Electric field in hard superconductors with arbitrary cross section and general critical current law

The induced electric field E(x) during magnetic flux entry in superconductors with arbitrary cross section Ω and general critical current law, has been evaluated by integration along the vortex penetration paths. Nontrivial flux motion streamlines are obtained from a variational statement of the critical state, which takes the form of an optimization problem on the finite element dis-cretization of Ω. The generality of the theory allows to deal with physical conditions not considered before. In particular, it is shown that the boundary condition to be used for determining E is the knowledge of the locus E = 0 within the sample. This is emphasized for anisotropic materials in which the electric field is not parallel to the surface. Both numerical and analytical evaluations are presented for homogeneous materials with different geometries: convex and concave contours, samples with holes, variable curvature contours, and for anisotropic samples. In the isotropic case, discontinuities in the electric current paths are shown to be related to changing curvature of the sample's surface. Anisotropic samples display the same kind of discontinuities, even for constant surface curvature.