Electromagnetism on anisotropic fractal media



Basic equations of electromagnetic fields in anisotropic fractal media are obtained using a dimensional regularization approach. First, a formulation based on product measures is shown to satisfy the four basic identities of the vector calculus. This allows a generalization of the Green–Gauss and Stokes theorems as well as the charge conservation equation on anisotropic fractals. Then, pursuing the conceptual approach, we derive the Faraday and Ampère laws for such fractal media, which, along with two auxiliary null-divergence conditions, effectively give the modified Maxwell equations. Proceeding on a separate track, we employ a variational principle for electromagnetic fields, appropriately adapted to fractal media, so as to independently derive the same forms of these two laws. It is next found that the parabolic (for a conducting medium) and the hyperbolic (for a dielectric medium) equations involve modified gradient operators, while the Poynting vector has the same form as in the non-fractal case. Finally, Maxwell’s electromagnetic stress tensor is reformulated for fractal systems. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions in three different directions and reduce to conventional forms for continuous media with Euclidean geometries upon setting these each of dimensions equal to unity.

Mathematics Subject Classification

28A80 35Q61 35A15 


Fractal media Anisotropy Electromagnetism Maxwell’s equations Faraday’s law Ampère’s law Variational principle Electromagnetic energy stress 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ostoja-Starzewski M., Li J.: Fractal materials, beams and fracture mechanics. ZAMP 60(6), 1194–1205 (2009)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Li, J., Ostoja-Starzewski, J.: Fractal solids, product measures and fractional wave equations. Proc. R. Soc. A 465, 2521-2536 (2009); Errata (2010)Google Scholar
  3. 3.
    Ignaczak J., Ostoja-Starzewski M.: Thermoelasticity with Finite Wave Speeds. Oxford University Press, Oxford (2009)CrossRefGoogle Scholar
  4. 4.
    Demmie P.N., Ostoja-Starzewski M.: Waves in fractal media. J. Elast. 104, 187–204 (2011)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Li J., Ostoja-Starzewski M.: Micropolar continuum mechanics of fractal media. Int. J. Eng. Sci. 49, 1302–1310 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Joumaa H., Ostoja-Starzewski M.: On the wave propagation in isotropic fractal media. ZAMP 62, 1117–1129 (2011)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Tarasov V.E.: Electromagnetic fields on fractals. Mod. Phys. Lett. A 21, 1587–1600 (2006)MATHCrossRefGoogle Scholar
  8. 8.
    Tarasov V.E.: Fractional vector calculus and fractional Maxwell’s equations. Ann. Phys. 323, 2756–2778 (2008)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Tarasov V.E.: Fractional Dynamics. Springer, HEP, Berlin, Beijing (2010)MATHCrossRefGoogle Scholar
  10. 10.
    Falconer K.: Fractal Geometry: Mathematical Foundations and Applications. Wiley, Chichester (2003)MATHCrossRefGoogle Scholar
  11. 11.
    Seliger R.L., Whitham G.B.: Variational principles in continuum mechanics. Proc. R. Soc. A 305, 1–25 (1968)MATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of Mechanical Science and Engineering, Beckman Institute, Institute for Condensed Matter TheoryUniversity of Illinois at Urbana-ChampaignUrbanaUSA

Personalised recommendations