Abstract
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 (ik)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.
Highlights
Several practical applications such as Earth’s subsurface electromagnetic probing for mineral exploration, geoelectromagnetism, or other physical cases related to the identification of buried metallic or nonmetallic objects of different shapes and sizes stand in the front line of the scientific research
This paper describes how to build a versatile set of mathematical tools in order to infer information on two unknown subsurface bodies, by solving the direct scattering problem and calculating the 3D magnetic and electric fields, scattered off when the bodies are illuminated by a magnetic dipole of arbitrary orientation
If we introduce the H-notation for the magnetic field and the E-notation for the electric field, the electromagnetic incident fields Hi, Ei, produced by the magnetic dipole 2.2, are scattered by the solid spheres, creating the correspondingly scattered fields Hs, Es, while the total magnetic and electric fields Ht, Et are given by summation of incident and scattered fields, that is, Ht Hi Hs, Et Ei Es for r ∈ V R3 − {r0}, 2.3 where we have excluded the singular point r0 from the scattering domain
Summary
Several practical applications such as Earth’s subsurface electromagnetic probing for mineral exploration, geoelectromagnetism, or other physical cases related to the identification of buried metallic or nonmetallic objects of different shapes and sizes stand in the front line of the scientific research. Under the basic aim of the present research activities, which is mineral exploration of the Earth by inductive means, unexploded ordinance investigations and exploration of natural structures like water-filled cavities and other possibly conductive materials in subsoil at shallow depths, useful results can be recovered from the already ample library of scattering by simple shapes e.g., spheroids via analytical methods in books 21, At this point, let us mention a successful numerical approach of the characterization of spheroidal metallic objects using electromagnetic induction and an interesting numerical spheroidal-mode approach of unexploded ordinance inversion under time harmonic excitations in the magnetoquasistatic regime 4. This paper describes how to build a versatile set of mathematical tools in order to infer information on two unknown subsurface bodies, by solving the direct scattering problem and calculating the 3D magnetic and electric fields, scattered off when the bodies are illuminated by a magnetic dipole of arbitrary orientation This primary source is considered to be time harmonic and operated at low frequencies to deeply penetrate in the conductive ground. For completeness, the necessary mathematical material concerning the associated Legendre functions of the first and of the second kind, as well as useful formulae associated with trigonometric and hyperbolic functions, are collected in the appendix, along with the presentation of various vector identities
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
