Vol. 46

Front:[PDF file] Back:[PDF file]
Latest Volume
All Volumes
All Issues

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


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.


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.


    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.

    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.

    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.

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

    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.

    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.

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

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

    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.

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

    14. Schenkel, G., "Das Vektorpotentialfeld stromumflossener Toroide," Annalen der Physik, Vol. 426, 541-560, 1939.

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

    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.

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

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

    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.

    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.

    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.

    26. Bhattacharyya, R., M. S. Janaki, and B. Dasgupta, "Minimum dissipative relaxed states in toroidal plasmas," Pramana --- J. Phys., Vol. 55, 947-952, 2000.

    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.

    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.

    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.

    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.

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

    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.

    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.