Research of the current distribution in solids using the Gibbs magnetodynamic principle

Abstract

Abstract To find equilibrium distribution of surface and volumetric currents in continuous bodies Gibbs’ thermodynamic hypothesis is used, which allows to solve a task using the magnetic principle of virtual work. Variation of the magnetic energy is considered under additional conditions, defining constancy of currents, two of which are of a differential nature, necessary and sufficient to solve the task in case of simply connected body. If the body under consideration is doubly-connected (e.g. a torus, a thick ring), one extra condition is necessary. It is shown in this paper that this condition, which makes the solution unique at the same time, can be the constancy of current flowing through the cross section of the torus or setting of a constant flux of magnetic induction through the torus hole. Setting up a problem, the first method is selected as more visual. The task is solved with the use of the method of Lagrange multipliers. The fundamental result of this paper is that magnetic induction and the volumetric current in a volume are vanished. Thus, the magnetic field with currents are extruded to the surface. Connections between obtained results, the Meissner-Oxenfield effect and the London equitation used in the theory of superconductivity are discussed, as well as the problem of connection of molecular currents and conduction currents.