The transverse critical-state susceptibility of rectangular bars

The complex ac susceptibility χ = χ′ - jχ″ of an infinitely long hard superconducting rectangular (2a × 2b) bar is calculated numerically with the uniform ac field applied along the 2b dimension, based on the critical-state model with a constant Jc. Normalized to the exact low-field limit of - χ′, χ0, - χ′ and χ″ as functions of the field amplitude Hm normalized to the exact full penetration field Hp are given in figures and tables for 0.01 ≤ b/a ≤ 100. The numerical results are compared with existing exact and approximate analytical results. It is found for any value of b/a that with increasing Hm/Hp, χ″ is proportional to and inversely proportional to Hm at Hm/Hp ≪ 1 and ≫ 1, respectively. The maximum χ″/χ0 at a certain intermediate Hm/Hp decreases and then increases with increasing b/a, showing a minimum value at b/a ≈ 0.5. The low-field linear dependence of χ″ on Hm at low b/a is qualitatively different from the prediction of a one-dimensional calculation for a thin strip, whose low-field χ″ varies as Hm2.