Vol. 71

Latest Volume
All Volumes
All Issues

Classical Power and Energy Relations for Macroscopic Dipolar Continua Derived from the Microscopic Maxwell Equations

By Arthur D. Yaghjian
Progress In Electromagnetics Research B, Vol. 71, 1-37, 2016


Positive semi-definite expressions for the time-domain macroscopic energy density in passive, spatially nondispersive, dipolar continua are derived from the underlying microscopic Maxwellian equations satisfied by classical models of discrete bound dipolar molecules or inclusions of the material or metamaterial continua. The microscopic derivation reveals two distinct positive semi-definite macroscopic energy expressions, one that applies to diamagnetic continua (induced magnetic dipole moments) and another that applies to paramagnetic continua (alignment of permanent magnetic dipole moments), which includes ferro(i)magnetic and antiferromagnetic materials. The diamagnetic dipoles are ``unconditionally passive'' in that their Amperian (circulating electric current) magnetic dipole moments are zero in the absence of applied fields. The analysis of paramagnetic continua, whose magnetization is caused by the alignment of randomly oriented permanent Amperian magnetic dipole moments that dominate any induced diamagnetic magnetization, is greatly simplified by first proving that the microscopic power equations for rotating permanent Amperian magnetic dipoles (which are shown to not satisfy unconditional passivity) reduce effectively to the same power equations obeyed by rotating unconditionally passive magnetic charge magnetic dipoles. The difference between the macroscopic paramagnetic and diamagnetic energy expressions is equal to a ``hidden energy'' that parallels the hidden momentum often attributed to Amperian magnetic dipoles. The microscopic derivation reveals that this hidden energy is drawn from the reservoir of inductive energy in the permanent microscopic Amperian magnetic dipole moments. The macroscopic, positive semi-definite, time-domain energy expressions are applied to lossless bianisotropic media to determine the inequalities obeyed by the frequency-domain bianisotropic constitutive parameters. Subtleties associated with the causality as well as the group and energy-transport velocities for diamagnetic media are discussed in view of the diamagnetic inequalities.


Arthur D. Yaghjian, "Classical Power and Energy Relations for Macroscopic Dipolar Continua Derived from the Microscopic Maxwell Equations," Progress In Electromagnetics Research B, Vol. 71, 1-37, 2016.


    1. Poynting, J. H., "On the transfer of energy in the electromagnetic field," Phil. Trans. R. Soc. Lond., Vol. 175, 343-361, January 1884.

    2. Maxwell, J. C., A Treatise on Electricity and Magnetism, Unabridged 3rd Ed., Dover, New York, 1954.

    3. Yaghjian, A. D., "Reflections on Maxwell’s treatise," Progress In Electromagnetics Research, Vol. 149, 217-249, November 2014; see also Yaghjian, A. D., ``An overview of Maxwell's treatise," FERMAT Multimedia, Vol. 11, 2015.

    4. Glasgow, S., M. Ware, and J. Peatross, "Poynting’s theorem and luminal total energy transport in passive dielectric media," Phys. Rev. E, Vol. 64, 046610(1-14), September 2001.

    5. Mansuripur, M., "On the foundational equations of the classical theory of electrodynamics," Resonance, Vol. 18, 130-155, February 2013.

    6. Welters, A., Y. Avniel, and S. G. Johnson, "Speed-of-light limitations in passive linear media," Phys. Rev. A, Vol. 90, 023847(1-17), August 2014.

    7. Felsen, L. B. and N. Marcuwitz, Radiation and Scattering of Waves, Wiley/IEEE Press, New York, 1994.

    8. Kittel, C., Introduction to Solid State Physics, 8th Ed., Wiley, Hoboken, NJ, 2005.

    9. Yaghjian, A. D., Relativistic Dynamics of the Charged Sphere: Updating the Lorentz-Abraham Model of the Electron, 2nd Ed., Springer, New York, 2006.

    10. Yaghjian, A. D., A. Alu, and M. G. Silveirinha, "Anisotropic representation for spatially dispersive periodic metamaterial arrays," Transformation Electromagnetics and Metamaterials, Ch. 13, Springer, 2014.

    11. Yaghjian, A. D., A. Alu, and M. G. Silveirinha, "Homogenization of spatially dispersive metamaterial arrays in terms of generalized electric and magnetic polarizations," Photonics and Nanostructures --- Fundamentals and Applications, 374-396, November 2013.

    12. Hansen, T. B. and A. D. Yaghjian, Plane-wave Theory of Time-domain Fields: Near- field Scanning Applications, Wiley/IEEE Press, New York, 1999.

    13. Yaghjian, A. D., "Electric dyadic Green’s functions in the source region," Proc. IEEE, Vol. 68 & 69, 248-263 & 282–285, February 1980 & February 1981.

    14. Jackson, J. D., Classical Electrodynamics, 3rd Ed., Wiley, New York, 1999.

    15. Chew, W. C., "Vector potential electromagnetics with generalized gauge for inhomogeneous media: formulation," Progress In Electromagnetics Research, Vol. 149, 69-84, November 2014.

    16. Stratton, J. A., Electromagnetic Theory, McGraw-Hill, New York, 1941.

    17. Russakoff, G., "A derivation of the macroscopic Maxwell equations," Am. J. Phys., Vol. 38, 1188-1195, October 1970.

    18. Tai, C. T., Generalized Vector and Dyadic Analysis, Wiley/IEEE Press, New York, 1996.

    19. Yaghjian, A. D., "Maxwell and cavity electromagnetic fields within continuous sources," Am. J. Phys., Vol. 53, 859-863, September 1985.

    20. Yaghjian, A. D., "Boundary conditions for electric quadrupolar continua," Radio Science, Vol. 49, 1289-1299, December 2014.

    21. Lorentz, H. A., "The fundamental equations for electromagnetic phenomena in ponderable bodies, deduced from the theory of electrons," Proc. Roy. Acad. Amsterdam, Vol. 5, 254-266, 1902.

    22. Rosenfeld, L., Theory of Electrons, Dover, New York, 1965.

    23. Van Vleck, J. H., The Theory of Electric and Magnetic Susceptibilities, Oxford University Press, Oxford, UK, 1932.

    24. Robinson, F. N. H., Macroscopic Electromagnetism, Pergamon, Oxford, UK, 1973.

    25. De Groot, S. R. and L. G. Suttorp, Foundations of Electrodynamics, 195-196, North-Holland, Amsterdam, 1972.

    26. Landau, L. D., E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd Ed., Butterworth Heinemann, Oxford, UK, 1984.

    27. Yaghjian, A. D., "Extreme electromagnetic boundary conditions and their manifestation at the inner surfaces of spherical and cylindrical cloaks," Metamaterials Journal, Vol. 4, 70-76, August–September 2010.

    28. Costa, J. T., M. G. Silveirinha, and A. Alu, "Poynting vector in negative-index metamaterials," Phys. Rev. B, Vol. 83, 165120(1-8), 2011.

    29. Drude, P., The Theory of Optics, Dover, New York, 2005.

    30. Simovski, C. R. and S. A. Tretyakov, "Local constitutive parameters of metamaterials from an effective-medium perspective," Phys. Rev. B, Vol. 75, 195111(1-10), 2007.

    31. Scher, A. D. and E. F. Kuester, "Boundary effects in the electromagnetic response of a metamaterial in the case of normal incidence," Progress In Electromagnetics Research B, Vol. 14, 341-381, 2009.

    32. Shore, R. A. and A. D. Yaghjian, "Traveling waves on two- and three-dimensional periodic arrays of lossless scatterers," Radio Science, Vol. 42, RS6S21(1-40), 2007.

    33. Yaghjian, A. D. and S. Maci, "Alternative derivation of electromagnetic cloaks and concentrators," New Journal of Physics, Vol. 10, 115022(1-29), November 2008.

    34. Yaghjian, A. D., "Power-energy & dispersion relations for diamagnetic media," Proceedings of the IEEE APS Symposium, Charleston SC, 4 pages, June 2009.

    35. Plonsey, R. and R. E. Collin, Principles and Applications of Electromagnetic Fields, McGraw-Hill, New York, 1961.

    36. Abraham, M. and R. Becker, The Classical Theory of Electricity and Magnetism, Blackie, London, UK, 1932.

    37. Penfield, P., Jr. and H. A. Haus, Electrodynamics of Moving Media, M.I.T. Press, Cambridge, MA, 1967.

    38. McDonald, K. T., "On the definition of `hidden' momentum,", http://physics.princeton.edu/mcdonald/examples/hiddendef.pdf, accessed 5 August 2015.

    39. Yaghjian, A. D. and S. R. Best, "Impedance, bandwidth, and Q of antennas," IEEE Trans. Antennas Propagat., Vol. 53, 1298-1324, April 2005; Correction, Vol. 55, 3748, December 2007.

    40. Yaghjian, A. D., "Internal energy, Q-energy, Poynting’s theorem, and the stress dyadic in dispersive material," IEEE Trans. Antennas Propagat., Vol. 55, 1495-1505, June 2007.

    41. Marques, R., L. Jelinek, M. J. Freire, J. D. Baena, and M. Lapine, "Bulk metamaterials made of resonant rings," Proc. IEEE, Vol. 99, 1660-1668, October 2011.

    42. Alu, A., A. D. Yaghjian, R. A. Shore, and M. G. Silveirinha, "Causality relations in the homogenization of metamaterials," Phys. Rev. B, Vol. 84, 054305(1-16), August 2011.

    43. Yaghjian, A. D., S. Maci, and E. Martini, "Characteristic wave velocities in spherical electromagnetic cloaks," New Journal of Physics, Vol. 11, 113011(1-14), November 2009.

    44. Boyd, R. W. and D. J. Gunther, "Controlling the velocity of light pulses," Science, Vol. 326, 1074-1077, November 2009.

    45. Wood, B. and J. B. Pendry, "Metamaterials at zero frequency," J. Phys. Condens. Matter, Vol. 19, 076208(1-9), 2007.

    46. Sohl, C., M. Gustafsson, and A. Bernland, "Some paradoxes associated with a recent sum rule in scattering theory," Proc. URSI General Assembly, Chicago, USA, August 2008.