The response to an alternating field of a slab of flux-permeated superconducting material with a local electrodynamic response law E~IJI J is obtained numerically. Fundamental and third-harmonic components of the susceptibility are reported as functions of 0. and slab thickness. The third-harmonic component is always smaller than in the critical-state model, but not much smaller if cr ))1. If the slab is thick enough, and 0 )0 the field is totally extinguished at a finite depth. In that case it gradually adopts a self-replicating, nonsinusoidal wave form whose amplitude diminishes according to a power law while the phase lag diverges logarithmically.Vinokur, Feigel'man, and Geshkenbein' gave an exact solution describing flux creep in high temperature superconductors, assuming the creep activation barrier grows logarithmically with decreasing current density J according to U=UQlnIJ, /JI. This law had been found as an exact result when the vortex motion is controlled by intrinsic pinning in a layered system with the field parallel to the layers, and approximates other theoretical results.It implies a power-law dependence of electric field on current density:where o = UolkT. This is often a good description of experimental data over a wide range of J.Only the product J, p needs to be specified.Vinokur, Feigel'man, and Geshkenbein' considered a simple slab geometry and the response to a suddenly applied uniform field. Initially they made the same simplifying assumption as Bean, viz. , that the field switched on at t =0 produces an extra magnetic induction B (z, t}«BQ, where the prior uniform induction, BQ --(MQHQ was induced by a constant field H" «HQ«H, 2. For incomplete flux penetration, B(z,t) was found to depend only on t ' ' + 'z, a scaling law which results from the dimensionality.A closely related problem has an even simpler analytic solution, as reported earlier. It is the case when B0 is perpendicular to the slab, which is of isotropic material, and B(z, t) rotates uniformly in its plane at angular ratẽ . The appropriate Bean's solution is immediately generalizable to the creep law, Eq. (1). The penetration is best expressed in terms of a scaling depth: (B / p )rr/(rr+2)(2 /p r0)1/(++2) Here B, is the magnitude of B(z,t) at the surface, z =0, and 5, is a hybrid between Bean's penetration depth and the Ohmic skin depth. The rotating perturbation is found to be totally extinguished at a depth -1( 3~+4) 1/2( +2 )(0 + 1)/(2rr+4) Z0 Cr X [ ( (~+ 1)] 1/(2(r+4)5 [In Ref. 7, p =1/(o +1). ] In the penetrated layer, 0 (z (z0. ' I B(z,t) I IB, = ( 1z Izo )('+ and the phase lag -(I)(z) is given by ())(z) = [(1+2/o )(2+2/0 )] ' ln(1z/z() ) . (4a) (4b)An alternating, plane-polarized B(z,t) is experimentally more practicable and more profitable, since odd harmonics are generated and can be analyzed (even harmonics too, if the conditions of Bean's simplifying assumption are not met). In this case it is diScult to envisage an analytic solution on account of the non-analytic points where H and J change sense, so we sought, an...
