The critical state in thin superconductors as a mixed boundary value problem: analysis and solution by means of the Erd\'elyi-Kober operators

TL;DR

This paper introduces a unified method using Erdélyi-Kober operators and Hankel transforms to solve electromagnetic boundary value problems in thin superconductors, simplifying derivations and enabling extensions to complex geometries.

Contribution

The paper develops a systematic operator-based approach to solve mixed boundary value problems in thin superconductors, streamlining derivations and facilitating extensions to more complex geometries.

Findings

Derived current density and magnetic field distributions for superconducting discs and tapes.

Provided a detailed analytical method applicable to various geometries.

Simplified the process of solving mixed boundary value problems in superconductivity.

Abstract

With this paper we provide an effective method to solve a large class of problems related to the electromagnetic behavior of thin superconductors. Here all the problems are reduced to finding the weight functions for the Green integrals that represent the magnetic field components; these latter must satisfy the mixed boundary value conditions that naturally arise from the critical state assumptions. The use of the Erd\'elyi-Kober operators and of the Hankel transforms (and mostly the employment of their composition properties) is the keystone to unify the method toward the solution. In fact, the procedure consists always of the same steps and does not require any peculiar invention. For this reason the method, here presented in detail for the simplest cases that can be handled in analytical way (two parts boundary), can be directly extended to many other more complex geometries (three or more parts), which usually will require a numerical treatment. In this paper we use the operator technique to derive the current density and field distributions in perfectly conducting and superconducting thin discs and tapes subjected to a uniform magnetic field or carrying a transport current. Although analytical expressions for the field and current distributions have already been found by other authors in the past by using several other methods, their derivation is often cumbersome or missing key details, which makes it difficult for the reader to fully understand the derivation of the analytical formulas and, more importantly, to extend the same methods to solve similar new problems. On the contrary, the characterization of these cases as mixed boundary conditions has the advantage of referring to an immediate and na\"ive translation of physics into a consistent mathematical formulation whose possible extension to other cases is self-evident.