Application of the Gibbs Magnetodynamic Principle to Calculation of the Distribution of Direct Currents in Solid Bodies

Authors: Aliev I.N., Gusarov A.I., Dokukin M.Yu., Samedova Z.A. Published: 08.08.2020
Published in issue: #4(91)/2020  
DOI: 10.18698/1812-3368-2020-4-73-88

Category: Physics | Chapter: Theoretical Physics  
Keywords: volume currents, superficial currents, two-dimensional vector analysis, variation analysis, Lagrange multipliers, stream of magnetic induction, ideal magnetic, London’s equation, Meissner --- Ochsenfeld effect

The thermodynamic hypothesis of Gibbs allowing to solve a problem by means of the magnetic principle of virtual works is applied to finding of equilibrium distribution of superficial and volume stationary currents in a continuous body. The variation of magnetic energy is considered with the additional conditions defining constancy of currents, two of which having a differential appearance are necessary and sufficient for the solution of a task in case of a one-coherent body. If the considered body two-coherent (torus, a thick ring) appears one more condition is necessary. In work it is shown what this condition which is also providing uniqueness of the decision can be or constancy of the current proceeding through cross section a torus, or a task of a constant stream of magnetic induction through an opening a torus. At problem definition the first option as more evident was chosen. The problem is solved with the help of a method of Lagrange multipliers. The main received result is that circumstance that induction of magnetic field and volume current in volume address in zero. Thus, magnetic field together with currents is squeezed out on a surface. Communication of the received results with Meissner --- Ochsenfeld effect and the London’s equation applied in the theory of superconductivity and also a problem of communication of molecular currents and currents of conductivity are discussed


[1] Green G. Mathematical papers. London, MacMillan and Co., 1871.

[2] Thomson K.W. Reprint of papers on electrostatics a magnetism. MacMillan and Co., 1872.

[3] Fiolhais M.C.N., Essen H., Providencia C., et al. Magnetic field and current are zero inside ideal conductors. PIERB, 2011, vol. 27, pp. 187--212. DOI: https://doi.org/10.2528/PIERB10082701

[4] Fiolhais M.C.N., Essen H. Magnetic field expulsion in perfect conductors --- the magnetic equivalent of Thomson’s theorem. Proc. Symp. Progress in Electromagnetic Research. Stockholm, 2013, pp. 1193--1197.

[5] Maksimov E.G. High-temperature superconductivity: current state. Phys. Usp., 2000, vol. 43, no. 10, pp. 965--990. DOI: https://doi.org/10.1070/PU2000v043n10ABEH000770

[6] Aliev I.N., Kopylov I.S. Use of Dirac monopoles formalism in some magnetism problems. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2015, no. 6 (63), pp. 25--39 (in Russ.). DOI: https://doi.org/10.18698/1812-3368-2015-6-25-39

[7] Meissner W., Ochsenfeld R. Ein Neurer Effekt bei Eintritt der Supraleitfahigkeit. Naturwissenschaften, 1933, vol. 21, no. 44, pp. 787--788.

[8] Hirsh J.E. The origin of the Meissner effect in new and old superconductors. Phys. Scr., 2012, vol. 85, no. 3, art. 035704. DOI: https://doi.org/10.1088/0031-8949/85/03/035704

[9] London F., London H. The electromagnetic equations of the supraconductor. Proc. Roy. Soc. A, 1935, vol. 149, iss. 866, pp. 71--88. DOI: https://doi.org/10.1098/rspa.1935.0048

[10] London H. Phase-equilibrium of superconductors in a magnetic field. Proc. Roy. Soc., 1935, vol. 152, iss. 876, pp. 650--663.

[11] Lynton E.A., McLean W.L. Type II superconductors. Advances Electronics and Electron Physics, 1967, vol. 23, pp. 1--37. DOI: https://doi.org/10.1016/S0065-2539(08)60059-1

[12] Peierls R.E. Quantum theory of solids. Clarendon Press, 1955.

[13] Aliev I.N., Kopylov I.S. About electrodynamic Londons’ model and Gorter --- Casimir theory. J. Synch. Investig., 2017, vol. 11, no. 2, pp. 238--245. DOI: https://doi.org/10.1134/S1027451016050670

[14] Schwinger J. A magnetic model of matter. Science, 1969, vol. 165, no. 3895, pp. 757--761. DOI: https://doi.org/10.1126/science.165.3895.757

[15] Schwinger J. Magnetic charge and quantum field theory. Phys. Rev., 1966, vol. 144, iss. 4, pp. 1087--1093. DOI: https://doi.org/10.1103/PhysRev.144.1087

[16] Schwinger J. Electric- and magnetic-charge renormalization. I. Phys. Rev., 1966, vol. 151, iss. 4, pp. 1048--1054. DOI: https://doi.org/10.1103/PhysRev.151.1048

[17] Schwinger J. Sources and magnetic charge. Phys. Rev., 1968, vol. 173, iss. 5, pp. 1536--1544. DOI: https://doi.org/10.1103/PhysRev.173.1536

[18] Aliev I.N., Melikyants D.G. On the magnetization of a superconducting ball. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2016, no. 3 (66), pp. 82--92 (in Russ.). DOI: https://doi.org/10.18698/1812-3368-2016-3-82-92

[19] Aliev I.N., Samedova Z.A. Minimum magnetic energy of the superconducting sphere. Elektromagnitnye volny i elektronnye sistemy [Electromagnetic Waves and Electronic Systems], 2017, vol. 22, no. 5, pp. 47--54 (in Russ.).

[20] Aliev I.N., Samedova Z.A. Current distribution in a superconducting segment of a cylindrical wire. Vestnik MGOU. Seriya: Fizika-matematika [Bulletin MSRU. Series: Physics and Mathematics], 2016, no. 1, pp. 88--98 (in Russ.). DOI: http://dx.doi.org/10.18384/2310-7251-2016-1-88-97

[21] Aliev I.N., Samedova Z.A. On the boundary conditions for the Maxwell and London electrodynamic equations. J. Synch. Investig., 2017, vol. 11, no. 6, pp. 1306--1312. DOI: https://doi.org/10.1134/S1027451017050238

[22] Aliev I.N., Samedova Z.A. Application of Gibbs variational principle and Lagrange multiplier method to problems of electrostatics. Part 2. General conditions of equilibrium. Elektromagnitnye volny i elektronnye sistemy [Electromagnetic Waves and Electronic Systems], 2018, vol. 23, no. 2, pp. 32--39 (in Russ.).

[23] Aliev I.N., Samedova Z.A. Surface current approximation in certain problems of the classical theory of superconductivity. Russ. Phys. J., 2018, vol. 61, no. 4, pp. 770--780. DOI: https://doi.org/10.1007/s11182-018-1458-7