a) Given that k = kx so the dir^n of B will be perpendicular to h Then that dir^n may be y or 2 But z is not possible because the z - component of B can'6 possibly depend on the dir^n of the current in the xy plane so must probability y dir^n Then we know that, According to Ampere's law contourintegral B.dl = 2 B l = mu_0 I enclosed implies 2 Bl = mu_0 h l implies = mu_0 k/2 (factor 2 comes due to extincting an equal distance above & the surface) we draw a rectangular amperian loop as shown in fig parallel to the y_2 plane & extending an equal distance above & below the surface \r\n Then B = { (mu_0/2 k y for z < 0 - (mu_0/2) k y for 2 > 0 Then we know that B = mu_0 H implies - B/mu_0 so H = { k/2 y for z < 0 - x/2 y for z > 0 Again we know