volume | PIER Journals

Abstract

We prove a theorem on the magnetic energy minimum in a system of perfect, or ideal, conductors. It is analogous to Thomson's theorem on the equilibrium electric field and charge distribution in a system of conductors. We first prove Thomson's theorem using a variational principle. Our new theorem is then derived by similar methods. We find that magnetic energy is minimized when the current distribution is a surface current density with zero interior magnetic field; perfect conductors are perfectly diamagnetic. The results agree with currents in superconductors being confined near the surface. The theorem implies a generalized force that expels current and magnetic field from the interior of a conductor that loses its resistivity. Examples of solutions that obey the theorem are presented.

1. Carr, Jr., W. J., "Macroscopic theory of superconductors," Phys. Rev. B, Vol. 23, 3208-3212, 1981.
doi:10.1103/PhysRevB.23.3208

2. Jackson, J. D., Classical Electrodynamics, 3 Ed., John Wiley & Sons, New York, 1999.

3. Coulson, C. A., Electricity, 3 Ed., Oliver and Boyd, Edinburgh, 1953.

4. Panofsky, W. K. H. and M. Phillips, Classical Electricity and Magnetism, 2 Ed., Dover, New York, 2005.

5. Landau, L. D. and E. M. Lifshitz, Electrodynamics of Continuous Media, 2 Ed., Butterworth-Heinemann, Oxford, 1984.

6. Bakhoum, E. G., "Proof of Thomson's theorem of electrostatics," J. Electrostatics, Vol. 66, 561-563, 2008.
doi:10.1016/j.elstat.2008.06.002

7. Kovetz, A., Electromagnetic Theory, Oxford University Press, Oxford, 2000.

8. Sancho, M., J. L. Sebastián, and V. Giner, "Distribution of charges on conductors and Thomson's theorem," Eng. Sci. Educ. J., Vol. 10, 26-30, 2001.
doi:10.1049/esej:20010104

9. Donolato, C., "An application of Thomson's theorem to the determination of induced charge density," Eur. J. Phys., Vol. 24, L1-L4, 2003.
doi:10.1088/0143-0807/24/3/101

10. Brito, L. and M. Fiolhais, "Energetics of charge distributions," Eur. J. Phys., Vol. 23, 427-431, 2002.
doi:10.1088/0143-0807/23/4/306

11. Sancho, M., J. L. Sebastián, S. Muñoz, and J. M. Miranda, "Computational method in electrostatics based on monte carlo energy minimization," IEE Proc., Sci. Meas. Technol., Vol. 148, 121-124, 2009.
doi:10.1049/ip-smt:20010449

12. Karlsson, P. W., "Inductance inequalities for ideal conductors," Arch. f. Elektrotech., Vol. 67, 29-33, 1984.
doi:10.1007/BF01574728

13. Badía-Majós, A., "Understanding stable levitation of super-conductors from intermediate electromagnetics," Am. J. Phys., Vol. 74, 1136-1142, 2006.
doi:10.1119/1.2338548

14. McAllister, I. W., "Surface current density K: An introduction," IEEE Trans. Elect. Insul., Vol. 26, 416-417, 1991.
doi:10.1109/14.85112

15. Dolecek, R. L. and J. de Launay, "Conservation of flux by a superconducting torus," Phys. Rev., Vol. 78, 58-60, 1950.
doi:10.1103/PhysRev.78.58

16. Hehl, F. W. and Y. N. Obukhov, "Dimensions and units in electrodynamics," Gen. Relativ. Gravit., Vol. 37, 733-749, USA, 2005.
doi:10.1007/s10714-005-0059-2

17. Landau, L. D. and E. M. Lifshitz, The Classical Theory of Fields, 4 Ed., Pergamon, Oxford, 1975.

18. London, F. and H. London, "The electromagnetic equations of the supraconductor," Proc. Roy. Soc. A, Vol. 149, 71-88, 1935.
doi:10.1098/rspa.1935.0048

19. Badía-Majós, A., J. F. Cariñena, and C. López "Geometric treatment of electromagnetic phenomena in conducting materials: variational principles," J. Phys. A: Math. Gen., Vol. 39, 14699-14726, 2006.
doi:10.1088/0305-4470/39/47/013

20. Woltjer, L., "A theorem on force-free magnetic fields," Proc. Nat. Acad. Sci., Vol. 44, 489-491, 1958.
doi:10.1073/pnas.44.6.489

21. Griffiths, D. J., Introduction to Electrodynamics, 3 Ed., Prentice Hall, New Jersey, 1999.

22. Meissner, W. and R. Ochsenfeld, "Ein neuer Effekt bei eintritt der Supraleitfähigkeit," Naturwiss., Vol. 21, 787-788, 1933.
doi:10.1007/BF01504252

23. Hirsch, J. E., "Charge expulsion, spin Meissner effect, and charge inhomogeneity in superconductors," J. Supercond. Nov. Magn., Vol. 22, 131-139, 2009.
doi:10.1007/s10948-008-0381-5

24. Forrest, A. M., "Meissner and Ochsenfeld revisited," Eur. J. Phys., Vol. 4, 117-120, 1983, Comments on and translation into English of Meissner and Ochsenfeld.
doi:10.1088/0143-0807/4/2/011

25. Alfvén, H. and C.-G. Fälthammar, Cosmical Electrodynamics, 2 Ed., Oxford University Press, Oxford, 1963.

26. Essén, H, "From least action in electrodynamics to magnetomechanical energy," Eur. J. Phys., Vol. 30, 515-539, 2009.
doi:10.1088/0143-0807/30/3/009

27. Gorter, C. J. and H. Casimir, "On supraconductivity I," Physica, Vol. 1, 306-320, 1934.
doi:10.1016/S0031-8914(34)90037-9

28. Essén, H., "Magnetic fields, rotating atoms, and the origin of diamagnetism," Phys. Scr., Vol. 40, 761-767, 1989.
doi:10.1088/0031-8949/40/6/012

29. Essén, H., "Darwin magnetic interaction energy and its macroscopic consequences," Phys. Rev. E, Vol. 53, 5228-5239, 1996.
doi:10.1103/PhysRevE.53.5228

30. Essén, H., "Magnetic dynamics of simple collective modes in a two-sphere plasma model," Phys. of Plasmas, Vol. 12, 122101-1-7, 2005.

31. Essén, H., "Electrodynamic model connecting superconductor response to magnetic field and to rotation," Eur. J. Phys., Vol. 26, 279-285, 2005.
doi:10.1088/0143-0807/26/2/007

32. Greiner, W., Classical Electrodynamics, Springer, New York, 1998.
doi:10.1007/978-1-4612-0587-6

33. Fock, V., "Skineffekt in einem Ringe," Phys. Z. Sowjetunion, Vol. 1, 215-236, 1932.

34. De Launay, J., "Electrodynamics of a superconducting torus,", Technical Report NRL-3441, Naval Research Lab, Washington DC, 1949.

35. Carter, G. W., S. C. Loh, and C. Y. K. Po, "The magnetic field of systems of currents circulating in a conducting ring," Quart. Journ. Mech. and Applied Math., Vol. 18, 87-106, 1965.
doi:10.1093/qjmam/18.1.87

36. Bhadra, D., "Field due to current in toroidal geometry," Rev. Sci. Instrum., Vol. 39, 1536-1546, 1968.
doi:10.1063/1.1683157

37. Haas, H., "Das Magnetfeld eines gleichstromdurchflossenen Torus," Arch. f. Elektrotech., Vol. 58, 197-209, 1976.
doi:10.1007/BF01600116

38. Belevitch, V. and J. Boersma, "Some electrical problems for a torus," Philips J. Res., Vol. 38, 79-137, 1983.

39. Ivaska, V., V. Jonkus, and V. Palenskis, "Magnetic field distribution around a superconducting torus," Physica C, Vol. 319, 79-86, 1999.
doi:10.1016/S0921-4534(99)00279-8

40. Zhilichev, Y. N., "Superconducting cylinder in a static transverse magnetic field," IEEE Trans. Appl. Supercond., Vol. 7, 3874-3879, 1997.
doi:10.1109/77.659441

41. Matute, E. A., "On the superconducting sphere in an external magnetic field," Am. J. Phys., Vol. 67, 786-788, 1999.
doi:10.1119/1.19126

Dr. Miguel C. N. Fiolhais

Prof. Hanno Essén

Dr. Constanca Providencia

Dr. Arne B. Nordmark