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The field of an arbitrary finite defect in a 3D semispace is calculated. This problem is solved by reduction to a 2D case that yields an integral equation on the surface of the defect alone. Calculation formulas for a spherical defect are presented. Results of calculations using these formulas for the case of a uniform external field normal or tangential to the surface of a magnetic semispace are presented in a graphical form.Solving the magnetostatic problem for different models of bodies containing defects remains a challenging task for practical use of magnetic testing methods. The internal defect is approximated mainly by an infinite elliptic cylinder placed either in an infinite magnetic space, in a semispace, or in a plate. (Such an approximation is used owing to its simplicity.). The symmetry of the model enables significant simplification of the problem, which can be reduced to a 2D case. Corresponding results can be found, e.g., in [5].A solution of the magnetostatic problem for a defect with finite dimensions has only been found for an infinite magnetic space containing a cavity (insert) with an ellipsoidal surface [5]. In continuation of [6], we solve in this study the problem of a semispace containing a defect with finite dimensions. A technique was used in [6] that allows reduction of the problem of finding the field of a defect with finite dimensions located in a semispace to calculation of the value of the normal field component on the defect's surface alone. Then, the case of a spherical defect was considered where the external field was assumed to be uniform and directed normally to the surface of the semispace. Publication [6] contains typographical errors. We apologize to the readers for them and briefly iterate the main stages of the study and the most important formulas since, moreover, they are of importance for this study as well.1. Let us use an integrodifferential approach to the magnetostatic problem. The equation for the strength vector of the magnetic field has the form (1) where Ω is the area occupied by a magnetic matter with magnetic permeability µ , H 0 ( r ) is the strength of the field created by external sources, and R = | r -r ' | .Using Eq. (1), let us consider the following problem: a defect characterized by a finite diameter and magnetic permeability µ d = const and confined within smooth surface S is located in the semispace z ≤ d ( d > 0) with µ = const. The frame origin is placed at the center of the defect. In this case, area Ω in (1) is a semispace containing the defect. Let us denote the area occupied by the defect as Ω d and Ω 1 = Ω / Ω d . The described situation where the defect is approximated by a sphere is illustrated by Fig. 1. It is worth stressing that, for the calculations made in the first part of this study to hold for a defect of any form, it only needs to be finite. The result of the first stage is formulated as Eq. (7) applicable to solution of the problem for a defect of any form. Moreover, Eq. (7) is preferable for practical usage since it requires the...

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Formulas for the asymptotic behavior of the electromagnetic field when r ∞ that are convenient for practical use are deduced on the basis of Maxwell's integro-differential equations. The uniqueness theorem is proven for the solution of Maxwell equations under an assumption that displacement currents can be disregarded.The asymptotic behavior of the solutions to mathematical models used for describing physical phenomena is of importance for determining the essential properties of these solutions. Scattering problems in classical and quantum physics are examples where such asymptotic behavior is used. In classical electrodynamics, the asymptotic properties (Sommerfeld conditions) are used as a basis for proving uniqueness of the solutions of Maxwell equations. In this study, integro-differential equations of classical electrodynamics, which are equivalent to the system of Maxwell equations, are used to deduce the asymptotic behavior for the electromagnetic field (EMF) when r ∞ . As a result, the Sommerfeld conditions are obtained showing that they are not a postulate. In addition, the normal field components are shown to fade more rapidly than the tangential ones. The proposed technique describes each term in the asymptotic expansion in a more efficient and simple way. Comparison of two methods for calculating the asymptotic behavior enabled deduction of new formulas for summing up spherical Bessel functions. The EMF's asymptotic behavior is also considered with displacement currents disregarded, and the proof of the uniqueness theorem for the solution of Maxwell equations is provided in this case.Let an inspected body, which occupies a limited area Ω , have dielectric permittivity ε i , magnetic permeability µ i , and electric conductivity σ . Outside Ω , these parameters have the following values: σ e = 0, ε e , and µ e = const. It is assumed as well that the external field is created by extraneous currents with density j extr ( r ) and frequency ω localized in a limited volume T ; ε 0 and µ 0 are the dielectric and magnetic constants, respectively.Further consideration is based on the system of integro-differential equations for the strength vectors of electric field E ( r ) and magnetic field ç ( r ). This system using the notation (1) can be represented in the formr r' , ( )v r' ( ) r', d Ω ∫ H r ( ) H 0 r ( ) = div k 2 + ( ) G r r' , ( )n r' ( ) r' d Ω ∫ + = -iωε 0 ε e curl G r r' , ( )u r' ( ) r', d Ω ∫ MAGNETIC METHODS