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Revista mexicana de física E
Print version ISSN 1870-3542
Rev. mex. fís. E vol.52 n.2 México Dec. 2006
Enseñanza
Complete electromagnetic multipole expansion including toroidal moments
A. Góngora T. y E. Ley-Koo
Centre for Interdisciplinary Research on Complex Systems, Physics Department, Northeastern University Boston M.A.
Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México D.F., México.
Recibido el 3 de marzo de 2006;
aceptado el 22 de mayo de 2006
Abstract
The electromagnetic multipole expansion presented in this paper is complete on two accounts: i) It is valid for all points in space, and ii) it recognizes the existence of toroidal moments. The electromagnetic field due to alternating poloidal currents in a toroidal solenoid is evaluated exactly via the solution of the inhomogeneous vector Helmholtz equations, using the outgoing wave Green function technique and the Debye potentials for sources and fields. The physical meaning of the toroidal moments can be appreciated when they are compared with the familiar electric and magnetic moments; the analysis of the long-wavelength limit of the exact results also explains the previous neglect of the toroidal moments. The magnetostatic limit and the point source limit are also physically and didactically interesting.
Keywords: Electromagnetic radiation; multipole expansion; electric, magnetic and toroidal moments.
Resumen
El desarrollo multipolar electromagnético que se presenta en este artículo es completo por dos razones: i) es válido para todos los puntos del espacio, y ii) reconoce la existencia de momentos toroidales. El campo electromagnético producido por corrientes poloidales alternas en un solenoide toroidal se evalúa exactamente a través de la solución de ecuaciones de Helmholtz inhomogéneas, usando la técnica de la función de Green de onda saliente y los potenciales de Debye para fuentes y campos. El significado físico de los momentos toroidales se destaca al compararlos con los momentos eléctricos y magnéticos familiares; el análisis del límite de longitud de onda grande de los resultados exactos también explica la ignorancia previa de los momentos toroidales. El límite magnetostatico y el límite de fuente puntual son también interesantes físicamente y didácticamente.
Descriptores: Radiación electromagnética; desarrollo multipolar; momentos eléctricos, magnéticos y toroidales.
PACS: 41.20 Jb
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