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• W2051338294 abstract "Abstract IN this paper we derive a strict solution of the problem of the diffraction of an electromagnetic field by two separated ideally conducting spheres on the assumption that the source of the primary field is an elementary oscillating dipole (electric or magnetic), situated at some point P of an unbounded space with physical constants e, μ, σ (σ = 0) such that its moment forms an arbitrary angle β with the plane Π passing through the point P and the polar axis which is common to the spheres (the axis of symmetry for the diffracting system). The distance between the centres of the spheres is ι, where ι 〉 a−1 + a+1, where a−1 and a+1 are the radii of the spheres (in the general case a−1 ≠ a+1). The problem of the diffraction of a field of a horizontal and vertical dipole by one sphere was first discussed in [1, 2]. Numerous results of numerical analysis, related to a spherical surface antenna, excited by a radial or horizontal dipole, have been given in [3], In the case of a plane electromagnetic wave the problem of the diffraction of the electromagnetic field by two spheres has been considered in [4–6]. As a particular case, the solution of this problem for an electric dipole, situated on the common axis of two spheres and orientated along it, is given in [7]. The problem is solved by separation of the variables in spherical coordinates using the results of [1, 2]. In the general case the dipole moment makes an angle β with the plane Π. In this connection the problem is divided into parts: problem (A) concerns the diffraction of the field of a dipole with moment lying in the plane Π, and problem (B) the diffraction of the field of a dipole with moment normal to the plane Π at the point P. The solution of the problem in the general case can then be found in the form of a linear combination of the solutions of problems (A) and (B). The coordinate system oxyz and local coordinate systems osxsyszs, s = ± 1 are introduced as shown in the figure (the axis oz and axes oszs, of the local systems are in the same direction and lie in the plane Π, coinciding with the plane oxz). At the same time a spherical system of coordinates r, θ, ϕ is introduced, connected with oxyz, and local spherical systems of coordinates rs, θs, ϕs connected with the s-th sphere, s = ± 1. The distance of the dipole from os is ιs, s = ± 1. The angles formed by the vectors ls with the axis oz are αs, s = ± 1. If p(m) is the moment of the electric (magnetic) dipole, then is projections on the plane Π and the normal to it at the point P are, respectively, the vectors p1 = p cos β (m1 = m cos β), p2 = p sin β (m2 = m sin β). Fom now on it will be assumed that the angle between the direction of the vector p1(m1) and the positive direction of the axis Oz is γ. The resulting electromagnetic field E = E0 + E1, H = H0 + H1 (the index 0 is assigned to the primary field and the index 1 to the secondary) is found in terms of Debye potentials u, υ(u = u0 + u1, υ = υ0 + υ1) from the relations E r = ∂ 2 (ru) ∂r 2 + k 2 (ru) , H r = ∂ 2 (rv) ∂r 2 + k 2 (rv) , E θ = 1 r ∂ 2 (ru) ∂r∂θ + ik 0 μ sinθ ∂v ∂ gj , H θ = 1 r ∂ 2 (rv) ∂r∂θ − ik 0 e sin θ ∂u ∂ϑ E ϑ = 1 r sin θ ∂ 2 (ru) ∂∂ϑ − ik 0 μ ∂v ∂θ , H ϑ = 1 r sin θ ∂ 2 (rv) ∂ ∂ gj + ik 0 e built∂u ∂θ (k = k0√ (μe), k 0 = ω c ), written in one of the spherical systems of coordinates (the time relation is determined by the factor exp(−iωt), which is omitted everywhere). The potentials u1, v1 of the secondary field are the solutions of the Helmholtz wave equations Δu1 + k2u1 = 0, Δv1 + k2v1 = 0, (1) and on the surface of each sphere they simultaneously satisfy the boundary conditions ∂ ∂r s [r s (u 0 + u 1 )] = 0 , v0 + v1 = 0, rs = as, s = ± 1 (2) (at infinity u1, v1 must satisfy the radiation condition). The method of solving the boundary value problems (1), (2), used below requires the previous expansion of the potentials of the primary field u0, v0 in a series of spherical wave functions in the coordinates of the s-th sphere, s = ± 1." @default.
• W2051338294 created "2016-06-24" @default.
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• W2051338294 date "1967-01-01" @default.
• W2051338294 modified "2023-09-25" @default.
• W2051338294 title "The diffraction of the field of dipole radiation by two spheres" @default.
• W2051338294 doi "https://doi.org/10.1016/0041-5553(67)90118-8" @default.
• W2051338294 hasPublicationYear "1967" @default.
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