Electric field in superconductors with rectangular cross section

The electric field E induced in type-II superconductors during the penetration of magnetic flux exhibits a strange profile when the sample has the usual square or rectangular cross section perpendicular to the applied field ${\mathit{B}}_{\mathit{a}}$(t). For example, in the Bean critical state with full penetration of the current density j, though j=${\mathit{j}}_{\mathit{c}}$ is constant, E is not constant but is a linear function of space directed along concentric rectangles (like j) with magnitude E exhibiting a zig-zag-folded profile with (a) sharp minima with E=0 along the discontinuity lines where the current flow bends, (b) sharp maxima along the median of the short sides, and (c) sharp folds along two lines which are at a distance of half the short side from the short side. This fold defines new discontinuity lines occurring in regions where the current flow is quasihomogeneous. The resulting electric charge density q=${\mathrm{\epsilon }}_0$divE is piecewise constant in twelve sections of the rectangle where it is either zero or q=±${\mathrm{\epsilon }}_0$∂${\mathit{B}}_{\mathit{a}}$/∂t. These exotic features of E apply to arbitrary specimen thickness, unlike the current density and the magnetic field, which during flux penetration or exit are qualitatively different in longitudinal and transverse geometries. From a nonlinear diffusion equation, which is local for longitudinal and nonlocal for transverse geometry, the current density, magnetic, and electric fields, vortex velocity, charge density, and local energy dissipation are calculated for penetration and creep of longitudinal or transverse flux in slabs, strips, rectangular bars, and rectangular films or plates of a superconductor which is characterized by a highly nonlinear resistivity.