Integro-differential equations for a quasi-stationary electromagnetic field in a nonmagnetic conductive body with a defect

S. Marvin

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 16 (2024) no. 3, pp. 38-44

Résumé

The initial-boundary value problem (IBVP) for the system of Maxwell’s equations for a quasi-stationary approximation is considered in relation to a non-ferromagnetic conductive body in the field of an external current. It is assumed, that the body is not homogeneous in its conductive properties and includes a volume defect in the form of a cavity (a non-conductive subdomain). The IBVP is considered in the classical formulation: the tensions of the electric and magnetic fields are supposed to be continuously derivatively outside the boundary between conductive and non-conductive domains, and continuous at the boundaries of these domains; in this case, the boundaries of the domains are Lyapunov surfaces. On these surfaces, the usual boundary conditions for the tensions of electric and magnetic field must be satisfied: their tangential components are continuous. In addition, tensions decrease quickly at infinity. Based on these assumptions, integro-differential equations for the tensions of electric and magnetic field are derived. These equations take into account the inhomogeneity of the conductor and the presence of the internal defect. The equivalence of the integro-differential equations and the IBVP for the system of Maxwell’s equations is proved for the electromagnetic field inside and outside the conductor.

Détail

Keywords: initial-boundary value problem, Maxwell’s equations, quasi-stationary approximation, volume potential, simple layer potential.

Bibliographie

[1] V. V. Dyakin, V. A. Sandovskiy, Electrodynamics Problems in the Nondestructive Testing, IFM UrO RAN Publ, Ekaterinburg, 2008, 389 pp. (in Russ.)

[2] S. V. Marvin, “Existence and the Uniqueness of Solution of Initial-Boundary Problem for the Uniform System of Equations of Maxwell in the Case of Nonferromagnetic Defective Metallic Body”, Proceedings of Voronezh State University. Series: Physics. Mathematics, 2016, no. 1, 105–117 (in Russ.) | Zbl

[3] S. V. Marvin, “An Initial-boundary Value Problem of Structurescopy of a Nonferromagnetic Metal Solid with Foreign Dielectric Inclusions using the Residual Field of an Instantaneously Cut-off Extraneous Current”, Russian Journal of Nondestructive Testing, 52:2 (2016), 85–94 | DOI

[4] O. V. Tozoni, I. D. Maergoyz, Calculation of Three-Dimensional Electromagnetic Fields, Tekhnika Publ, Kiev, 1974, 352 pp. (in Russ.)

[5] O. V. Tozoni, Secondary Sources Method in the Electrical Engineering, Energiya, M., 1975, 295 pp. (in Russ.)

[6] N. E. Kochin, The Vector Calculus and Basics of the Tensor Calculus, Nauka, M., 1965, 426 pp. (in Russ.) | MR

[7] N. M. Gyunter, The Theory of Potential and it's Applications in Problems of the Mathematical Physics, GITTL, M., 1953, 416 pp. (in Russ.) | MR

[8] E. B. Bykhovskiy, N. V. Smirnov, “About Vector-Functions Space Orthogonal Decomposition, which Quadratically Summed in a Specified Domain, and the Vector Analysis Operators”, Matematicheskie voprosy gidrodinamiki i magnitnoy gidrodinamiki dlya vyazkoy neszhimaemoy zhidkosti, sbornik rabot, Tr. MIAN SSSR, 59, Izd-vo AN SSSR Publ., M.–L., 1960, 5–36 (in Russ.)