Reduction of the classical electromagnetism to a two-dimensional curved surface

ABSTRACT

The reduction of the three-dimensional classical electromagnetism to a two-dimensional curved surface is performed in a twofold way. In the first case, the ordinary two-dimensional electromagnetism is obtained with sources in the form of conserved electric currents flowing along the surface. The electric field is a two-vector tangent to the surface and magnetic field is a scalar quantity. In the second approach, the reduced theory is that of the two-vector magnetic field and a scalar electric one. The only source coupled to the fields is now a scalar, subject to no conservation law. In the redefined theory, this scalar source may be converted into an eddy magnetic current flowing in the surface. No magnetic monopoles appear. Our results can find some applications in the electrodynamics of thin layers and of metal-dielectric interfaces.

KEYWORDS:

• Electromagnetism
• wave propagation
• thin layers
• low-dimensional theory

Acknowledgments

The author would like to thank to Professors I. Bialynicki-Birula and J. Kijowski for interesting and elucidating discussions.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

The work was supported by the Polish National Science Center [grant number 2012/07/B/ST1/03347].