Advances in Mathematical Physics

Volume 2018 (2018), Article ID 9587972, 20 pages

https://doi.org/10.1155/2018/9587972

## Dipolar Excitation of a Perfectly Electrically Conducting Spheroid in a Lossless Medium at the Low-Frequency Regime

Department of Chemical Engineering, University of Patras, 26504 Patras, Greece

Correspondence should be addressed to Panayiotis Vafeas; rg.sartapu.gnemehc@saefav

Received 2 September 2017; Accepted 13 March 2018; Published 26 April 2018

Academic Editor: Antonio Scarfone

Copyright © 2018 Panayiotis Vafeas. 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.

#### Abstract

The electromagnetic vector fields, which are scattered off a highly conductive spheroid that is embedded within an otherwise lossless medium, are investigated in this contribution. A time-harmonic magnetic dipolar source, located nearby and operating at low frequencies, serves as the excitation primary field, being arbitrarily orientated in the three-dimensional space. The main idea is to obtain an analytical solution of this scattering problem, using the appropriate system of spheroidal coordinates, such that a possibly fast numerical estimation of the scattered fields could be useful for real data inversion. To this end, incident and scattered as well as total fields are written in a rigorous low-frequency manner in terms of positive integral powers of the real-valued wave number of the exterior environment. Then, the Maxwell-type problem is converted to interconnected Laplace’s or Poisson’s equations, complemented by the perfectly conducting boundary conditions on the spheroidal object and the necessary radiation behavior at infinity. The static approximation and the three first dynamic contributors are sufficient for the present study, while terms of higher orders are neglected at the low-frequency regime. Henceforth, the 3D scattering boundary value problems are solved incrementally, whereas the determination of the unknown constant coefficients leads either to concrete expressions or to infinite linear algebraic systems, which can be readily solved by implementing standard cut-off techniques. The nonaxisymmetric scattered magnetic and electric fields follow and they are obtained in an analytical compact fashion via infinite series expansions in spheroidal eigenfunctions. In order to demonstrate the efficiency of our analytical approach, the results are degenerated so as to recover the spherical case, which validates this approach.

#### 1. Introduction

The fundamental principles of classical electromagnetism [1], initially introduced by the pioneer in the field James Clerk Maxwell, represent the basic foundation of the low-frequency scattering theory [2]. Many physical applications that are associated with the response of receivers with arbitrary shape, embedded within various media, when they are excited by several primary sources, still stand in the frontline of scientific research. Indeed, let us give emphasis to some of them like two-phase composites, Earth’s subsurface probing for mineral exploration, identification of cavities or other underground detection such as UneXploded Ordnance and buried objects, and scattering by chiral material in either chiral or nonchiral environments. Obviously, the range of the applied frequencies varies according to each case. However, low-frequency interactions exhibit a particular interest for numerous real-life problems that occur in physical areas, which are related to the present contribution. By deciphering the implicated fields to each case, information about main parameters like orientations, sizes, shapes, and magnetic and electric properties of the anomalies brings insight into the field behavior. Nevertheless, this is not at all an easy task, since the situation of an inversion scheme [3] cannot be tackled in a robust-like manner, unless efficient models of the field distribution and strong effective mathematical tools [4] are available, since in practice the measurement and the identification of the scatterer are needed simultaneously.

The large amount of vector data, the electromagnetic and geometrical complexity of the different media involved, the many configurations of sources and receivers, and the uncertainty resulting from datasets containing both the contribution of the incident and scattered fields explain the interest of solving forward and inverse electromagnetic scattering problems, first at the modeling stage and later at the inversion stage. Therefore, the already ample library of scattering by simple shapes using analytical methods is open to accept new and useful semianalytical and, why not, analytical results. In the last decade, a lot of articles were published in that direction and many references are available in the literature. Collectively, we pertain to the low-frequency scattering by a perfectly electrically conducting sphere in a conductive environment, illuminated by an arbitrarily orientated and settled magnetic dipole source field [5]. There, the 3D scattered electromagnetic fields have assumed a complete analytical formalism in terms of closed-type solutions, while a numerical implementation has validated almost perfectly the results via a comparison with the exact Mie series expansion [6]. However, this work was a first introduction to scattering by metallic objects that reflect the complete isotropy of the three-dimensional space. Moving to more complicated geometries for the same physical consideration such as spheroids [7] or ellipsoids [8], one can realize that the orientation of the body, save the position or the size, plays a crucial role not only to the mathematical treatment but also to the computational elaboration of the corresponding fields. Moreover, it is obvious that the more general the shape becomes, the more difficulty is added to a possible inversion, which, by the way, is the final goal when we deal with real-life applications. It is to this end that the utilization of elaborate numerical techniques is inevitable, as it is demonstrated in [9] for the identification of an orebody of approximately infinite conductivity, which is buried in Earth, by using an equivalent perfectly conducting triaxial model ellipsoid. High-contrast cases, where the target has an extremely large conductivity with respect to the surrounding host environment and hence is considered to be impenetrable, motivated researchers to embark upon much more complex patterns, such as the two-sphere case for two closely adjacent metallic objects [10] or the circumstance of a nonpenetrable, that is, perfect conductor, ring torus [11], both articles presuming a conductive occupied medium. These reports, each one containing a sufficient reference list, are a small but representative sample of displaying the inconvenience induced in performing analytical techniques when incorporating with different geometrical models [12, 13]. As a matter of fact, the difficulty increases due to the emergence of intricate eigenfunctions [14, 15], dealing with the involved potentials.

This project is focused now on the adjustment of the low-frequency diffusive scattering theory in solving the particular problem of identifying impenetrable metallic bodies in a lossless, that is, perfect dielectric, medium, aiming to follow the similar successful path of other contributions to the field. Bearing in mind the few, but with quite good results, attempts of simulating the environment as lossless and not conductive (e.g., see [16] for the case of two metallic spheres almost touching and [17] for the toroidal case in this direction), we realize the necessity of dealing with such brute-force analytical models. Seeking for different body structures in order to attain the representation of all kind of anomalies in lossless media, as much effectively as possible, the present paper deals with the spheroidal configuration. Although research on quasi-static fields around spheroids is vast, there are still elementary properties to be discovered; thus the demand for comparable investigations stays on continuously even nowadays. Consequently, here we extend the research to a perfectly reflective spheroid, embedded in a lossless ambient, where the appropriate and best fitting system for the particular modeling purposes is the spheroidal coordinate system [12, 13]. The prolate spheroidal geometry is considered, since the oblate one is readily obtained via a simple analytical transformation, while limiting cases are also available. On the other hand, a time-harmonic magnetic dipole source, acting as the primary source, operates at the low-frequency regime and generates the three-dimensional incident electromagnetic waves, which propagate towards the scatterer. Then, the spheroidal-shaped body responds to the excitation and generates the scattered magnetic and electric fields, where their summation comprises the total fields.

The physical nature of the problem urges us to cope with the following low-frequency technique, where the incident, the scattered, and the total electromagnetic fields are expanded in terms of positive integral powers of , where denotes the real wave number of the exterior medium at the operation frequency. Here, we elaborate the 3D vector fields at each order of for every , whereas our problem is transformed into a sequence of coupled boundary value scattering problems. The model is mathematically formulated in terms of second-order Laplace’s and Poisson’s partial differential equations, accompanied by the appropriate perfectly electrically conducting boundary conditions, that is, cancellation of normal magnetic and tangential electric fields, reflecting the impenetrable character of the boundary of the spheroid-type metallic targets. The Silver–Müller radiation conditions at infinity are applied as well. We restrict our analysis to the important terms of the expansions, confined by the static Rayleigh term for and the dynamic terms for , since terms of higher orders, that is, , are small contributors due to the low frequencies of which the source operates and, consequently, they are neglected. The analytical procedure sketched above leads either to closed relationships or to infinite linear algebraic systems for the determination of the unknown constant coefficients of the implied potential functions, where the latter can be solved approximately by enforcing usual cut-off methods. The three-dimensional scattered magnetic and electric fields for each admit closed-type forms via infinite series expansions in view of spheroidal eigenfunctions [14, 15]. The key to the validation of our approach is the degeneration of the obtained formulae in order to recover the spherical case.

The rest of the paper itself is organized as follows. In Section 2, a detailed physical interpretation and the theoretical basis via an analytical mathematical formulation are sketched, which incorporate the general features, independently of any geometry. Thereafter, a brief configuration of the prolate spheroidal system and the eigenfunctions associated with it is developed in Section 3, whereas the sought solution is constructed step by step and the electromagnetic low-frequency fields assume infinite series expansions in terms of prolate spheroidal eigenfunctions. Though the transformation of these geometrical and analytical attributes to the oblate spheroidal case is presented, some interesting limiting occasions are also discussed. Section 4 is devoted to the validation of the aforementioned analytical method via a proper reduction to the corresponding spherical problem that illustrates the above results. Finally, a short conclusion and discussion, containing an outline of our work and future steps, follow in Section 5, while an updated reference list is displayed immediately after.

#### 2. Physical and Mathematical Interpretation

Suppose a solid spheroidal body, either prolate or oblate, which by physical means has an approximate conductivity with respect to the surrounding homogeneous, isotropic, and nonmagnetic lossless medium of conductivity . The constructed body-medium system is assumed to occupy the three-dimensional space , while the interface that distinguishes the two regions is actually the smooth surface of the perfectly electrically conducting object. Hence, the area of electromagnetic scattering is spread to the unbounded domain of dielectric permittivity and magnetic permeability , approximated by the corresponding one of the vacuum, outside the spheroid, yielding exterior-type problems. By definition of the given low circular frequency , the real-valued wave number of the medium iswhere (1) secures the conjecture of low frequencies. Harmonic time-dependence is implied for the fields via multiplication of their spatial part by (imaginary unit ); consequently they admit expressions as a function of , which is expressed via the Cartesian basis , in Cartesian coordinates , where this dependence will be omitted henceforth for reasons of writing convenience.

The impenetrable spheroid is excited by a primary vector source field, for example, a magnetic dipole for the purposes of this project, having an arbitrary orientation that yieldsbeing located at . The fixed singular point is excluded from the domain of scattering, since the region of interest is at a reasonably long distance from the source. Thus, we adopt the explicit definition as for the area of electromagnetic activity. According to the simple notation and , the magnetic dipolar source radiates the primary fields [2]where the symbol “” denotes dyadic product. In terms of Maclaurin’s series expansions of the exponent and implying (1), we perform some extended algebraic calculations by collecting terms as integral powers of and we retain the adequate number of them that correspond to the static Rayleigh approximation and to the first three dynamic factors. Therefore, the incident magnetic field (3) is rewritten aswhile the corresponding primary incident electric field (4) renderswhere the input of the fields of order is minor, since and consequently are very small. On the other hand, the surviving electromagnetic fields for all enjoyas far as the magnetic components are concerned, whileobtained for the electric elements, where , as it is indicated by the performed analysis. Expressions (7)–(11) are provided in view of the unit dyadic , while the handy forms, depicted on the right-hand side of (7), (8), and (10), are based on straightforward computations with the aid of trivial differential identities and by virtue ofgiven that . They contain the fundamental quantity, captured under the action of the gradient operator (note that the Laplacian is ). Though these operators could also be applied on , however for reasons of technical clarity we distinguish this feature by introducing the above notation; otherwise it will be instantly designated. In the aftermath, the primary incident fields are perturbed by the solid object, producing the corresponding scattered fields and , while the total fields emerge as the summationwhere, since the spheroidal metal body is nonpenetrable, the wave propagation is not possible inside it. By definition of the simple notation , corresponding to primary (), scattered (), and total () electromagnetic fields, all pairs satisfy the well-known Maxwell’s equations [2]. Yet, as for the scattering components that interest us in the sought solution within , they becomewhich represent a deduced form for our particular circumstance and they are divergence-free; that is,implying their solenoidal attribute, such that (14) is readily sufficed.

The low-frequency interaction high-contrast cases, as dictated by the physics that we address here, are inherited to the scattered fields and they are postulated by the serieswhere their written fashion is referenced in [2]. Substituting expansions (16) into Maxwell’s equations (14), accompanied by (15), we retrievewhich stand for the low-frequency equivalent expressions. However, for all the reasons described earlier, the behavior of the incident fields (5) and (6) with (7)–(11) reflects exactly the very same physical and mathematical treatment to the scattered fields (16) with (17) and (18); that is, we hold the corresponding nonvanishing terms, attaining fields up to the third order, since the rest of them consist of a minor correction. As a consequence, the scattered magnetic field readswhile scattered electric field is assumed to beconsidering , in the absence of the relative incident components. Obviously, taking this into account, the fundamental relations (17) and (18) are immediately fulfilled by the surviving low-frequency fields , , , , and arising from (19) and (20) for , where their evaluation specifies our final goal. Aiming to understand the contribution of each low-frequency term within (19) and (20), we reinforce (1) and we separate the real from the imaginary part, concluding tothe electric field being purely imaginary, while the magnetic field that is practically measured is a complex-valued function.

In order to proceed to the solution, we are obliged to complete the aforementioned analysis by the exposition of the related to the problem boundary and limiting conditions. Under this aspect, the properly fitted perfectly electrically conducting boundary conditions on the surface of the spheroidal target are provided in terms of the corresponding outward unit normal vector and involve the total fields viawhich demand cancellation of the normal component of the total magnetic field and of the tangential component of the total electric field. The low-frequency analogy of (23) turns out to be a combination of it with (16), whereas by virtue of (13) we are led towhile relations (24) refer only to the remaining electromagnetic terms, since for any . Additionally, for the implicated functions, the Silver–Müller radiation conditions at infinity for the scattered fields, those beingmust be instantly satisfied, also for the low-frequency regime through (16), givingsince, for , it holds that and as it is verified from (17). Exterior-type solutions, like in our case, satisfy (25) automatically, as a consequence of the appropriate construction of the potentials via the corresponding eigenfunctions [14, 15].

During our analysis, it is supposed that the conductivity of the spheroid approaches infinity. Nevertheless, herein we should include a small discussion about the transition zone, that is, how to treat materials with large, or even comparable with respect to the exterior medium, but not infinite conductivity. In this case, the problem becomes slightly more complicated, since the body becomes penetrable; hence the interior electromagnetic fields exist and contribute to the final solution via the transmission conditions, which are applied on the material’s surface instead of (23); see [2] for more details. There, it is shown that conductive or even dielectric particles can resonate with an incident electromagnetic wave over a wide range of modes related to surface excitation. This is due to the excess surface charge that causes the particles of arbitrary composition obtaining metallic properties, a fact especially important for spheres and spheroids. However, this is a completely different situation from the present investigation, concerning applications of importance in other physical areas, for example, in optics.

In summary, we have to manipulate a little bit further the partial differential equations (17) with (18) to obtain easy-to-handle elliptic equations with partial derivatives and, then, use the spheroidal coordinate geometry and analysis, whereby the conditions (24) and (26) are applied, to reach the final desired solution.

#### 3. Spheroidal Low-Frequency Electromagnetic Fields

We intend to derive closed analytical formulae for the main nonvanishing scattered electromagnetic fields , , , , and , since . To achieve that, we have to recover from (17), provided (18), known partial differential equations. To this direction, we apply on both sides of Maxwell’s equations (17) and we utilize repeatedly the very useful identity with being any vector, where, along with (18), we arrive at a set of boundary value problems, which are coupled to one another, from the static one at to all the rest of the dynamic ones at higher orders up to . Those are written explicitly as functions of the scalar , and the vector , harmonic potentials,providing independentlywhile utilizing (28); then the right-hand side of what follows is known; that is,from which we may recoverwhich serves as the nonhomogeneous part of the equation that follows; that is,Moreover, we work similarly to the zeroth-order field to obtainwhere we recall that the scattered fields , , , , and must be determined in the confined scattering region of wave propagation . Some extended analysis reveals that, for , which corresponds to the Rayleigh approximation term, an uncoupled Laplace’s equation is solved for the magnetic field, while there exists no electric field, since the corresponding term for the incident electric field is absent. On the contrary, the dynamic term for reveals the existence of a vector harmonic electric field with a zero-term magnetic field. As for the case, it involves a Poisson’s equation for the magnetic field, coupled to the static term. Exploiting appropriately the problem, we solve Poisson’s equation by introducing a special form of a particular solution, which is actually the outcome of the application of a standard vector identity, using the fact that and . The incident electric field does not exist, inheriting the same attribute to the scattered one. Finally, at both magnetic and electric fields appear. As far as the magnetic field is concerned, we simply solve an uncoupled Laplace’s equation. However, this is not the case for the problem corresponding to the electric field for which the solution is introduced in the form of an integral representation in terms of the fundamental solution of Laplace’s equation [14]. Problems (28)–(32) with (27) are supplemented by the perfectly reflecting boundary conditions (24) and the proper limiting behavior (26). It is obvious that the low-frequency magnetic terms of order vary like , while the corresponding electric ones vary like as increases to infinity. Thus, implying to obtain , integral within (31) is of order , which converges as .

##### 3.1. The Prolate Spheroidal Coordinate System with Geometry and Analysis

For the sake of completeness, we provide a brief analysis of the spheroidal case and especially the prolate spheroidal geometry, since the oblate one can be readily recovered as it will be seen afterwards. Given a fixed positive number , which we consider to be the semifocal distance of our system, we define the transformed prolate spheroidal coordinates for every , , and (see Figure 1) aswith position vector , where = (, , ) designate the prescribed source’s location. In the limit as the prolate spheroid degenerates to the standard focal segment , while as it approaches a sphere located at infinity. A key point in this analysis is the exploitation of the common -axis of symmetry between the applicable coordinate system and the spheroidal body under investigation. In fact, this is a quite practical technique, which simplifies the mathematical treatment, though it retains the generality unaltered and we still cope with a genuine three-dimensional problem, since the arbitrary orientation of the magnetic dipole, which is implied by (2), defends the 3D configuration until the final solution. Nevertheless, in the very special case where the magnetic dipole is set on the axis of symmetry of the proposed prolate geometry, that is, the -axis, which means and is orientated to the -direction, yielding , then the problem under consideration turns into being axisymmetric, since any kind of dependence from the azimuthal angular is omitted. This restriction is imposed in every field in addition to the vanishing of the other two components of the magnetic dipole () and of its position () to the analytical results. However, the general nonaxisymmetric case that is examined here is concerned with a source of arbitrary location and orientation.