Low-frequency modeling of the interaction of a magnetic dipole and two perfectly conducting spheres in a conductive medium
Résumé
This work concerns the interaction of a time-harmonic magnetic dipole, arbitrarily orientated in the three-dimensional space, with two perfectly conducting spherical bodies embedded in a homogeneous conductive medium. For many practical applications involving two buried obstacles nearly located such as Earth's subsurface electromagnetic probing or other physical cases (e.g. geo-electromagnetics), the bispherical geometry might provide a very good approximation. The particular physics here concerns two solid impenetrable bodies under a magnetic dipole excitation, where the scattering boundary value problem is attacked via rigorous low-frequency expansions in terms of integral powers of (ik), k being the complex wavenumber of the exterior medium, for the incident, scattered and total electric and magnetic fields. Our goal is to obtain the most important terms of the low-frequency expansions of the electromagnetic fields, that is the static (for n = 0 ) and the dynamic (n = 1, 2, 3) terms. In particular, for n = 1 there are no incident fields and thus no scattered ones, while for n= 0 the Rayleigh electromagnetic term is obtained in terms of infinite series. Emphasis is given on the calculation of the next two nontrivial terms (at n = 2 and at n = 3) of the aforementioned fields. Consequently, those are found in closed form from exact solutions of coupled (at n = 2, to the one at n = 0) or uncoupled (at n = 3) Laplace equations and they are given in compact fashion, as infinite series expansions for n = 2, 3. Let us notice that the difficulty of the Poisson's that has to be solved for is not minor and for this case we present only the methodology to obtain the solution. Finally, our analytical approach calls for the use of the well-known cut-off method in order to obtain properly closed solutions.