ARTICLE IN PRESS JID: PHYSC [m5G;January 12, 2016;13:55] Physica C: Superconductivity and its applications 000 (2016) 1–4 Contents lists available at ScienceDirect Physica C: Superconductivity and its applications journal homepage: www.elsevier.com/locate/physc Electric field obtained from an elliptic critical-state model for anisotropic type-II superconductors C. Romero-Salazar ∗ , O.A. Hernández-Flores Escuela de Ciencias, Universidad Autónoma “Benito Juárez” de Oaxaca, Apdo. Post. No. 76, Oaxaca de Juárez, Oaxaca, C.P. 68120, México article info Article history: Received 27 September 2015 Accepted 8 December 2015 Available online xxx Keywords: Critical state Elliptic model Anisotropy Electric field abstract The conventional elliptic critical-state models (ECSM) establish that the electric field vector is zero when it flows a critical current density in a type-II superconductor. This proposal incorporates a finite electric field on the ECSM to study samples with anisotropic-current-carrying capacity. Our theoretical scheme has the advantage of being able to dispense of a material law which drives the electric field magnitude, however, it does not consider the magnetic history of the superconductor. © 2015 Elsevier B.V. All rights reserved. 1. Introduction The macroscopic approach for the superconducting-critical-state employing elliptic models has been successful to describe materials with structural anisotropy [1] as well as the anisotropy produced by flux-line cutting [2–4]. The elliptic model and others start with the hypothesis that a material in electromagnetic stationary state flows a critical current with no electric field. However it is well established that in type-II superconductors there is dissipation and therefore a non-zero electric field. For this reason, Clem proposed an extension for the elliptical-type models [5] incorporating the electric field to the critical ellipse as a new element, he preserved the use of a vertical law to calculate the electric field magnitude. A comparison between the elliptic and the extended models was performed to study the collapse of the remanent magnetization of a PbBi specimen by a sweeping ex- ternal transverse magnetic field [6]. The fundamental idea about the origin of an electric field is that the magnetic field penetration, in vor- tex form, is prevented by a random distribution of pinning centers, thus the magnetic flux distribution deals with a variety of metastable states from which can exit when the current density exceeds a criti- cal value J c , which corresponds to the threshold between pinning and flux transport. Due to the Lorentz force per volume F L = J × B over the vortex array, they would move if F L exceeds the average pinning force F p = J c B [5]. The other mechanism that can generate an electric field is the entanglement of vortices in such a way that when the current density reaches a threshold value J c|| , it starts the flux-line cutting [7] and a vortex reconfiguration [8]. On the other hand, it has been stud- ∗ Corresponding author. Tel.: +529515154743. E-mail address: cromeros@ifuap.buap.mx, cromero@uabjo.mx (C. Romero-Salazar). ied, experimental and numerically, the electric field in thin films [9]. In this work, the electric field was reconstructed using a magneto- optical technique. The theory that we will expose here works at the parallel geom- etry, it does not take into account transport currents neither the ex- istence of an initial magnetic configuration previous to any external field. In this paper we show how to incorporate an electric field to the elliptic-type theories. The outline of this paper is as follows, in Section 2 we describe the response of an infinite superconducting plate under the influence of a external field parallel to the sample plane. It is defined the geom- etry, the structural properties, the set of equations that describe the electromagnetic field behavior as well as the boundary conditions. Additionally, it is shown how to incorporate a finite value of the elec- tric field and its direction in the so-called critical state, and finally is postulated the critical state of an anisotropic type-II superconduc- tor under an elliptic approximation with non-zero electric field. In Section 3 we present an example to obtain numerically the magnetic induction and current density profiles as well as their orientations. In Section 4 is synthesized the relevant points of this work. 2. Theory The system under study is a plate of thickness x = d with in- finite surface extended along −∞ < y < ∞ and −∞ < z < ∞. We are interested at the so-called parallel geometry, where every ex- ternal magnetic field H a is applied parallel to the yz − plane, thus, H a = H a (sin α a ˆ y + cos α a ˆ z), with the angle α a defined respect to the z−axis. In this case the electromagnetic fields inside the supercon- ducting sample lay on the yz − plane depending on the variable x only. Due to the symmetry of the system, the planar super-current http://dx.doi.org/10.1016/j.physc.2015.12.002 0921-4534/© 2015 Elsevier B.V. All rights reserved. Please cite this article as: C. Romero-Salazar, O.A. Hernández-Flores, Electric field obtained from an elliptic critical-state model for anisotropic type-II superconductors, Physica C: Superconductivity and its applications (2016), http://dx.doi.org/10.1016/j.physc.2015.12.002