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Applying the Lagrange multipliers method to the calculation of DC magnetic field


I.N. Aliev – Dr. Sc. (Phys.-Math.), Professor, Department «Physics», Bauman Moscow State Technical University I.S. Kopylov – student, Bauman Moscow State Technical University

To find the equilibrium distribution is stationary surface and bulk currents in the solid body applied thermodynamic Gibbs hypothesis, allowing to solve a problem using the magnetic principle of virtual work. The variation of the magnetic energy is considered with the additional conditions that determine the consistency of the currents, two of which, named the differential nature that are necessary and sufficient for solving the problem in the case of a simply connected body. If you consider the body doubly linked (Thor, thick ring) turns out you need one more condition. It is shown that this condition, at the same time ensuring the uniqueness of the solution, or may be a constant current flowing through the cross section of the torus or the job of the permanent flow of magnetic flux through the hole of the torus. When the formulation was chosen the first option as more graphic, the Problem is solved using the method of Lagrange multipliers, the Main obtained result is the fact that the magnetic field and the bulk current in the volume go to zero. I.e. the magnetic field with the currents is squeezed out to the surface. Discussed the relationship of the results obtained with the Meissner effect-Axenfeld and the London equation used in the theory of superconductivity. Discussed the problem of the relationship of molecular currents and conduction currents.


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