Progress In Electromagnetics Research B, Vol. 46, 101{118, 2013 THE FIELD OF A MAGNETIC DIPOLE AND THE PO- LARIZABILITY OF A SUPERCONDUCTING OBJECT EMBEDDED IN THE INTERFACE BETWEEN MAG- NETIC MATERIALS Mohamed A. Mohamed1, *, Edward F. Kuester1, and Mustapha C. E. Yagoub2 1University of Colorado at Boulder, Boulder, CO 80309-0425, USA 2University of Ottawa, 161, Louis Pasteur, Ottawa, Ontario K1N 6N5, Canada Abstract|In this paper, a careful study is made of the magnetostatic potential and ¯eld of a magnetic dipole embedded in, and with dipole moment parallel to, the interface between two magnetic regions. Unlike the case of a magnetic dipole perpendicular to the interface, the detailed position of the current of the dipole relative to the location of the interface has a profound e®ect on the value of the ¯eld produced away from the dipole. As a consequence, the question of de¯ning and determining the magnetic polarizability of a superconducting object partially embedded in a magnetic interface is examined. The results of this paper are important for the proper modeling of arrays of scatterers embedded in an interface, such as frequency-selective surfaces (FSSs) and meta¯lms. 1. INTRODUCTION In recent work [1, 2], Generalized Sheet Transition Conditions (GSTCs) for the electromagnetic ¯elds at a meta¯lm (a surface array of small scatterers) have been obtained. Using these, plane wave reflection and transmission coe±cients from the meta¯lm have been derived and their dependencies on scatterer geometry and incidence angle were investigated [3, 4]. The coe±cients in the GSTCs depend on the density and polarizabilities of the scatterers that make up the meta¯lm. The analysis in [1, 3, 4] assumed that the meta¯lm is embedded in an in¯nite homogeneous medium, in which case the meaning of dipole moments Received 3 July 2012, Accepted 15 November 2012, Scheduled 25 November 2012 * Corresponding author: Mohamed Abed Mohamed ([email protected]). 102 Mohamed, Kuester, and Yagoub and polarizabilities is well understood. When the meta¯lm is placed at an interface between two di®erent media, the GSTCs must be modi¯ed to account for the influence of the interface. If the scatterers are in¯nitely thin, so that only tangential electric currents can be induced in them, the meta¯lm can only produce tangential electric dipoles and normal magnetic dipoles at the interface. In such a case, a simple modi¯cation of the GSTCs for a meta¯lm in a homogeneous medium will provide the correct result for the same meta¯lm at an interface. For the case when the scatterers are not thin, currents can be induced in the direction normal to the interface, and it is not clear what modi¯cations must be made to the GSTCs. In a previous paper [5], we have carried out a detailed study of the problem of a static electric dipole placed in the interface between dielectric media. In this study, it was found that either the excess or the net dipole moment may be the most important characteristic of an embedded dipole, depending on its orientation. The concepts learned from the study of the dipole moments were then used to provide a clear de¯nition for the electric polarizability of a scatterer partially embedded in an interface. Our general result was illustrated for the case of a perfectly conducting spherical scatterer. Unlike the electric dipole case, however, little previous work has been done on the problem of a magnetic dipole in an interface between magnetic media [6{8], and a complete analysis is lacking. In this paper, we perform a detailed study of this problem. Our goals are to ¯nd a suitable unambiguous de¯nition for the magnetic dipole moment and to determine correctly the resulting magnetic ¯eld. Based on these results, we will determine a proper speci¯cation of the polarizability of a superconducting object partially embedded in the interface between di®erent magnetic media. The paper is organized as follows. In Section 2, we obtain expressions for the magnetic vector potential and magnetic ¯eld for a certain model of a magnetic dipole located in a magnetic material interface. In Section 3, the distributions of free and magnetization currents resulting from this magnetic dipole are carefully considered and various possible de¯nitions for the magnetic dipole moment are obtained. The jump condition for the ¯elds at a surface distribution of magnetization in the interface is then studied in Section 4. In Section 5, the question of a proper de¯nition for the magnetic polarizability is studied. For the special case of a superconducting object, symmetric with respect to a plane but otherwise arbitrary, whose magnetic ¯eld is known when placed in a uniform incident static magnetic ¯eld in free space, we ¯nd the ¯eld when the object is embedded symmetrically in the interface. From this, we are able to obtain the magnetic polarizability of the object in the interface in terms of its free space Progress In Electromagnetics Research B, Vol. 46, 2013 103 value. The case of a sphere is then presented as an illustrative example. We conclude with a discussion of the results and how they might apply to various electromagnetic modeling problems. 2. STATIC MAGNETIC DIPOLE AT THE INTERFACE BETWEEN TWO MEDIA Let us ¯rst recall the static vector potential in an in¯nite homogeneous medium (whose permeability is ¹) of a line element of current I flowing along a line segment described by the vector d. As shown in many texts (see, for example, [9]): (¯ ¯ ¯ ¯ ) ¹Id ¯r + d ¯ + ¯r ¡ d ¯ + d A(r) = ln ¯ 2 ¯ ¯ 2 ¯ (1) 4¼d ¯ d ¯ ¯ d ¯ r + 2 + r ¡ 2 ¡ d where r is the position vector directed from the center of the line segment to the observation point. From the basic result (1) we can use superposition of several such current segments to obtain the vector potential of a closed current loop of area S. If we let S ! 0 and I ! 1 in such as way that m = IS remains constant, we obtain the vector potential of a magnetic dipole with magnetic dipole moment m = ISan (where an is a unit vector perpendicular to the plane of the loop, in a direction given by the right-hand rule with respect to the current). This result will also be the far-¯eld limit for the loop with nonzero area. Omitting the details of the derivation (which can be found, e.g., in [10, p. 15, 11] or [12, p. 244]), we have in vector form ¹m £ r A(r) ' (2) 4¼r3 for r = jrj À h; d, where r is the observation point measured from the center of the loop. The magnetic ¯eld can be calculated from the magnetic vector potential using B = r £ A (r) (3) Since m is a constant, we may use a vector identity to evaluate ³ r ´ ³ r ´ r r r£ m £ =m r ¢ ¡(m ¢ r) =¡(m ¢ r) + 4¼m±(r) (4) r3 r3 r3 r3 r 2 1 having used the fact that r ¢ r3 = ¡r ( r ) = 4¼±(r), where ±(r) = ±(x)±(y)±(z) is the three-dimensional Dirac delta function. We thus have r B = ¡¹ (m ¢ r) for r 6= 0 (5) 4¼r3 In the remainder of this section, we examine how these results are a®ected by the presence of a plane interface between two magnetic materials. 104 Mohamed, Kuester, and Yagoub 2.1. Vector Potential of a Tangentially Oriented Magnetic Dipole We will consider ¯rst a particular model for a magnetic dipole partially embedded in the interface between two media, with dipole moment parallel to the interface (a tangentially oriented dipole). The permeability is ¹1 in the upper medium (z > 0), and ¹2 in the lower medium (z < 0). We take the magnetic dipole to be a closed loop of six line current segments (labeled 1{6) carrying a current I in the directions shown in Fig. 1. The current loop lies in the xz plane, and the width of the loop is d. The height of the portion of the loop in the upper medium is h1, and h2 in the lower medium, so that the partial loop areas are S1 = h1d and S2 = h2d in the upper and lower half-spaces respectively. We will ¯nd the static magnetic ¯eld as h1, h2 and d ! 0 and I ! 1 such that the partial dipole moments m1 = S1I and m2 = S2I remain ¯nite. To facilitate the calculation of the ¯eld, we will insert two additional current segments (labeled 7 and 8) just above and below the interface as shown, so that two loops (segments 1-2-3-7 and segments 4-5-6-8) are formed, each of which lies completely in one of the half-spaces. The contributions of these extra segments to the total ¯eld cancel each other out. Equation (2) and image theory can now be used to determine the vector potential of a current loop near a material interface. Consider ¯rst a current distribution J(x; y; z) in z > 0 corresponding to the loop 1-2-3-7. Its ¯eld at an observation point in the upper half-space can d (1) h (2) (3) 1 R µ 1 µ 2 h (6) 2 (4) (5) Figure 1. Current loop segments for a magnetic dipole in, and with dipole moment parallel to, a magnetic interface. Progress In Electromagnetics Research B, Vol. 46, 2013 105 be found as the superposition of that due to the actual currents J and image currents Ji1 given by [10, pp. 347{349, 13]: µ ¶ ¹2 ¡¹1 Ji1 = [axJx (x; y; ¡z)+ayJy (x; y; ¡z)¡azJz (x; y; ¡z)] (6) ¹2 +¹1 acting in an in¯nite homogeneous space with permeability ¹1.