# Electrodynamics of non- and slow moving layered media with boundary condition of full current continuity

### Abstract

The consistent physical and mathematical model of propagation of an electromagnetic wave in a heterogeneous medium is constructed using the generalized wave equation and the Dirichlet theorem. Twelve conditions at the interfaces of adjacent media are obtained and justified without using a surface charge and surface current in explicit form. The conditions are fulfilled automatically in each section of calculation schemes. A consistent physical and mathematical model of interaction of nonstationary electric and thermal fields in a layered medium with allowance or mass transfer is constructed.

### References

Y. Eremin and T. Wriedt, “Large dielectric non-spherical particle in an evanescent wave field near a plane surface,” Optics Communications, vol. 214, no. 1–6, pp. 39–45, 2003.

W. Hu and H. Guo, “Ultrashort pulsed Bessel beams and spatially induced group-velocity dispersion,” Journal of the Optical Society of America A, vol. 19, no. 1, pp. 49–52, 2002.

D. Danae, P. Bienstman, R. Bockstaele, and R. Baets, “Rigorous electromagnetic analysis of dipole emission in periodically corrugated layers: the grating-assisted resonant-cavity lightemitting diode,” Journal of the Optical Society of America A, vol. 19, no. 5, pp. 871–880, 2002.

J.I. Larruquert, “Reflectance enhancement with sub-quarterwave multilayers of highly absorbing materials,” Journal ofthe Optical Society of America A, vol. 18, no. 6, pp. 1406–1414, 2001.

B.M. Kolundzija, “Electromagnetic modeling of composite metallic and dielectric structures,” IEEE Transactions onMicrowave Theory and Techniques, vol. 47, no. 7, pp. 1021–1032, 1999.

R.A. Ehlers and A.C.R. Metaxas, “3D FE discontinuous sheet for microwave heating,” IEEE Transactions onMicrowaveTheory and Techniques, vol. 51, no. 3, pp. 718–726, 2003.

O. B´arta, J. Piˇstora, J. Vlˇcek, F. Stanˇek, and T. Kreml, “Magneto-optics in bi-gyrotropic garnet waveguide,” Opto-ElectronicsReview, vol. 9, no. 3, pp. 320–325, 2001.

J. Broe and O. Keller, “Quantum-well enhancement of the Goos-Hanchen shift for p-polarized beams in a two-prism configuration,” Journal of the Optical Society of America A, vol. 19, no. 6, pp. 1212–1222, 2002.

O. Keller, “Optical response of a quantum-well sheet: internal electrodynamics,” Journal of the Optical Society of America B, vol. 12, no. 6, pp. 997–1005, 1995.

O. Keller, “Sheet-model description of the linear optical response of quantum wells,” Journal of the Optical Society ofAmerica B, vol. 12, no. 6, pp. 987–997, 1995.

O. Keller, “V: Local fields in linear and nonlinear optics of mesoscopic system,” Progress in Optics, vol. 37, pp. 257–343, 1997.

G.A. Grinberg and V.A. Fok, “On the theory of Coastal Refraction of Electromagnetic Waves,” in:Investigations onPropagation of Radio Waves, ed. by B. A. Vvedenskii, AN SSSR, Moscow-Leningrad, 1948.

N.N. Grinchik and A.P. Dostanko, “Influence of Thermal and Diffusional Processes on the Propagation of Electromagnetic Waves in Layered Materials,” A.V. Luikov HMTI Press,NASof Belarus, Minsk, 2005.

M. Born, Principles of Optics, Mir, Moscow, Russia, 1970.

L. Kudryavtsev, “Mathematical Analysis,” Mir, Moscow, Russia, 1970.

A. Frumkin, Electrode Processes, Nauka,Moscow, Russia, 1987.

A.N. Tikhonov and A.A. Samarskii, Equations of Mathematical Physics, Nauka, Moscow, Russia, 1977.

A.F. Kryachko et al., Theory of Scattering of Electromagnetic Waves in the Angular Structure, Nauka, Moscow, Russia, 2009.

M. Leontovich, On the Approximate Boundary Conditions for the Electromagnetic Field on the Surface of Well Conducting Bodies,On the approximate boundary conditions for the electromagnetic field on the surface of well conducting bodies, Moscow, Russia, 1948.

N.N. Grinchik, A.P. Dostanko, I.A. Gishkelyuk, and Y.N. Grinchik, “Electrodynamics of layered media with boundary conditions corresponding to the total-current continuum,” Journal of Engineering Physics and Thermophysics, vol. 82, no. 4, pp. 810–819, 2009.

Z.P. Shul’man and V.I. Kordonskii, “Magnetorheological Effect”, Nauka i Tekhnika, Minsk,Belarus, 1982.

M. Khomich, Magnetic-Abrasive Machining of the Manufactured Articles, BNTU Press, Minsk, Belarus, 2006.

M.N. Levin et al., “Activation of the surface of semiconductors by the effect of a pulsed magnetic field,” Zh. Tekh.Fiz., vol. 73, no. 10, pp. 85–87, 2003.

A.M. Orlov et al., “Magnetic-stimulated alteration of the mobility of dislocations in the plastically deformed silicon of n-type,” Fiz.Tverd.Tela, vol. 43, no. 7, pp. 1207–1210, 2001.

V.A. Makara et al., “On the influence of a constant magnetic field on the electroplastic effect in silicon crystals,” Fiz.Tverd.Tela, no. 3, pp. 462–465, 2001.

A.P. Rakomsin, Strengthening and Restoration of Items in an Electromagnetic Field, Paradoks, Minsk, Belarus, 2000.

Yu.I. Golovin et al., “Influence of weak magnetic fields on the dynamics of changes in the microhardness of silicon initiated by low-intensity beta-irradiation,” Fiz. Tverd.Tela, vol. 49, no. 5.

V.A. Makara et al., “Magnetic field-induced changes in the admixture composition and microhardness of near-surface layers of silicon crystals,” Fiz. Tekh.Poluprovadn, vol. 42, no. 9, pp. 1061–1064, 2008.

A.M. Orlov et al., “Dynamics of the surface dislocation ensembles in silicon in the presence of mechanical and magnetic perturbation,” Fiz.Tverd.Tela, vol. 45, no. 4, pp. 613–617, 2003.

N.S. Akulov, “Dislocations and Plasticity,”ANBSSR Press, Minsk, Belarus, 1961.

N.S. Akulov, Ferromagnetism,ONTI, Leningrad, Russia, 1939.

I.P. Bazarov, Thermodynamics: Textbook for Higher Educational Establishments,Vysshaya Shkola, Moscow, Russia, 1991.

N.N. Grinchik et al., “Electrodynamic processes in a surface layer in magneto-abrasive polishing,” Journal of EngineeringPhysics and Thermodynamics, vol. 83, no. 3, pp. 638–649, 2010.

A. Einstein, Elementary Theory of Brownian Motion, Collected Papers (3), 1966.

V.E. Golant et al., Fundamental Principles of Plasma Physics, Moscow, Russia, 1977.

Y. Kharkats, “Dependence of the limiting diffusion-migration current on the degree of electrolyte dissociation,” Elektrokhimiya, vol. 24, no. 4, pp. 539–541, 1988.

A. Sokirko and Yu.Kharkats, “The limiting diffusion ion and migration currents as functions of the rate constants of electrolyte dissociation and recombination,” Elektrokhimiya, vol. 25, no. 3, pp. 331–335, 1989.

D.V. Gibbs, Thermodynamics. Statistical Mechanics, Moscow, Russia, 1982.

L.N. Antropov, Theoretical Electrochemistry,Moscow, Russia, 1984.

H.G. Hertz, Electrochemistry, New York, NY, USA, 1980.

V. Levich et al., Physicochemical Hydrodynamics, Moscow, Russia, 1959.

J. Neumann, Electrochemical Systems, Moscow, Russia, 1977.

V. Skorcheletti, Theoretical Electrochemistry, Leningrad, Russia, 1969.

L.D. Landau and E.M. Lifshits, Theoretical Physics, Vol. 8, Electrodynamics of Continuous Media, Moscow, Russia, 1982.

V. Tsurko, Proc. Institute of Mathematics of NAS of Belarus, vol. 3, pp. 128–133, 1999.

A.I. Shvab, Elektrichestvo, no. 4, pp. 59–67, 1994.

A.I. Shvab and F.Imo, Elektrichestvo, no. 5, pp. 55–59, 1994.

N.N. Grinchik et al., Proc. NAS of Belarus, Ser. Fiz.-Mat. Nauk, vol. 2, pp. 66–70, 1997.

I.E. Tamm, Principles of the Theory of Electricity, Moscow, Russia, 1976.

J.A. Stratton, The Theory of Electromagnetism, Leningrad, Russia, 1938.

N.N. Grinchik, “Diffusional-electrical phenomena in electrolytes,” Journal of Engineering Physics and Thermophysics, vol. 64, no. 5, pp. 497–504, 1993.

N.N. Grinchik, “Electrodiffusion phenomena in electrolytes,” Inzhenerno-Fizicheskii Zhurnal, vol. 64, no. 5, pp. 610–618, 1993.

N.N. Grinchik et al., “Interaction of thermal and electric phenomena in polarized media,” Mathematical Modeling, vol. 12, no. 11, pp. 67–76, 2000.

N.N. Grinchik, Modeling of Electrical and Thermophysical Processes in Layered Medium, Belorusskaya Nauka Press,Minsk, Belarus, 2008.

N.N. Grinchik, A.N. Muchynski, A.A. Khmyl, and V.A. Tsurkob, “Finite-difference method for modeling electric diffusion phenomena,” Matematicheskoe Modelirovanie, vol. 10, no. 8, pp. 55–66, 1998.

L.N. Antropov, Theoretical Electrochemistry, Moscow, Russia, 1989.

N. A. Kostin and O. V. Labyak, Elektrokhimiya, vol. 31, no. 5, pp. 510–516, 1995.

A.I. Dikusar et al., Thermokinetic Phenomena in High-Frequency Processes, Kishinev, 1989.

F. Bark, Yu. Kharkats, and R. Vedin, Elektrokhimiya, vol. 34, no. 4, pp. 434–44, 1998.

N.N. Grinchik and V.A. Tsurko, “Problem of modeling of the interaction of nonstationary electric, thermal, and diffusion field in layered media,” Journal of Engineering Physics andThermodynamics, vol. 75, no. 3, pp. 693–699, 2002.

P. Kolesnikov, Theory and Calculation of Waveguides, Light-guides, and Integral-Optoelectronics Elements.Electrodynamics and Theory of Waveguides,’HMTI Press, Minsk, Belarus, 2001.

T. Skanavi, Dielectric Physics (Region of Weak Fields), Gostekhizdat, Moscow, Russia, 1949.

P. Perre and I. W. Turner, “A complete coupled model of the combined microwave and convective drying of softwood in an oversized waveguide,” in Proceedings of the 10th InternationalDrying Symposium (IDS’96), vol.A, pp. 183–194, Krakow, Poland, 1996.

J. Jaeger,Methods ofMeasurement in Electrochemistry, Moscow, Russia, 1977.

Y. Barash and V.L. Ginzburg, “On the expressions of energy density and the release of heat in electrodynamics of a dispersing and absorbing medium,” Uspekhi FizicheskikhNauk, vol. 118, no. 3, p. 523, 1976.

D.E. Vakman and L.A. Vanshtein, “Amplitude, phase, frequency are the principal notions in the theory of oscillations,” Uspekhi Fizicheskikh Nauk, vol. 123, no. 4, p. 657, 1977.

B.-T. Choo, Plasma in aMagnetic Field and Direct Thermal-to-Electric Energy Conversion, Moscow, Russia, 1962.

I.V. Antonets, L.N. Kotov, V.G. Shavrov, and V.I. Shcheglov, “Energy characteristics of propagation of a wave through the boundaries of media with complex parameters,” Radiotekhnika i Elektronika, vol. 54, no. 10, pp. 1171–1183, 2009.

I.E. Tamm, Foundations of Electricity Theory, Nauka, Moscow, Russia, 2003.

V. Golant, A.P. Zhilinski, and I.E. Sakharov,Fundamentals of Plasma Physics,Moscow, Russia, p. 383, 1977.

D.I. Blokhintsev, Zh. Tekh Fiz., 15, nos.1–2, pp. 72–81, 1945.

D.I. Blokhintsev, Acoustics of an Inhomogeneous Moving Medium,Moscow, Russia, pp. 84–87, 1981.

O.A. Godin, Acoustics of the Ocean Medium,Moscow, Russia, pp. 217–220, 1989.

S.L. Odintsov, Akustich. Zh.,36, no. 2, pp. 337–339, 1990.

I.A. Miniovich, A.D. Pernik, and V.S. Petrovski, Hydrodynamic Sources of Sound,Leningrad, Russia, pp. 26–28, 1971.

N.N. Grinchik, P.V. Akulich, P.S. Kuts et al., “Aero-acoustics of Moving Media”, Proc. NAS of Belarus, Minsk, Belarus, no 3, pp. 91–95, 1995.

N.N. Grinchik, P.V. Akulich, P.S. Kuts et al., “Modeling of Unsteady Wave Processes in Moving Media”, J. Eng. Phys. and Thermophysics, vol. 68, no. 6, pp. 812–817, 1995.

M. Laue, Zs. Phys., Bd. 128, p. 387, 1950.

A. Einstein, Collected Works, Nauka, Moscow, Russia, vol. 1, p. 12, 1965.

A. Sommerfeld, Ann. Phys.,Bd. 44, p. 177, 1987.

N.N. Grinchik and Yu. Grinchik,“Fundamental Problems of the Electrodynamics of Heterogeneons Media”, Hindawi Publishing Corporation, Physics Research International volume 2012, Article ID 185647,28 pages Doi:10.1155/2012/185647 (GOOGLE.com).

N.N. Grinchiket. al.,“Electromagnetic wave propagation in complex matter”, Edited by Ahmed Kishk, Published by Intech, Janeza Trdine 9,5100 Rijeka, Croatia, 291 p, 2011.

Slepyan L. Unipolar induction problem.IZV Pertrograd Polytechnical Institute, 1914, Vol. XXII, Issue 1, pp. 55–175.

Djurie J. J. Appl. Phys., 1975, Vol. 46, No. 2, p. 679.

Bartlett D.F. et al. Phys. Rev. D, 1977, Vol. 16, No. 12, p. 3459.

Rodin A.O. Unknown electromagnetic induction experiments. Electricity, 1994, No. 7 (in Russian).

Luparev V.V., Kharitonov V.I. One erroneous statement associated with the uniporlar induction phenomenon. Electricity, 2012, No. 7, pp. 69–72 (in Russian).

Tarnovsky A.S. Definition of the concepts ‘potential’ and ‘potential field’. Electricity, 2000, No. 1, pp. 63–64.

Bogach V.A. Hypothesis of the existence of a static electromagnetic field and its properties. Preprint of the Joint Institute for Nuclear Research, 1996, No. 13-96-463, p. 36.

Sommerfeld А. Electrodynamics. Мoscow, 1958, 327 p.

I. P. Bazarov, Thermodynamics. Textbook, 5th edition, Saint-Petersburg: Publishing House “Lan”, 2010, 384 p.

B.M. Vul, Physics of dielectrics and semiconductors, Moscow: Nauka, 1988, 372 p.

V. G. Polevoi, Heat transfer by fluctuation electromagnetic field, Moscow: Nauka, 1990, 90 p.

S. М. Rytov, Theory of electric fluctuations and thermal radiation, Moscow: Publishing House of the USSR Academy of Sciences, 1953, 231 p.

М. А. Leontovich, Statistical Physics. # 19, # 26, Moscow-Leningrad: Publishing House “OGIZ”, 1944.

R. Siegel, J. Howell, Thermal radiation heat transfer, McGraw-Hill Book Company, Toronto, 1972, p. 534.

M.N. Özisik. Radiative transfer and interactions with conduction and convection, John Wiley & Sons, New York, London, Sydney, Toronto, 1976.

M. Planck, Thermal radiation laws and hypothesis on the elementary quantum of action, Collected Papers, Thermodynamics, Poiseuille’s theory, Moscow: Nauka, 1975, 778 p.

K. Bohren, D. Huffman, Absorption and scattering of light by small particles, John Wiley & Sons, A Wiley-Interscience Publication, New York, Toronto), Moscow: Nauka, 1986, 658, .

H.C. Vann de Hulst. The scattering of light by small particles, Moscow: PublishingHouse “IL”, 1961, 536 p.

А. P. Ivanov, Application of multiple scattering for the study of optical properties of substances, in: Theoretical and Applied Problems of Light Scattering, Minsk: Nauka i Tekhnika, 1971, 486 p.

А. S. Тоporets, Light reflection by rough surfaces, in: Theoretical and Applied Problems of Light Scattering, Minsk: Nauka i Tekhnika, 1971, 486 p.

G. М. Gorodinsky, Spectroscopy methods and devices for study and industrial quality control of glass and metal surfaces, optical and semiconductor crystals, in: Theoretical and Applied Problems of Light Scattering, Minsk: Nauka i Tekhnika, 1971, 486 p.

А.V. Luikov, Application of irreversible thermodynamics for the study of heat and mass transfer processes, J. Eng. Phys. Thermophys., Vol. IX, No. 3, 1965, pp. 287–304.

V. A. Bubnov, Theory of heat waves, J. Eng. Phys. Thermophys., No. 3, 1982, pp. 431–437.

А. G. Shashkov, V.A. Bubnov, and S.Yu. Yanovsky, Heat conduction wave phenomena, Minsk: Nauka i Tekhnika, 1993, p. 274.

V. А. Кudinov and I.V. Kudinov, Obtaining and analysis of an exact analytical solution of the hyperbolic heat conduction equation for a flat wall, Teplofiz. Vysok. Temp., 2012, Vol. 50, No, 1, pp. 118–125.

V. А. Кudinov and I. V. Kudinov, Study of heat conduction with regard to a finite heat propagation velocity, Teplofiz. Vysok. Temp., 2013, Vol. 51, No. 2, pp. 301–310.

С. Cattaneo, Sur ube forme de l’equation de la chaleur eliminant le paradoxe d’une propagation instantance, Com. Press Rendus, 1958, Vol. 247, No. 4, p. 431.

А. N. Tikhonov and А. А. Samarsky, Mathematical physics equations, Moscow: Nauka, 5th edition, 1977, 734 p.

V. I. Коrzyuk, Mathematica physics equations, Minsk: BSU Press, 2011, 459 p.

I. L. Коlpashchikov and А. I. Shnip, Thermodynamics and models of non-plastic media, Minsk: A.V. Luikov HMTI NAS Belarus, preprint No. 13, 1986, 34 p.

V. L. Bonch-Bruevich and S. G. Kalashnikov, Physics of semiconductors, Moscow: Nauka, 1990, 622 p.

W.Shockley. Theory of electronic semiconductors,Moscow: PublishingHouse “IL”, 1953, 714 p.