Regularization of the electrostatics problem for three spheres and an electrostatic charge
A numerical-analytical algorithm for investigation of the potential of a sphere with a circular hole, surrounded by external and internal closed ribbon spheres, is constructed. The number of ribbons on the spheres is arbitrary. The ribbons on the spheres are separated by non-conductive, infinitely thin partitions. The partitions are located in planes parallel to the shear plane of the sphere with a hole. Each ribbon has its own independent potential. An electrostatic charge is placed between the outer sphere and the sphere with a hole in the axis of the structure. The full potential must satisfy, in particular, Maxwell’s equations, taking into account the absence of magnetic fields, satisfy the boundary conditions, have the required singularity at the point where the charge is placed. To solve this problem, we first used the method of partial domains and the method of separating variables in a spherical coordinate system. In this case, for the Fourier series, we use power functions and Legendre polynomials of integer orders. From the boundary conditions, using an auxiliary system of 3 equations with 4 unknowns, a pairwise system of functional equations of the first kind with respect to the coefficients of the Fourier series is obtained. The system is not effective for solving by direct methods. The method of inversion of the Volterra integral operator and semi-inversion of the matrix operators of the Dirichlet problem for the Laplace equation are applied. The method is based on the ideas of the analytical method of the Riemann - Hilbert problem. In this case, integral representations for the Legendre polynomials are used. A system of linear algebraic equations of the second kind with a compact matrix operator in the Hilbert space l`2 is obtained. The system is effectively solvable numerically for arbitrary parameters of the problem and analytically for the limiting parameters of the problem. Particular variants of the problem are considered.
H. Nakano, T. Shimizu, H. Kataoka, J. Yamauchi. Circularly and linearly polarized waves from a metamaterial spiral antenna. IEEE Antennas and Propagation Society International Symposium. – 2014. – July 6-11. – P. 226.2. DOI: https://doi.org/10.1109/APS.2014.6904599
D. B. Kyruliak, Z. T. Nazarchuk, O. B. Trishchuk. Axially-symmetrical TM – waves diffraction by sphere-cone cavity. Progress in electromagnetics research. B. — 2017. – Vol. 73.– P. 1-17. DOI: https://doi.org/10.2528/PIERB16120904
K. Mei, M. Meyer. Solution to spherical anisotropic antennas. IEEE Transactions on Antennas and Propagation. – 1964. – AP-12. – P.459-463. DOI: https://doi.org/10.1109/TAP.1964.1138250
L. B. Felsen, N. Marcuvitz. Radiation and scattering of waves. Prentice-Hall Microwaves and Fields Series. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1973. xxxii+888 pp.
V. A. Rezunenko. Electrostatic field of a segment which is shielded by sectional spheres. Visnyk of V.N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics. — 2010. – N. 931. – P. 59–72.
V. A. Rezunenko. The field of the vertical electrical dipole placed above the spiral conductive unclosed sphere. Visnyk of V.N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics. — 2015. – Vol. 81. – P. 10–19. DOI: https://doi.org/10.26565/2221-5646-2015-81-02
V. A. Rezunenko. The field diffraction of current ring on a spiral conductive sphere with a hole. The X-th International Conference on Antenna Theory and Techniques ICATT’2015, KhNURE, Kharkiv, 21—24 April, 2015. Proceedings. – P. 129—131. DOI: https://doi.org/10.1109/ICATT.2015.7136803
V. A. Rezunenko. Difraction of the field of vertical electric dipole on the spiral conductive sphere in the presence of a cone.Visnyk of V.N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics. — 2018. – Vol. 88. – P. 17–26. DOI: 10.26565/2221-5646-2018-88-02
O. A. Ladyzhenskaya. The boundary value problems of mathematical physics. Translated from the Russian by Jack Lohwater. Applied Mathematical Sciences. – New York. – 1985. 322p. DOI: https://doi.org/10.1007/978-1-4757-4317-3
V. A. Rezunenko, I. A. Vyazmitinov, L. V. Udyanskaya, V. P. Shestopalov. Antennas characteristics of spherical reflector which is working at Helmholtz resonance excitation. International Scientific and Technical Conference "Modern Radar". – Kiev. – 1994. –P. 72-76.
V. A. Rezunenko, S. V. Roshchupkin, E. I. Radchenko. Diffraction field of the vertical dipole from sphere with aperture, screening by the dielectical layer. 2007 6th International Conference on Antenna Theory and Techniques. Sevastopol. 2007. - P. 128-130. DOI: https://doi.org/10.1109/ICATT.2007.4425133
B. M. Singh, J. G. Rokne, R. S. Dhaliwal. Two-dimensional electrostatic problem in a plane with earthed elliptic cavity due to one or two collinear charged electroststic strips. International Journal of Mathematics and Mathematical Sciences. Vol. 2007, Article ID 60595, 9 pages, 2007. DOI: https://doi.org/10.1155/2007/60595
V. P. Shestopalov, Yu. A. Tuchkin, A. E. Poedinchuk, Yu. K. Sirenko. New methods for solving direct and inverse problems of the theory of diffraction. Analytical regularization of boundary value problems in electrodynamics.- Kharkiv: -Osnova. -. 1997. - 284p. (in Russian).
V. A. Doroshenko, V. F. Kravchenko. Excitation open conical and biconical structures. Electromagnetic waves and electronic systems. – 2003. - Vol. 8. - P.4-78.
V. A. Sadovnichy. Theory of the operators. –M.: –High School. –1999. –368p, (in Russian).
V. F. Kravchenko, Yu. K. Sirenko, K. Yu. Sirenko. Electromagnetic waves transformation and radiation by the open resonant structures. Modeling and analysis of transient and steady-state processes. - M. – Physmatlit. – 2011. – 320 p. (in Russian).
Z. S. Agranovich, V. A. Marchenko, V. P. Shestopalov. Electromagnetic wave diffraction on plane metallic gratings. Journal of Technical Physics. – 1962. – T.32. - Issue 4. - P. 381–394, (in Russian).
V. P. Shestopalov, L. N. Litvinenko, S. A. Masalov, V. G. Sologub. Diffraction of waves by gratings. Kharkov. – Kharkov University Press. – 1973. – 288 p. (in Russian).
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