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2012-12-10

Magnetic Energy of Surface Currents on a Torus

By Hanno Essen, Johan Sten, and Arne B. Nordmark
Progress In Electromagnetics Research B, Vol. 46, 357-378, 2013
doi:10.2528/PIERB12102909

Abstract

The magnetic energy and inductance of current distributions on the surface of a torus are considered. Specifically, we investigate the in°uence of the aspect ratio of the torus, and of the pitch angle for helical current densities, on the energy. We show that, for a fixed surface area of the torus, the energy experiences a minimum for a certain pitch angle. New analytical relationships are presented as well as a review of results scattered in the literature. Results for the ideally conducting torus, as well as for thin rings are given.

Citation


Hanno Essen, Johan Sten, and Arne B. Nordmark, "Magnetic Energy of Surface Currents on a Torus," Progress In Electromagnetics Research B, Vol. 46, 357-378, 2013.
doi:10.2528/PIERB12102909
http://www.jpier.org/PIERB/pier.php?paper=12102909

References


    1. Tonomura, A., N. Osakabe, T. Kawasaki, J. Endo, S. Yano, and H. Yamada, "Evidence for Aharonov-Bohm effect with magnetic ¯eld completely shielded from electron wave," Phys. Rev. Lett., Vol. 56, 792-795, 1986.
    doi:10.1103/PhysRevLett.56.792

    2. Osakabe, N., T. Matsuda, T. Kawasaki, J. Endo, A. Tonomura, S. Yano, and H. Yamada, "Experimental confirmation of Aharonov-Bohm effect using a toroidal magnetic field confined by a superconductor," Phys. Rev. A, Vol. 34, 815-822, 1986.
    doi:10.1103/PhysRevA.34.815

    3. Carron, N. J., "On the fields of the torus and the role of the vector potential," Am. J. Phys., Vol. 64, 717-729, 1995.
    doi:10.1119/1.17842

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

    5. 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

    6. Doinikov, N. I., "Determination of magnetic fields set up by currents flowing on the surface of a torus," Sov. Phys. --- Tech. Phys., Vol. 9, 1367-1374, USA, 1965, Translated from: Zhurnal Tekhnicheskoi Fiziki, Vol. 34, 1769-1779, 1964.

    7. Gyimesi, M. and D. Lavers, "Magnetic field around an iron torus," IEEE Transactions on Magnetics, Vol. 28, 2799-2801, 1992.
    doi:10.1109/20.179631

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

    9. Hansen, R. C. and R. D. Ridgley, "Fields of the contrawound toroidal helix antenna," IEEE Trans. Ant. Prop., Vol. 49, 1138-1141, 2001.
    doi:10.1109/8.943308

    10. Haubitzer, W., "Das magnetische Feld eines Toroids und einer mehrlagigen Zylinderspule," Z. elektr. Inf. Energietech., Vol. 4, 129-136, 1974.

    11. McDonald, K., "Electromagnetic fields of a small helical toroidal antenna,", Dec. 2008, URL: http://www.physics.princeton.edu/~mcdonald/examples/cwhta.pdf.

    12. Page, C. H., "External field of an ideal toroid," Am. J. Phys., Vol. 39, 1039-1043, 1971.
    doi:10.1119/1.1986365

    13. Page, C. H., "On the external magnetic field of a closed-loop core," Am. J. Phys., Vol. 39, 1206-1209, 1971.
    doi:10.1119/1.1976606

    14. Schenkel, G., "Das Vektorpotentialfeld stromumflossener Toroide," Annalen der Physik, Vol. 426, 541-560, 1939.
    doi:10.1002/andp.19394260604

    15. Sy, W. N.-C., "Magnetic field due to helical currents on torus," J. Phys. A: Math. Gen., Vol. 14, 2095-2112, 1981.
    doi:10.1088/0305-4470/14/8/031

    16. Rayleigh, L., "On the self-induction of electric currents in a thin anchor-ring," Proc. Roy. Soc. A, Vol. 86, No. 590, 562-571, 1912.
    doi:10.1098/rspa.1912.0046

    17. Haas, H., "Ein Beitrag zur Berechnung der Selbstinduktivitateines Torus," Arch. f. Elektrotech., Vol. 58, 305-308, 1976.
    doi:10.1007/BF01584576

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

    19. Kliem, B. and T. Torok, "Torus instability," Phys. Rev. Lett., Vol. 96, 255002-1-255002-4, 2006.

    20. Salingaros, N. A., "Optimal current distribution for energy storage in superconducting magnets," J. Appl. Phys., Vol. 69, 531-533, 1991.
    doi:10.1063/1.347701

    21. Tayler, R. J., "The distribution of currents on the surface of a toroidal conductor,", Technical Report AERE-M-563, Atomic Energy Research Establishment, Harwell, 1960.

    22. Zic, T., B. Vrsnak, and M. Skender, "The magnetic flux and self-inductivity of a thick toroidal current," J. Plasma Physics, Vol. 73, 741-756, 2007.

    23. Buck, G. J., "Force-free magnetic-field solution in toroidal coordinates," J. Appl. Phys., Vol. 36, 2231-2235, 1965.
    doi:10.1063/1.1714456

    24. Romashets, E. P. and M. Vandas, "Force-free field inside a toroidal magetic cloud," Geophys. Res. Lett., Vol. 64, 144505-1-144505-7, 2003.

    25. Miller, G. and L. Turner, "Force free equilibria in toroidal geometry," Phys. Fluids, Vol. 24, 363-365, 1981.
    doi:10.1063/1.863351

    26. Bhattacharyya, R., M. S. Janaki, and B. Dasgupta, "Minimum dissipative relaxed states in toroidal plasmas," Pramana --- J. Phys., Vol. 55, 947-952, 2000.
    doi:10.1007/s12043-000-0064-7

    27. Miura, Y., M. Sakota, and R. Shimada, "Force-free coil principle applied to helical winding," IEEE Transactions on Magnetics, Vol. 30, 2573-2576, 1994.
    doi:10.1109/20.305804

    28. Aliferov, A. and S. Lupi, "Skin effect in toroidal conductors with circular cross section," COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 27, 408-414, 2008.
    doi:10.1108/03321640810847698

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

    30. 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

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

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

    33. 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

    34. Malmberg, J. H. and M. N. Rosenbluth, "High frequency inductance of a torus," Rev. Sci. Instr., Vol. 36, 1886-1887, 1965.
    doi:10.1063/1.1719491

    35. Irons, M. L., "The curvature and geodesics of the torus,", 2005, URL: http://www.rdrop.com/~half/math/torus/torus.geodesics.pdf.

    36. Hayt Jr., W. H. and J. A. Buck, Engineering Electromagnetics, McGraw-Hill, New York, 2006.

    37. Grover, F. W., Inductance Calculations --- Working Formulas and Tables, Van Nostrand, New York, 1946.

    38. Snow, C., Formulas for Computing Capacitance and Inductance, National Bureau of Standards, Washington DC, 1954.

    39. Knoepfel, H. E., Magnetic Fields: A Comprehensive Theoretical Treatise for Practical Use, Wiley-Interscience, New York, 2000.

    41. Paul, C. R., "Inductance --- Loop and Partial," John Wiley, Hoboken, NJ, 2010.

    42. Field Theory Handbook --- Including Coordinate Systems, Di®erential Equations and their Solutions, P. and D. E. Spencer, D. E. Spencer, Springer, Berlin, 1961.

    43. Becker, R., Electromagnetic Fields and Interactions, Blaisdell, New York, 1964, Reprinted: Dover, New York, 1982.

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

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

    46. Cohen, E. R., The Physics Quick Reference Guide, AIP Press, Woodbury, NY, 1996.

    47. Frank, N. H. and W. Tobocman, "Electromagnetic theory," Fundamental Formulas of Physics, D. H. Menzel (ed.), Vol. 1, 307-354, Dover, New York, 1960.

    48. Fiolhais, M. C. N., H. Essen, C. Providentia, and A. B. Nordmark, "Magnetic ¯eld and current are zero inside ideal conductors," Progress In Electromagnetics Research B, Vol. 27, 187-212, 2011.

    49. Neumann, F. E., "Allgemeine Gesetze der inducirten elektrischen Strome," Abhandlungen der Koniglichen Akademie der Wissenschaften zu Berlin, Phys. Klasse., 1845.

    50. Weisstein, E. W., CRC Concise Encyclopedia of Mathematics, 2nd Ed., Chapman & Hall/CRC, Boca Raton, 2003.