Journal of Applied Mathematics
Volume 2012 (2012), Article ID 628261, 37 pages
Research Article

Electromagnetic Low-Frequency Dipolar Excitation of Two Metal Spheres in a Conductive Medium

1Division of Applied Mathematics & Mechanics, Department of Engineering Sciences, University of Patras, 265 04 Patras, Greece
2Département de Recherche en Electromagnétisme, Laboratoire des Signaux et Systèmes, CNRS-Supélec-Univ Paris Sud, 91192 Gif-sur-Yvette, France

Received 17 August 2011; Accepted 12 November 2011

Academic Editor: Mina Abd-El-Malek

Copyright © 2012 Panayiotis Vafeas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This work concerns the low-frequency interaction of a time-harmonic magnetic dipole, arbitrarily orientated in the three-dimensional space, with two perfectly conducting spheres embedded within a homogeneous conductive medium. In such physical applications, where two bodies are placed near one another, the 3D bispherical geometry fits perfectly. Considering two solid impenetrable (metallic) obstacles, excited by a magnetic dipole, the scattering boundary value problem is attacked via rigorous low-frequency expansions in terms of integral powers ( 𝑖 𝑘 ) 𝑛 , where 𝑛 0 , 𝑘 being the complex wave number of the exterior medium, for the incident, scattered, and total non-axisymmetric electric and magnetic fields. We deal with the static ( 𝑛 = 0 ) and the dynamic ( 𝑛 = 1 , 2 , 3 ) terms of the fields, while for 𝑛 4 the contribution has minor significance. The calculation of the exact solutions, satisfying Laplace’s and Poisson’s differential equations, leads to infinite linear systems, solved approximately within any order of accuracy through a cut-off procedure and via numerical implementation. Thus, we obtain the electromagnetic fields in an analytically compact fashion as infinite series expansions of bispherical eigenfunctions. A simulation is developed in order to investigate the effect of the radii ratio, the relative position of the spheres, and the position of the dipole on the real and imaginary parts of the calculated scattered magnetic field.