Методика побудови розв’язку двовимірної задачі електродинаміки для електропровідного тіла з плоскопаралельними межами за дії зов-нішнього нестаціонарного електромагнітного поля

Abstract

A two-dimensional initial-boundary value problem of electrodynamics for an electroconductive non-ferromagnetic body with planar-parallel boundaries is formulated under the action of an external non-stationary electromagnetic field. The electromagnetic field is given by the values of the inhomogeneous longitudinal coordinate of the component of the magnetic field intensity vector at its bases. To construct a general solution of the formulated initial boundary value problem under such electromagnetic action, a cubic approximation by thickness coordinate and an integral Fourier transform along a longitudinal coordinate was used for a key function — a given component of the magnetic field intensity vector. As a result, the initial two-dimensional initial boundary value problem for the key function is reduced to the one-dimensional initial boundary value problem. This initial boundary value problem is on the integral key function. These characteristics are the functions of time variable and the thickness coordinate. The coefficients of the polynomial approximating the key function are given by the transformants of the integral characteristics of the key function and given its values on the bases of the body as corresponding functions of time and longitudinal coordinate. The general solutions of the one-dimensional initial boundary value problem are obtained as a convolution of functions. These functions describe homogeneous solutions and key function boundary values on the bases of the body. Using the inverse Fourier transforms, the solution of the original electrodynamics problem under the action of an arbitrary variable in time and by the longitudinal coordinate of a non-stationary electromagnetic field is written. On the basis of such a common solution, as a partial case, we also write the solutions of the original two-dimensional initial boundary value problem under the action of a stationary harmonic in time variable electromagnetic field. In order to improve the accuracy of the approximate solution of the problem under consideration, an independent approximation of the corresponding component of the electric field intensity vector along the thickness coordinate is proposed. Systems of equations are formulated to determine the integral characteristics of this component, both under the action of an arbitrary variable in time and for a longitudinal coordinate non-stationary and the stationary electromagnetic field.