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AIJ-199
American International Journal of Contemporary Scientific Research
ISSN 2349 – 4425
Electrodynamics of non- and slow moving layered
media with boundary condition of full current
continuity
N.N. Grinchik1 and A.D. Chorny2
1,2
A.V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences, Minsk, Republic
of Belarus, 15 P. Brovka Str., 220072
[email protected]
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. The model is based
on the methods of thermodynamics and on the equations of an electromagnetic field and is formulated without explicit separation
of the charge carriers and the charge of an electric double layer. The influence of a slowly moving medium on the electromagnetic
wave propagation is considered. The calculation results show the absence of the influence of the medium’s motion on the phase
shift of waves, which is consistent with experimental data.
Keywords
Electromagnetic wave, heterogeneous medium, surface-impedance matrix, induced surface charge, boundary conditions,
electrodiffusion phenomena in electrolytes, energy density, heat in electrodynamics, aeroacoustics.
1. Introduction
Let us consider a boundary surface S between two media having different electrophysical properties. On each of its side, magnetic
field and magnetic inductance vectors, as well as electric field and electric displacement vectors are finite and continuous.
However, at the boundary surface S they can experience first-kind discontinuity. Furthermore, at the boundary surface there arise
surface charges σ and surface currents i (whose vectors lie in the plane tangential to the boundary surface S) induced when acted
upon by an external electric field.
There are no closing relations for induced charge and surface current, and this fact forces to introduce a surface-impedance matrix.
This matrix is determined experimentally or, in some cases, theoretically from a quantum representation where n is the normal to
the boundary surface directed to the field region and i and k are the subscripts for orthogonal coordinate directions at this
boundary surface.
In radio physics, for the problem of propagation of electromagnetic waves in the presence of “perfectly conducting” bodies, a
surface impedance of the form as the boundary condition in outside the conductor (M.A. Leontovich, 1948) is often used
2


Ei  z ik  nH 
k 1
k
.
The continuity condition of tangential components of electric and magnetic field strength, i.e., the absence of a surface current, is
assumed to be satisfied.The induced surface charge and current not only define surface properties, but they are also the functions
of the process. Surface impedances therefore hold true for the conditions under which they have been determined.Under other
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AIJ-199
American International Journal of Contemporary Scientific Research
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experimental conditions, surface impedances cannot be used, since they depend on the structure of the field determined and are
not boundary conditions in the conventional sense. When the concept of “surface impedance” is used, there appear a number of
constraints and assumptions. The penetration depth of electromagnetic wave into a medium and its wavelength must be small as
compared to the wavelength in the ambient space in comparison with both distances from field sources and curvature radii of the
boundary surface. Changes both in the relative permittivity along the body length and in the magnetic permeability at a distance of
the order of the wavelength (or along the length equal to the penetration depth) are small.
Furthermore, wave absorption is absent or negligible; it is therefore difficult to use the existing impedance conditions, since in the
body waves can travel not only from the boundary surface, but also from the body.
By virtue of the afore-said, the propagation of electromagnetic waves by a group of close-packed bodies assumes the use of exact
boundary conditions involving surface charge and surface current.
Numerical codes under the general title “Finite Difference Time Domain (FDTD)” are currently available and allow a wave field
to be calculated within the framework of initial and boundary-value problems in a three-dimensional and unsteady statement in
inhomogeneous dissipative media possessing similar time dispersion. It should be noted that these numerical codes are also
expedient for calculation of fields not only in linear, but also in nonlinear media. There is an ample amount of the FDTD code
literature, but a practical implementation of the method indicated here is hindered by a need to prescribe a surface-impedance
matrix at the boundary surface of adjacent media. Unsteady thermal and diffusion processes, and also the dispersion process affect
the surface-impedance matrix; fundamentally, this makes the numerical simulation problem of wave propagation in a
heterogeneous ion conductor or in a conductor with mixed conduction more complicated. When the surface-impedance matrix is
used at the boundary surface of adjacent media, it is also difficult to obtain expressions for energy density and heat release in the
absorbing medium.
In dielectrics, when conduction currents are negligible, induced surface charge and surface current are also insignificant; the wellknown geometric laws of reflection and rarefaction in light propagation through the boundary surface of two media and the Frenel
formula hold true; here, the amplitude of an incident wave is assumed to be equal to a sum of amplitudes of transmitted and
reflected waves. This condition is widely used in optics and radio physics, since for a monochromatic wave, the solving of the
problem is reduced to the Helmholtz control, e.g., light propagation in a photon crystal. It should be emphasized here that a plane
monochromatic wave is infinitesimal. However, while considering the electromagnetic properties of composite material structures
representing homogeneous matrices with a large number of distributed small material in- homogeneities, mathematical models
must be take in account chiral, biisotropic, bianisotropic, Faraday, and gyrotropic media and metamaterials, induced surface
charge and surface current, whose prescription involves fundamental difficulties.
Invariably topical are the problems of radio location and radio communication. A successful solving of any radio location problem
requires that space–time characteristics of diffraction fields of electromagnetic waves scattered by an object of location into the
ambient space be known. Irradiated objects have a very intricate architecture and geometric shape of a surface consisting of
smooth portions, surface fractures, sharp edges, and edges with corner radii much smaller than a wavelength of a probing signal;
the regular Leontovich impedance conditions cannot then be used. An object has larger geometrical dimensions as compared to an
incident electromagnetic wave length and numerous coatings with different radio physical properties. Therefore, to solve radio
location problems calls for the methods of calculation of diffraction fields of electromagnetic waves excited and scattered by
different surface portions of an object, in particular, by their domains with different corners, since the latter are some of principal
sources of scattered waves.
The problems of electromagnetic wave diffraction in domains with corners have been the focus of numerous investigations.
Representation of a diffraction field in a domain with a corner in the form of A. Sommerfeld’s integral is the most promising
technique to solve the task of a perfectly conducting and impedance corner in the quasi-optical wavelength range.
Using A. Sommerfeld’s method, G.D. Malynzhinets obtained an asymptotic solution to the scalar problem for plane acoustic wave
excitation of an impedance corner with semi-infinite sides and an arbitrary opening angle; this author justified this method and a
later date it was called the “Sommerfeld–Malynzhinets method”. It is noteworthy that finding exact solutions to a generalized
wave equation for impedance and dielectric structures by the Sommerfeld–Malynzhinets method involves exceptional
mathematical difficulties; these solutions were obtained only for monochromatic waves for a semi-infinite corner; the
corresponding magnetic or electric vectors were oriented parallel to the corner edge.
It is difficult to use this method forboth normal components of the electromagnetic vector and non-monochromatic waves. As
noted above, it is possible, in principle, to use the surface-impedance matrix for numerical modeling. In our opinion, the numerical
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AIJ-199
American International Journal of Contemporary Scientific Research
ISSN 2349 – 4425
modeling technique and the physical and mathematical model of scattering of a signal, e.g., a ballistic object,cannot be affected by
the object geometry and empirical coefficients that depend on the electromagnetic field structure.
The existence of a surface charge at the boundary surface S between the two media having different electrophysical properties is
clearly demonstrated by the following example. A current transport through a flat capacitor filled with two dielectric materials
with relative permittivities ε1 and ε2 and electrical conductivities λ1 and λ2 will be considered. A current voltage U is applied to the
capacitor plates; R is the total capacitor resistance (Fig. 1). It is necessary to calculate the surface electric charge induced by the
electric current.
The electriccharge conservation law yields that the flux is constant in a circuit; the following equation is then obtained:
1 En1  2 En2  U /  RS 
(1)
where E n1 and E n2 are the normal components of the electric field vector.
At the boundary surface between dielectrics acted upon by the electric field, the normal components of the electricinductance
vector spasmodically change by a value equal to that of the induced surface charge σ:
 01 En1   0 2 En2  
(2)
Fig. 1. Dielectric media inside a flat capacitor
Solving the system of the equations (see Equations (1)–(2)) arrives at the expression for σ
  U / RS   0 1 / 1    2 / 2  
(3)
Then it follows (see Equation (3)) that the charge σ is determined by the current and the multiplier accounting for the medium
properties. If
1 / 1    2 / 2   0
(4)
the surface charge σ is not formed. And more, the recent trends are focused upon an increased application of micro machines and
engines manufactured from plastic materials, where the emergence of surface charge is undesirable. To oil elements of such
machines, it is better to use oil with permittivity
 oil obeying the relation
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1   oil   2
ISSN 2349 – 4425
(5)
The application of this oil allows the electrification of machine movable parts manufactured from dielectric materials to be
reduced. In addition to the charge σ, a contact potential difference arises always regardless of the current.
An electric field interacting with a material is investigated with the use of the Maxwell equation. For convenience and a further
correct representation, the axial vector of magnetic field strength and induction will be denoted as H⃑and B⃑, respectively
jtotal    H,   D  
(6)
B
   E,   B  0
t
(7)

where jtotal   E 
D
; B  0H; D   0 E . In this case, at the boundary surface S the above system of equations is
t
supplemented with the boundary conditions [1, 2] (Monzon, I.; Yonte,T.; Sanchez-Soto, L., 2003; Eremin,Y. & Wriedt,T., 2002)
(8)
Dn1  Dn2  
(9)
E 1  E 2  0
(10)
Bn1  Bn2  0
(11)
 
H1  H 2   i  n 
The subscripts n and τ denote the normal and tangential components of the vectors to the boundary surface S, and subscripts 1 and
2 stand for the adjacent media with different electrophysical properties. The subscript τ denotes any direction tangential to the
discontinuity surface. At the same time, a closing relation is absent for the induced surface charge σ, and this generates a need to
introduce an impedance matrix [3–7](Wei Hu & Hong Guo, 2002; Danae, D. et al., 2002; Larruquert, J. I., 2001; Koludzija, B. M.,
1999; Ehlers, R. A. & Metaxas, A. C., 2003). It is determined from experiments or, in some cases, theoretically from quantum
representations [8–12](Barta, O.; Pistora, I.; Vesec, I. et al., 2001; Broe, I. & Keller, O., 2002; Keller, 1995; Keller, O., 1995;
Keller, O., 1997). It should be noted that the electromagnetic Maxwell equation does not coincide completely with its modern
form (6)–(7) that was first given by Hertz (1884) and independently of him by Heaviside (1885). It should be borne in mind that
an electric field strength E is a polar vector. Concentration and temperature gradients are also polar vectors. On the contrary, a
magnetic field strength H⃑, as well as H⃑, are axial vectors, or pseudovectors. This is well seen from the formula for the Lorentz
force F = q[VH⃑]. The Biot–Savart–Laplace law establishes the value and direction of the axial vector H⃑at an arbitrary point of a
magnetic field produced by a semiconductor element de⃗ (the vector coincides with the current direction).
dH⃑ =
i dl⃑r⃗
4 πr
Indeed, the behavior of Н⃑upon reflection at the coordinate origin and upon substitution ofr (–r) is determined through the
behavior of the polar vectors F and V and the properties of their vector product. On substitution ofr (–r), the vectors F and V
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AIJ-199
American International Journal of Contemporary Scientific Research
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reverse their direction; the vector product reverses its sign. Consequently, on substitution of r (–r), the vector Н⃑ must remain
constant. This property is assumed to be an attribute of the axial vector.
The induced surface charge σ not only characterizes the properties of a surface, but also represents a function of the process, i.e.,
σ(E(∂E/∂t, H(∂H/∂t))); the surface impedances [3–7](Wei Hu & Hong Guo, 2002; Danae, D. et al., 2002; Larruquert, J. I., 2001;
Koludzija, B. M., 1999; Ehlers, R. A. & Metaxas, A. C., 2003) are therefore true for the conditions, under which they have been
determined. These impedances cannot be used in experiments being conducted under other experimental conditions.
The problem of determining surface charge and surface current at the metal–electrolyte boundaries becomes even more complex
whilst investigating and modeling unsteady electrochemical processes, e.g., pulse electrolysis, when lumped parameters L, C, and
R cannot be used in principle.
It will be shown that σ can be calculated using the phenomenological macroscopic electromagnetic Maxwell equations and the
electriccharge conservation law with regard to the features of the boundary surface between adjacent media.
A separate consideration will be given to ion conductors. To construct a physical and mathematical model, it should be taken into




account that E and H are not independent functions; the wave equation for E or H is then more preferable than the system of
equations (see Equations (6)–(7)). It will be shown that in considering wave propagation and interference in the heterogeneous
medium it is preferable to use the wave equation for a polar vector of electric field strength.
2. Electron Conductors. New Closing Relations for the Boundary Surfaces of
Adjacent Media

2.1 Generalized Wave Equation for E and Boundary Conditions in the Presence of
Strong Electromagnetic Field Discontinuities
2.1.1 Physical and Mathematical Model
A physical and mathematical model for propagation of an electromagnetic field in a heterogeneous medium will be formulated
here. Let us multiply the left- and right-hand sides of the equation for total current (see Equation (6)) by μμ0 and differentiate it
with respect to time. Acting by the operator rot on the left- and right-hand sides of the first equation of the equations (see
Equation (7)), provided that μ=const it can be obtained
1 2
1
jtotal

 E
grad  divE
t
0
0
(12)
In the Cartesian coordinates, the equations (see Equation (12)) will take the form
jtotalx
1   2 E x  2 E x  2 E x  1   E x E y Ez 


 2 


t
0  x 2
y 2
z  0 x  x
y
z 
(13)
2
2
2
1   Ey  Ey  Ey 
1   E x E y E z 





 2
2
2 
0  x
y
z  0 y  x
y
z 
(14)
jtotaly
t

1   2 Ez  2 E z  2 Ez  1   Ex E y Ez 
jtotalz


 2 


t
0  x 2
y 2
z  0 z  x
y
z 
(15)
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For the boundary surface, the following relation [2](Eremin,Y. & Wriedt,T., 2002) is also true:
divi  I qx1  I qx2  

t
(16)
Let us write the conditions (see Equations (8)–(11)) in the Cartesian coordinate system:
Dx1  Dx2  
(17)
E y1  E y2  0
(18)
E z1  E z2  0
(19)
Bx1  Bx2  0
(20)
H y1  H y2  iz
(21)
H z1  H z2  i y
(22)
whereiτ = iyj + izk is the surface current density, and the x coordinate is directed along the normal to the boundary surface. The
densities iy and iz of surface currents represent the electric charge per unit time per unit length by a segment positioned at the
boundary surface aligning the current perpendicular to its direction.
The order of the system of differential equations (see Equations (13)–(15)) is equal to 18. At the boundary surfaceS, it is therefore
necessary to set, by and large, nine boundary conditions. Moreover, the three additional conditions (see Equations (17), (21)–(22))
containing (prior to the solving) unknown quantities should be met at this boundary surface. Hence, the total number of
conjugation conditions at the boundary surfaceS should be equal to 12 for a correct solution of the problem.
As an example, a one-dimensional case of heat propagation in a layered material will be considered. As known, to solve the
problem, two conditions should be specified at the boundary surface: the equality of heat fluxes and the equality of temperatures,
since the heat conduction equation has the second derivative with respect to the coordinate.
Differentiating the expression (see Equation 17) with respect to time and using the relation (see Equation 16), the following
condition for the normal components of the total current at the medium–medium boundary surface can be obtained:
divi  jtotalx1  jtotalx2
that allows the surface charge σ to be disregarded Let us introduce the arbitrary function f:
(23)
 f  x
 f1 x  0  f 2
x  0
. In this
case, the expression (see Equation 23) will take the form
(24)
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 divi  jtotalx1  x   0
At the medium–medium boundary surface, Ex is assumed to be a continuous function of y and z. Then, differentiating the equation
(see Equation 23) with respect to y and z, it can be obtained
  divi 



 y jtotalx1 
y

 x 
(25)
  divi 


j


totalx
1
 z
z
 x 
(26)
Let us differentiate the conditions (see Equations (20)–(22)) for magnetic induction and magnetic field strength with respect to
time, provided that B=μμ0H
i
 1 B y 
i  1 Bz 
 Bx 
 0, 
 z,
 y


 t 
 0 t  x  t  0 t  x  t
x 
(27)
Using the equation (see Equation (7)) and expressing it (see Equation (27)) in terms of electric field rotorprojections arrives at
 E z E y 

0
z  x 
 y
 rotxE x  0 and 
(28)
 1

 1  E x E z  
i
i
 z or 

 z


  rot y E

x   x  t
 0  z
 0
 x  t
(29)
 1  E y E x  
 1

i
i
 z or 

 z



  rotz E
y  
t
 0
 x   t
 0  x
x 
(30)
Here, the equation (see Equation (28)) is the normal projection of the electric field rotor, the equation (see Equation (29)) is the
tangential projection of the rotor on y, and the equation (see Equation (30)) is the rotor projection on z.
Assuming that Ey and Ez are continuous differentiable functions of the y and z coordinates, from the conditions (see Equations
(18)–(19)),it can be obtained
 E y 
 E 
 0,  y 
0
 y 

 x 
 z  x 
(31)
 E z 
 E 
 0,  z 
0
 y 
 z  x 

 x 
In accordance with the condition that the tangential electric field projectionson the z and y coordinates are the same and in
accordance with the conditions (see Equations (18)–(19), the expressions for the densities of the surface currents iz andiy take the
form
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iz   E z
American International Journal of Contemporary Scientific Research
x 
, iy   Ey
x 
ISSN 2349 – 4425
(32)
where

1
 1  2 
2
x 
(33)
isan average value of the electrical conductivity at the boundary surface between the adjacent media according to the Dirichlet
theorem for a piecewise-smooth, piecewise-differentiable function.
Consequently, the formulas (see Equations (31)–(33)) yield

 divi 

 x   0
(34a)
The relation (see Equation (34)) and, hence, the equality of total current normal components were obtained differently by G.A.
Grinberg and by V.A. Fok [13](Grinberg, G.A. & Fok, V.A., 1948). In this work, it is shown that condition (34a) leads to the
equality of the derivatives of the electric field strength along the normal to the surface
 E x 
 x   0
x 
(34b)
In virtue of the above-said, twelve conditions are available at the boundary surface between adjacent media that are necessary for
solving a complete system of equations (see Equations (13)–(15)):
a) Eyand Ez are determined from the equations (see Equations (18)–(19));
b) Ex is determined from the condition (see Equation (24));
c) ∂Ex⁄∂y, ∂Ex⁄∂z, and ∂Ex⁄∂x values are determined from the relations (see Equations (25)–(26)) with the use of the continuity
conditions of a total current normal component at the boundary surface (see Equation (24)) and of the total current derivative with
respect to the x coordinate
d) ∂Ey⁄∂y, ∂Ey⁄∂z, and ∂Ez⁄∂z values are determined from the conditions (see Equations (31)–(32)) as a consequence of the
continuity of electric field tangential components along the y and z coordinates
e) ∂Ey⁄∂x and ∂Ez⁄∂x derivatives are determined from the conditions (see Equations (29)–(30)) as a consequence of the equality of
the tangential components of the electricfield rotor along the y and z coordinates.
It should be noted that the condition (see Equation (23)) was used in [14] (Grinchik, N. N. & Dostanko, A. P., 2005)for numerical
simulation of pulsed electrochemical processes in the one-dimensional case. The condition (see Equation (28)) for the normal
component of the electric field rotor represents a linear combination of the conditions (see Equations (31)–(32)); therefore, rotxE
= 0 and there is no need to use it in a subsequent discussion. The specificity of the expression for the general law of electric charge
conservation at the boundary surface is that the ∂Ey⁄∂y and ∂Ez⁄∂zcomponents are determined from the conditions (see Equations
(31)–(32)) that follow from the equality and continuity of the tangential components of Ey and Ez at the boundary surfaceS.
At the boundary surface between adjacent media, the following conditions are then met: the equality of total current normal
components; the equality of electric field rotor tangential projections; the electric charge conservation law; the equality of electric
field tangential components, and their derivatives in the tangential direction; the equality of the derivatives of the total current
normal components in the direction tangential to the boundary surface between adjacent media determined with regard to surface
currents and with no regard to a surface charge introduced in an explicit form. These are true over each cross section of a sample
under investigation.
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2.1.2 Calculation Features of Electromagnetic Wave Propagation in Layered Media
Electromagnetic effects arising at the boundary surface between different media acted upon by plane electromagnetic waves have
a profound impact on equipment because all real devices are bounded by surfaces and are inhomogeneous in space. At the same
time, studies of the propagation of waves in layered conducting media and, according to [15](Born, 1970), in thin films are
reduced to calculating reflection and transmission coefficients; the function E(x) is not determined for the film thickness, i.e., the
geometrical optics approximation is used.
The physical and mathematical model as proposed herein allows the electromagnetic wave propagation in a layered medium to be
investigated, not using the assumptions adopted in [3–5, 7] (Wei Hu & Hong Guo, 2002; Danae, D. et al., 2002; Larruquert, J. I.,
2001; Ehlers, R. A. & Metaxas, A. C., 2003).
Since the conditions (see Equations (23)–(32)) are true over each cross section of a layered medium, the capturing method-based
schemes with no explicit definition of the boundary surface between media will be used. In this case, it is proposed to calculate Ex
at the boundary surface in the following way.
In accordance with the equation (see Equation (17)), Ex1≠Ex2, i.e., Ex(x) experiences a first-kind discontinuity. Let us determine an
electric field strength at the discontinuity point x = ξ on condition that Ex(x) is a piecewise-smooth, piecewise-differentiable
function having finite one-sided derivatives E x  ( x ) and E x  ( x ) . At the discontinuity points xi,
E  xi  xi   E  xi  0 
xi 0
xi
(35)
E  xi  xi   E  xi  0
xi 0
xi
(36)
E x  ( xi )  lim
E x  ( xi )  lim
In this case, following the Dirichlet theorem [16] (Kudryavtsev, 1970), the Fourier series of the function E(x) at each point x,
including the discontinuity point ξ, converges and its sum is equal to
E x  
1
 E   0   E   0  
2
(37)
The Dirichlet condition (see Equation (37)) is also a reasonable assumption. In the case of contact of two solid conductors, e.g.,
dielectrics or electrolytes in different combinations (metal–electrolyte, dielectric–electrolyte, metal–vacuum, and so on), at the
boundary surface between adjacent media there always arises an electric double layer (EDL) with an unknown (as a rule) structure
that, however, substantially influences electrokinetic effects, a rate of electrochemical processes, and so on. It is significant that in
reality, the electrophysical characteristics λ, ε, and E(x) change continuously in the electric double layer; this is therefore true for
the case (see Equation (37)) where the thickness of the electric double layer, i.e., the thickness of the boundary surface, is much
smaller than the characteristic dimension of a homogeneous medium. In a composite, e.g., in a metal embedded with dielectric
balls, where the concentration of both the components is fairly large and their characteristic dimensions are small, the boundaries
of the boundary surface can overlap and the condition (see Equation (37)) cannot be satisfied.
If the thickness of the electric double layer is much smaller than the characteristic dimension, L, of an object, then (see Equation
(37)) it also follows from the condition that E(x) changes linearly in the EDL domain. In reality, the thickness of the electric
double layer depends on the kind of contacting materials and can be several tens of angstroms [17] (Frumkin, 1987). In
accordance with the modern views, the metal surface of the electric double layer consists of two parts, the first of which is formed
by ions immediately attracted to the metal surface (a "dense" or a "Helmholtz" layer of thickness h), and the second is formed by
ions separated by a distance larger than the ion radius from the layer surface, and the number of these ions decreases as the
distance between them and the boundary surface (a "diffusion layer") increases. The potential distribution in the dense and
diffusion parts of the electric double layer is exponential in actual practice (Frumkin, 1987), i.e., the condition that E(x) changes
linearly, is not satisfied; in this case, the sum of the charges of the dense and diffusion parts of the metal surface of the electric
double layer is equal to the charge of its metal surface. However, if the thickness, h, of the electric double layer is much smaller
than the characteristic dimension of an object, then the series expansion of E(x) is valid, and consideration can be restricted to a
linear approximation. In accordance with the more general Dirichlet theorem (1829), knowledge of this function in the EDL
domain is not of necessity in effort to justify the equation (see Equation (37)). Nonetheless, the above-indicated physical features
of the electric double layer provide support to the validity of the condition (see Equation (37)).
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The condition at boundary surfaces, similar to the equation (see Equation (37)), has earlier been obtained [18] (Tikhonov, A. N. &
Samarskii, A. A., 1977) for the potential field (where rot E = 0) by introducing the surface potential, using the Green’s formula,
and considering the discontinuity of the double layer potential. In [18] (Tikhonov, A. N. & Samarskii, A. A., 1977), it is also
noted that in general, the consideration of the thickness of the double layer and the change in its potential at h/L≪1 has no sense;
it is then advantageous to consider the surface potential of any densityrather than the volume potential. The condition (see
Equation (37)) can be obtained, as shown in [16] (Kudryavtsev, 1970), from the more general Dirichlet theorem for a nonpotential vorticity field [18] (Tikhonov, A. N. & Samarskii, A. A., 1977).
Thus, the foregoing and the validity of the conditions (see Equations (17)–(19) and (25)–(32)) over each cross section of a layered
medium show that, for a numerical solution of the problem under consideration, it is advantageous to use capturing method-based
schemes and to discretize a medium in such a way that the boundaries of the layers would have common points.
The medium was divided into finite elements so that finite-element grid nodes lying at the boundary surface between media with
different electrophysical properties were shared by these media at a time. In this case, total currents or current fluxes at the
boundary surface should be equal if the Dirichlet condition (see Equation (37)) is satisfied.
2.1.3 Numerical Modeling for Electromagnetic Wave Propagation in Layered Media
Let us analyze the electromagnetic wave propagation through a layered medium that consists of several layers with different
electrophysical properties when an electromagnetic radiation source is positioned in the upper plane of the medium. It is assumed
that the normal component of the electric field vector Ex = 0 and its tangential component Ey = a sin (ωt) where a isthe
electromagnetic wave amplitude (Fig. 2).
In this example, to correctly specify the conditions at the lower boundary of the medium, an additional layer is added downstream
layer 6; this layer has larger conductivity and the electromagnetic wave therefore attenuates rapidly in it. In this case, the condition
Ey = Ez = 0 can be set at the lower boundary of the medium. The above manipulations were made to limit the dimension of the
medium under consideration because in the general case, the electromagnetic wave attenuates completely at an infinite distance
from an electromagnetic radiation source.
The electromagnetic wave propagation in the layered medium with electrophysical parameters ε1 = ε2 = 1, λ1 = 100, λ2 = 1000, and
μ1 = μ2 = 1 was calculated numerically. Two values of the cyclic frequency ω = 2π/T were used: in the first case, the
electromagnetic wave frequency was assumed to be equal to ω = 1014 Hz (infrared radiation), and, in the second case, the cyclic
frequency was taken to be ω = 109 Hz (radio frequency radiation).
Fig. 2. Scheme of a layered medium: layers 1, 3, and 5 are characterized by the electrophysical parameters ε1, λ1, and μ1, and
layers 2, 4, and 6 – by ε2, λ2, μ2
As a result of numerical solution of the system of the equations (see Equations (13)–(15)) under the conditions of the surfaceS (see
Equations (24)–(34)) at boundary surfaces, the time dependences of electric field strength at different distances from the surface
of the layered medium (Fig. 3) were obtained.
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Fig. 3. Time variation of the tangential component of electric field strength at a distance of 1 μm (1), 5 μm (2), and 10 μm (3)
from the medium surface at λ 1 = 100, λ 2 = 1000, ε1 = ε 2 = 1, μ1 = μ2 = 1, and ω= 1014 Hz. t, s
Our numerical modeling results (Fig. 4) show that a high-frequency electromagnetic wave propagating in a layered medium
attenuates, whereas a low-frequency electromagnetic wave penetrates to a larger depth into such a medium. The model developed
herein is also used for calculating the propagation of a modulated signal with a frequency of 20 kHz in a layered medium. Our
numerical results in Fig. 5 have found electric field strength changes at different depths of the layered medium. So the model
proposed herein can be used to advantage the calculation of the propagation of polyharmonic waves in layered media; such
calculation cannot be performed using the Helmholtz equation.
The physical and mathematical model as developed here can also be used to advantage the numerical modeling of the propagation
of electromagnetic waves in media with complex geometric parameters and large electromagnetic field discontinuities (Fig. 6).
Figure 6a shows the cross sectional view of the cellular structure representing a set of parallelepipeds with different cross sections
in the form of squares. The parameters of materials in large parallelepipeds are denoted by subscript 1, and the parameters of
materials in small parallelepipeds (the squares in the figure) – by subscript 2.
An electromagnetic wave propagates in parallelepipeds (channels) in the transverse direction. It is seen from Fig. 6b that the
cellular structure has "silence zones," where the electromagnetic wave strength amplitude is close to zero, and also inner zones,
where the signal has a marked value downstream the "silence" zone formed due to interference.
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Fig. 4. Distribution of the electric field strength amplitude
over the cross section of the layered medium: ω = 1014 (1)
and 109 Hz (2). y, μm
(a) 2D medium distribution of the electric field strength
amplitude
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Fig. 5. Time variation of electric field strength at a
distance of 1 (1), 5 (2), and 10 μm (3) from the medium
surface. t, s
(b) Depth distribution of the electric field strength
amplitude
Fig. 6. 2D medium and depth distributions of the electric field strength amplitude at ε1= 15, ε2 = 20, λ1 = 10-6, λ2= 10, μ1 = μ2 = 1,
and ω= 109 Hz (the dark background denotes medium 1 and the light background – medium 2). x, y, mm; E, V/m
2.1.4 Numerical Modeling Results of the Electromagnetic Wave Scattered by Structures
with Different Corners
These are radio location and radio communication problems that are among the main challenges for problems to be solved using
radio engineering devices.
Knowledge of space–time characteristics of diffraction fields of electromagnetic waves scattered by an object of location into the
environment is needed to successfully solve any radio location problem. An irradiated object has a very intricate architecture and
geometric shape of its surface consisting of smooth portions and numerous different-type corner domains of smooth portions,
surface fractures, sharp edges, etc. – with a rounded radius much smaller than a probing signal wavelength. Therefore, the solving
of radio location problems requires that the methods for calculation of diffraction fields of electromagnetic waves excited and
scattered by different surface portions of objects, in particular, by domains with corners, be known, since the latter are among the
main sources of scattered waves.
For another urgent problem, i.e., the problem of radio communication between objects, of most difficulty are the tasks of
designing antennas to be put on an object, since their operating efficiency is closely connected with the geometric and radio
physical properties of its surface.
The problems of electromagnetic wave diffraction in domains with corners are focused on numerous tasksdealing with a perfectly
conducting and impedance corner for monochromatic waves and on the representation of the diffraction field in a domain with a
corner in the form of a Sommerfeld integral [19] (Kryachko, A.F. et al., 2009).
Substituting Sommerfeld integrals into the system of boundary conditions gives a system of recurrence functional equations for
unknown analytical integrands. The system coefficients are the Fresnel coefficients defining the reflection of plane media or their
refraction into the opposite medium. From the system of functional equations, the sequences of integrand poles and residues at
these poles are determined in a recurrence manner.
The edge diffraction field in both media is determined using a pair of second-kind Fredholm-type singular integral equations that
have been obtained from the above-indicated systems of functional equations with a subsequent calculation of Sommerfeld
integrals by the saddle-point approximation. The branching points of the integrands make for the presence of creeping waves
excited by the edge of the dielectric corner.
The method proposed herein is only valid for monochromatic waves and the approximate Leontovich boundary conditions when
the electromagnetic wave field slowly varies from point to point on a wavelength scale (Leontovich, 1948).
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It should be noted that the existing approximate Leontovich conditions have a number of other constants and should be used with
caution [20] (Leontovich, 1948).
Actually, the proposed calculation method does not work, e.g., in the presence of two corners of an optical knife, or during
electromagnetic wave diffraction on a system of parallel lobes when their gap is of the order of microns, and the electromagnetic
field is strongly “cut” throughout the space with a step more than a wavelength.
A) Optical Knife
Figure 7 shows the electromagnetic wave field during its diffraction on the optical knife. The inlet and outlet parameters of the
wave are Ex=104sin (1010t), Ey=104cos (1010t).
The electrophysical characteristics are as follows: a corner is manufactured from aluminum: ε=1; μ=1; σ=3.774·107S/m; air is a
surrounding medium.
The computational domain sizes are 0.1×0.05 m. The computational time is 10-9 s, and the time step is 10-11 s.
Numerical solution of the system of equations (13)–(15) yields the optical knife-dependent distributions of Ex(x,y) and Ey(x,y).
The computation results are in good agreement with the available experimental data.
Optical laser experiments were carried out and were intermediate, but the diffraction field photographs obtained from experiment
and the computation results seemed to be in good qualitative agreement.
(a) Mesh
(b)Ex amplitude
(d)Ex isolines
(e)Ey amplitude
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(f)Ey amplitude
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(g)Ey isolines
Fig. 7. Mesh, Ex amplitude and isolines, Ey amplitude and electromagnetic field strength isolines
The known works by L.D. Landau and E.M. Lifshits also give the solution to the problem on the light diffraction from the edge of
a perfectly conducting corner bounded by two intersecting half-planes (A. Sommerfeld). A full representation of this sophisticated
mathematical theory involving special mathematical techniques is beyond the scope of this book. Theoretical fundamentals of
solution of problems of electromagnetic wave scattering in dielectric and impedance corner-shaped domains for monochromatic
waves are contained in [19] (A.F. Kryachko, et al., 2009). Using the Sommerfeld–Malyuzhinets method, the authors were able to
obtain exact and asymptotic solutions for the diffraction of a plane wave oblique with respect to the impedance corner edge. The
analytical calculation results were compared to the experimental data [19, p.162]. The numerical results in Fig. 7 are in qualitative
agreement with the data [19] (A.F. Kryachko, 2009); in the both cases, the electromagnetic radiation intensity is maximum at the
corner point. The present work has no specific electrophysical properties of the corner and the surface impedance of the dielectric
corner; it is therefore impossible, in practice, to correctly compare different computational techniques. The concentration of
electromagnetic energy on a structure with a different corner is a well-known fact.
B) Diffraction Grating
The wave parameters and the boundary surface conditions are the same, as those in the case of the “optical knife”. The
electrophysical characteristics are as follows: 2D lobes, ε=12; and σ=100 S/m; the ambient medium is air; the characteristics of
the prism and the square are identical to those of the lobes.
In Fig. 8 the calculation time is 10-10 s; the time step is 10-12 s. In Fig. 9 the calculation time is 10-9 s; the time step is 10-11 s.
(a)Ex amplitude
(b) Ex isolines
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(c)Ey amplitude
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(d) Ey isolines
Fig. 8.Ex amplitude and isolines, Ey amplitude and electromagnetic field strength isolines
It is seen from the numerical modeling results that the proposed “comb” can be used as a filter of a high-frequency signal.
Furthermore, numerical modeling of a modulated signal was made at a frequency of 20 kHz. The modulated signal results are not
given. The difference scheme for stability was analyzed using the initial data. When time and space steps are large, there appear
oscillations of the grid solution and of its “derivatives” (“ripple”) which strongly decrease the accuracy of the scheme.
Undoubtedly, this issue calls for separate consideration. The proposed algorithm of solution of Maxwell equations allows the
circuitry-engineering modeling of high-frequency radio engineering devices and the investigation of the propagation of
electromagnetic waves in media of intricate geometry in the presence of strong electromagnetic field discontinuities.
The results of Sect. 2.1 were published in parts [21] (Grinchik, N.N et al., 2009).
(a)Ex amplitude
(b) Ex isolines
(c)Ey amplitude
(d) Ey isolines
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Fig. 9.Ex amplitude and isolines, Ey amplitude and electromagnetic field strength isolines

2.2 Wave Equation for H and Boundary Conditions at Strong Electromagnetic Field
Discontinuities. Numerical Modeling of Electrodynamic Processes in the Surface Layer
2.2.1 Introduction
When an external magnetic field and magnetic abrasive particles are interacting, the particles undergo magnetization, and
magnetic dipoles with the moment mainly aligned along the field are formed. "Chains" along the force lines of the magnetic field
[22–23]are formedand periodically act on a processed surface with the frequency ω=l/v. A fixed area of material is periodically
acted upon by the magnetic field of one direction. Actually, the pulse frequency and its duration will be still higher because of the
rotation of a magnetic abrasive particle due to the presence of the moment of forces at contact and of the friction of the particle
against a surface part to be processed. In what follows, the particle rotation effect will not be taken into account.
It is assumed that the particle velocity on a polisher is denoted as v. If the particle radius is designated as r, then the angular
frequency is ω=2πv/r. This frequency exactly determines the frequency of variable magnetic field-induced influences since for a
ferromagnetic μ>1. The magnetic permeability, μ, of ferromagnetics, which are usually used in magnetoabrasive polishing, is
measured by thousands of units in weak magnetic fields. However, in polishing, a permanent external magnetic field is strong and
amounts to 105–106 A/m. In this case, the value of μ for iron and nickel compounds and for Heusler alloy decreases substantially.
When a strong magnetic field H0 is applied, a "small" absolute value of μ of an abrasive particle periodically "increases" and
"decreases" the normal component of magnetic induction near the processed surface. In the present work, neodymium magnets
(neodymium–iron–boron) with H0>485,000 A/m were used. The magnetic permeability of a carbonyl iron-based magnetic
abrasive particle was assumed in this case to be equal to μ1=100.
Because of the continuity of the normal component of magnetic induction Bn1=Bn2, where Bn1=μ1μ0H1; Bn2=μ2μ0H2. For example,
in glasses (μ2=1; then at the contact boundary of glass and magnetic abrasive particle an additional variable magnetic field of
strength H1>H0 appears.
In [24–27] (Levin, M. N. et al., 2003; Orlov, A. M. et al., 2001; Makara, V. A. et al., 2001; Rakomsin, 2000), magnetic fieldinduced influences in silicon are considered: the non-monotonic change in crystal lattice parameters in the silicon surface layer,
the defect gettering at the surface, the change in the sorption properties of the silicon surface, and the change in the mobility of
edge dislocations and in the silicon micro hardness.
In [28–30] (Golovin, Yu. I. et al.; Makara ,V. A. et al., 2008; Orlov, A. M. et al., 2003), it is established that the electromagnetic
field influenced domain boundaries, plasticity, strengthening, and reduced metals and alloys.
In view of the foregoing, it is of interest to find a relationship between discrete pulse actions of a magnetic field of one direction
upon the surface layer of processed material withinclusions (domains). According to [22] (Shul’man, Z. P. & Kordonskii ,V. I.,
1982), the domain size is as follows: 0.05 μm in iron, 1.5 μm in barium ferrite; 8 μm in a MnBi compound, and 0.5–1 μm in
acicular gamma ferric oxide. According to [31] (Akulov, N. S., 1961), the domain size may be 10-6 cm3 (obtained by the method
of magnetic metallography).
An abrasive,as a rule, exhibits distinct shape anisotropy, whereas the frequency of magnetic field-induced influences is
determined by the concentration of abrasive particles in a hydrophobic solution and by its velocity. The magnetic field strength at
the processed crystal surface is assumed to be H(t)=H1 sin4(ωt) + H0.
It is required that the magnetic field strength in the surface layer, having characteristics λ1, ε1, and μ1 and domains with
electrophysical properties λ2, ε2, and μ2, be found. Domains may be shaped as a triangular prism, a bar, a cylinder, etc.

2.2.2 Physical and Mathematical Model. Wave Equation for H
The interaction of a non-stationary magnetic field with substance will be considered using a specific technology of magnetoabrasive polishing based on magneto-rheological suspension. A physical and mathematical model for propagation of
electromagnetic waves in a heterogeneous medium will be formulated. Media in contact are considered homogeneous. First, the
operator rot on the left- and right-hand sides of the first equation for total current (see Equation (6)) is chosen and multiplied by
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μ0μ; then the second equation in the system of equations (see Equation (7)) will be differentiated with respect to time. Taking into
consideration the magnetic field solenoidality (see Equation (7)) and the rule of a repeated application of the operator ∇ to the
vector H yields
0 0
 2H
H 1 2
 0
 H
2

t
t
(38)
In the Cartesian coordinates, the equation (see Equation (38)) will have the form
 0
2 H x
H x 1   2 H x  2 H x  2 H x 

 



0
t 2
t
y 2
z 2 
  x 2
(39)
,
 0
2 H z
H z 1   2 H z  2 H z  2 H z 

 



0
t 2
t
y 2
z 2 
  x 2

One fundamental equation for electromagnetic field is represented by the equation divB  0 . Theuseofthe Dirichlet theorem for
approximation of the magnetic fieldstrength at the boundary surface between adjacentmedia similar to that of the electric fieldstrength
does not necessarilyguarantee the solenoidality condition of the magnetic field; furthermore, the magnetic properties of
heterogeneous media have been assumed constant in derivingthe generalizedwave equation.
The experience of numerical calculations shows that, when unsteady magnetic phenomena
 are
 to be modeled, in many cases it is



better to use the generalized wave equation for E , accordingly,denoting H  t , r  as E  t , r  and, ifnecessary, again tocalculate
 
H  t , r  . This approach is difficult to apply to the modeling of heterogeneous media with different magnetic properties, when the
magnetic permeability µ is coordinate-dependent.
In media with weak heterogeneity, where μ (x, y, z) is a piece-wise continuous quantity, the application of the proposed capturing
method is quite justified. Indeed, the system of equations (see Equations (13)–15, (39)) yields that the function discontinuity at the
boundary surface between adjacent media is determined by the complexes that will be called the generalized permeability
 *   0 0 and the generalized conductivity  *  0 . Using the Direchlet theorem for  * and  * , their values at the
boundary surface between adjacent media and electric field strength values at the discontinuity point (see Equation (37)) are
obtained; here, it should be noted that the electric field strength value is obtained without solving the Maxwell equations. In fact,
at the discontinuity point, the linear interpolation of the function is used to obtain the values of
E x  
 * ,  * , and
1
 E   0   E   0   . Consequently, for the piece-wise continuous quantity μ (x, y, z), the application of the
2
proposed capturing method is justified. It should be noted that the equality of the electric field strength derivatives along the
normal to the surface at the discontinuity point according to the equation (see Equation (34b)) is valid. When the wave equation

for H is used for media with different magnetic permeabilities, the equality condition of the derivatives is not met, i.e.,
H x
x

x  0
H x
x
x   0
This follows from the equation (see Equation (10)); it is therefore difficult to use the capturing method-based schemes for the

wave equation for H .
Moreover, modifying the boundary conditions for equations (1.39) with the electric field strength being omitted from them, does

not allow the surface current direction to be found. Indeed, the magnetic field strength H is an axial vector, which on substitution
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of r⃗ → ( −r⃗) must remain constant. The surface current value cannot also be determined without knowledge of the polar vector of
electric field strength.

The generalized wave equation for E contains the term

grad divE that directly allows for the influence of induced surface
charges on the wave propagation; the right-hand side of the generalized wave equation has the same form as that from theory of
linear elasticity, hydrodynamics. It should be noted that the proposed calculation method can be used on condition that there are
no space charges and external electromotive forces [13] (Grinberg, G.A. & Fok, V.A., 1948).
By virtue of what has been stated above, to model the propagation of electromagnetic waves in glasses having roughness and
defects, the system of equations (see Equations (13)–(15)) with the boundary conditions (see Equations (24)–(34)) is used.
2.2.3 Numerical Modeling Results
The physical and mathematical model developed can also be effectively used in modeling the propagation of electromagnetic
waves in media with complex geometry and strong electromagnetic field discontinuity.
The crosssectional view of the cellular structure represents a set of different- cross section parallelepipeds and triangular prisms,
as depicted in Figs. 10a and 11a. An electromagnetic wave is propagating across the direction of parallelepipeds and triangular
prisms (channels) along the x coordinate.
(a) Hx amplitude
(b) Isolines
Fig. 10.Hx amplitude and magnetic field strength isolines
The size of a two-dimensional object under study is 14×20∙10-6 m and the domain sizes are 2–4 μm. The frequency of the
magnetic field influence is ω= 2π∙106 and the field strength is
H x  21  105 sin 4  2  106 t  A / m
(40)
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(a) Hy amplitude
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(b) Isolines
Fig. 11.Hy amplitude and magnetic field strength isolines
The electrophysical properties are as follows: for a large parallelepiped, μ= 1, ε= 8, σ = 10-9 Ω∙m; for domains, μ= 1, ε= 6, σ= 10-8
Ω∙m. They correspond to the electrophysical properties of glasses.
In a 15–20 μm thick layer an electromagnetic wave is assumed to propagate without attenuation; on all sides of the large
parallelepiped, the condition (see Equation (40)) is then considered valid. On the parallelepiped sides parallel to the OXaxis, the
condition (see Equation (40)) corresponds to the "transverse" tangential component of the wave; on the sides parallel to the OY
axis the condition (see Equation (40)) corresponds to the normal component of the field.
The computations have been carried out with a time step of 10-13 s up to a time moment of 10-10 s.Figures 10a and 11a depict the
magnetic field strengthsHxand Hy, whereas Figures 11b and 11b – their corresponding isolines. The analysis of these figures
shows that at the discontinuity places – on the corners, the force lines of the electromagnetic field concentrate. According to [32]
(Akulov, N. S., 1939), corners are often the sources and sinks of vacancies that determine, for example, the hardness and plasticity
of a solid body.
The propagation of waves in media was also modelled for the case of a domain with magnetic properties. In computations it was
assumed that μ=100; the remaining parameters corresponded to the previous example of solution (Fig. 12).
(a) Hx amplitude
(b) Hy amplitude
Fig. 12.Hx and Hy amplitudes of the magnetic field strength
Of interest is the interaction of an electromagnetic wave with a rough surface shown in Figs. 13 and 14. As in the previous
examples, the electromagnetic energy concentration on structures with different corners is observed.
From Figs. 13 and 14 it is seen that electromagnetic heating of tapered structures may occur in addition to mechanical heating in
magnetic abrasive machining.
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As mentioned above, to investigate the propagation of electromagnetic waves in nonmagnetic substances, it is more expedient to

adopt the generalized equation for E . For the purpose of illustration, the example of numerical computation of the optical knife

involving the application of the wave equation for H (Fig. 15) is presented here.
(a) Hx amplitude
(b) Isolines
Fig. 13.Hxamplitude and magnetic field strength isolines
(a) Hx amplitude
(b) Isolines
Fig. 14.Hy amplitude and magnetic field strength isolines
(a) Hx amplitude
(b) Hy amplitude
Fig. 15.Hxand Hyamplitudes of the magnetic field strength
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From Fig. 15 it is seen that the actual problem of diffraction on the optical knife remains to be solved, i.e., there is no “glow” on
the optical-knife section. This agrees with experimental data.
As known [33](Bazarov, 1991), in thermodynamically equilibrium systems the temperature Tand the electrical φ and chemical μc
potentials are constant along the entire system:
grad T  0, grad   0, grad c  0
(41)
If these conditions are not satisfied (grad T≠0, grad φ≠0, grad μc≠0), irreversible processes of mass, energy, electrical charge
transport, etc. appear in the system.
The chemical potential of the jth component is determined, for example, as the free energy change with varying the number of
mols:
 cj   F / n j T ,V
(42)
dF   SdT  PdV  HdB
(43)
where
The last term in the equation (see Equation (43)) takes into account the free energy change of a dielectric with varying magnetic
induction. In this case, the free energy of a unit volume of the dielectric in the magnetic field is of the form
F T , D   F0  0
H2
2
(44)
Changes in the dielectric temperature and volume are assumed to be small. The mass flux is then determined by the magnitude
proportional to the chemical potential gradient. According to the equation (see Equation (43)), it can be obtained
qi   Dc grad  HdB    Dc grad W
where W
 0
(45)
H2
is the magnetic field density per unit volume of the dielectric.
2
In the case of magnetic abrasive polishing on sharp edges of domains, the magnetic energy gradients are large, and this can cause
vacancy flows to appear.
The analysis of the results obtained shows that the nonstationary component of total electromagnetic energy is also concentrated
in the domain with fractures and corners, i.e., on the sharp edges of domains. As a result, this may improve the structure of the
processed surface sublayer due to the "micromagnetoplastic" effect. A maximum value of the nonstationary part of total
electromagnetic energy Wmax in the sublayer corresponds to that of the function sin (2π∙106t) and occurs for the time moment t =
(n/4)10-6 s, where nis the integer, the value of Wmax for a neodymium magnet and a carboxyl iron-based magnetoabrasive particle
amounting to a value of the order of 106 J/m3 (see Equations (5)–(6)). Having multiplied Wmax by a domain volume, vacancy, or
atom, the appropriate energy value can be obtained approximately. The electromagnetic energy density in all of the cases is much
smaller than the bonding energy of atoms, 10-18–10-19 J. However, a periodic change in the magnetic field of one direction leads to
a ponderomotive force that may influence various defects and dislocations in order to create a stable and equilibrium structure of
atoms and molecules during magnetic abrasive polishing and, in a long run, in obtaining a surface with improved characteristics
due to the "micromagnetoplastic" effect. The results of Sect. 2.2 were published in parts [34].
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3. Ion Conductors. New Closing Relations at the Boundary Surface of
Adjacent Media
3.1 Diffusion Electric Phenomena in Electrolytes
A medium under study is assumed to consist of a non-ionized solvent, an electrolyte in the form of ions, and uncharged
components. First, infinitely diluted electrolytes are considered when molecules are completely dissociated into cations and
anions.
In the solution, ions are transferred by convection, diffusion, and migration when an electric field is applied. The derivation of the
equation for ion migration is based on the following considerations. In the solution with molar concentration n and diffusion
coefficient Di let there be ions with charge zi. When an ion is exposed to an electric field with intensity E applied to the solution,
the ion experiences the force zieE, which brings it into motion. The ion velocity is related to this force by the usual expression
known for the motion of particles in the viscous medium:
u   i* zi eE
(46)
where  i is the mobility of ions. The latter may be expressed in terms of the diffusion coefficient using the known Einstein
*
relation:
 i* 
Di F
RT
(47)
The total flux of ith-kind ions in a moving medium in the presence of diffusion and migration is determined by the Nernst-Planck
equation:
qi  ni v  Di ni 
Di zi FE
ni
RT
(48)
The formulas (see Equations (46)–(47)) have, in fact, a limited sphere of applicability. Indeed, A. Einstein's work [35] (Einstein,
1966) studies only the diffusion of neutral admixture with its small concentration in a solution when the usual relations of
hydrodynamics are valid for a flow around a sphere.
In plasma physics, the formulas (see Equations (46)–(47) are based on other considerations provided that plasma is weakly
ionized, i.e., particles are moving independently of each other. Here, only the collision of charged particles with neutral ones is
taken into account [36](Golant, V. E. et al., 1977).
In [37, 38](Kharkats, 1988; Sokirko, A. & Kharkats, Yu., 1989), for diffusion and migration of ions in a partially dissociated
electrolyte to be described, it is suggested to take into consideration their transfer by neutral molecules. According to the theory
[37] developed by Yu.I. Harkats (Kharkats, 1988), the expression for the total flux of ions takes the form
qi  ni v  Di ni 
Di zi FE
ni  Dzi n A
RT
(49)
The last term in the equation (see Equation (49)) takes into account ion transfer by a flux of neutral molecules; here the
concentrations of anions, cations, and neutral dissociated molecules are determined from the chemical equilibrium conditions

n 
z
n 
z
(50)
nA
where β is the dissociation equilibrium constant. If the dissociation degree at the prescribed total concentration n is α, then nA=(1α)n and, consequently,
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 2n
1
(51)
The dissociation constant β, unlike the dissociation degree, must not be concentration-dependent. However, in real partially
dissociated electrolytes those arguments for α and β are in rather poor agreement with experiment. In [39] (Gibbs, 1982), the HCl
dissociation degree is listed in Table 1 as a function of concentration calculated in terms of measured electric conductivity α1 and
e.m.f. α2.
CHCl, mol/liter
0.003
0.08
0.3
3.0
6.0
16.0
α1
0.986
0.957
0.903
–
–
–
α2
0.99
0.88
0.773
1.402
3.4
13.2
Table 1. HCl electrolyte dissociation degree in terms of measured electric conductivity α1and e.m.f. α2
The inspection of Table 1 reveals that the dissociation degrees obtained by different experimental methods best coincide in the
case of dilute solutions. In the high-concentration electrolyte range, α2 even exceeds unity, and, naturally, this has no physical
sense.
According to the Arrhenius theory, the dissociation constant β for a given electrolyte at prescribed temperature and pressure must
remain constant independently of the solution concentration. In [40](Antropov, L. N., 1984), dissociation constants of some
electrolytes are given at their different concentrations. Only for very weak electrolytes (ammonia and acetic acid solutions)
dissociation remains more or less constant upon dilution. For strong electrolytes (potassium chloride and magnesium sulfate), it
changes several times and in no way may be considered to beconstant.
Formally, the dependence of the dissociation constant on the electrolyte concentration can, of course, be taken into account and
used in the modified Nernst-Planck equation (see Equations (49)–(50)), but in D.I. Mendeleev's opinion, the main drawback in
describing separately diffusion and migration of ions in the electrolyte lies in the fact that the interaction of particles of dissolved
substance with one another, as well as with solvent molecules is neglected. Also, he pointed out that not just the processes of
formation of new compounds with solvent molecules are of importance for solutions. D.I. Mendeleev's viewpoints were extended
by A.I. Sakhanov (Antropov, L. N., 1984)who believed that in addition to the common dissociation reaction in the electrolyte
solution, simple molecules underwent association. In turn, molecular associations dissociate into complex and simple ions. In this
case, the equation (see Equation (49)) will not be valid mainly because of the fact that the expressions for diffusion, migration,
and non-dissociated flows of molecules must be determined with respect to some average liquid velocity. In concentrated
solutions, this velocity does not coincide with the solvent one and must be determined from the fluid dynamics equations for a
multicomponent mixture, whose characteristics of its components (physical density, charge, and diffusion coefficient of a complex
ion) are, in fact, unknown. This is a reason why the theory of diffusion and migration of ions for a partially dissociated electrolyte
solution, with a current flow through it, encounters crucial difficulties. Besides, it is rather difficult to take an account of the force
interaction of complex cations and anions with one another with an external electric field. So Einstein's formula (see Equation
(46)) will also change its form.
The main drawbacks of the theory of electrolyte dissociation are fully defined in the collected papers "Fundamental Principles of
Chemistry" by D.I. Mendeleev, as well as in [40, 41](Antropov, L. N., 1984; Hertz, 1980).
In our opinion, the main drawback of the approaches considered above lies in the fact that the Nernst-Planck equation, as well as
its modified form for a partially dissociated electrolyte (see Equation (49)) are based on the hydrodynamic theory of diffusion.
Below it will be proved that it is more reasonable to use irreversible thermodynamics equations.
While using the Nernst-Planck equation (see Equation (48)) and its modified forms (see Equation (49)), it is postulated in implicit
form that cations and anions in the electrolyte solution are different components. The transport equations for cations and anions
are written separately. On the other hand, according to Gibbs [39](Gibbs, 1982), the commonly used equation of electroneutrality
does not ensure the independence of differentials of cations dni and anions dne in a volumetric electrolyte solution; their changes
are functionally related and are therefore identical, indiscernible components.
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3.1.1 Current Flow Througha Binary Fully Dissociated Electrolyte
Let us write a system of equations (see Equation (48)) in the form
z FD
 ne
 v gradne  De ne  e e div  ne E 

RT
z FD
ni
 v gradni  Di ni  i i div  ni E 

RT
ze ne  zi ni  0
(52)
(53)
(54)
As in [39](Gibbs, 1982), the concentrations ne and nihave been replaced by the molar concentration related to ne and ni by the
expressions
C
ne ni

zi ze
(55)
Expressing ne and ni in the equations (see Equations (52)–(53)) in terms of the molar concentration arrives at the following
expression for the function C:
C
 vC  Def C

(56)
De Di  ze  zi 
ze De  zi Di
(57)
where
Def 
is the diffusion coefficient of salt or the effective diffusion coefficient of binary electrolyte. Expressing the concentrations ne and
ni in terms of the molar concentration C gives an expression for the vector of current density J(Levich, 1959) [42]:
F 2 zi ze
J   Di  De  Fzi zeC 
 zi Di  ze De  CE
RT
(58)
In [43](Neumann, 1977), it is shown that by definition,Def is the molecular diffusion coefficient. For instance, diffusion
coefficients of copper sulfate have an intermediate value between those of copper and sulfate ions; if De = 0.173·10-9 m2/s and Di
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= 1.065·10-9 m2/s, then Def = 0.854·10-9 m2/s. However, the expression (see Equation (57)) does not offer an explanation of the
abnormally high mobility of H and OH ions when they are travelling by a croquet-like mechanism (Skorcheletti, 1969) [44].
The system of the equations (see Equations (56)–(58)) has no symmetry of cross terms; it is therefore difficult to allow for
"superposition". Indeed, the equation (see Equation (58)) evidences that the molar concentration gradients VC exert an influence
on the current J. On the other hand, from the equation (see Equation (56)) it can be seen that an electric field has no influence on
the diffusion of molecules, and as a result, Onsager’s reciprocal relation is violated.
It can be argued that according to the Nernst-Planck equations, it is evident that the concentration gradients of cations and anions
influence a potential distribution of an electric field (Levich, 1959) [42]:
   R    F  z  Di Ci 
(59)
i
However, the relation (see Equation (59)) was derived by Neumann [43](Neumann, 1977) on the assumption that resultant current
is independent of mass flow.
While deriving the equations for diffusion and migration of ions in a volumetric electrolyte solution the relations of irreversible
thermodynamics will be employed.
Finally, it is worthy of note that in plasma physics ,"superposition effects" have no symmetry in describing the ambipolar
diffusion of ions, and the theory is based on the hydrodynamic model [36](Golant, V. E. et al., 1977).
In investigating the plasma dynamics of an electron-hole semiconductor the hydrodynamic model (see Equations (52)–(54)) is
also used.
In our opinion, the electric field-induced charge transport causes a transfer of their kinetic energy, as well as of heat and mass. On
the contrary, at the same time mass or heat transfer may cause charge transport, if a system of charged particles is concerned, and
may give rise to an electromotive force.
3.2 Landau Model
In what follows, consideration will be made of ion conductors containing electrolytic plasma with a fairly high concentration of
charged particles. Theoretically, the interaction of plasma charged particles with the external electromagnetic field and their
collective interaction can be fully explored only using the Boltzmann kinetic equation involving a self-consistent field. In this
probabilistic approach to the phenomena under study, characteristics averaged over a large ensemble of particles are introduced
and are always in direct relation to both the additional hypotheses on the properties of particles and their interactions and with the
simplification of these properties and interactions. It should be noted that in many cases, e.g., when electrolytic plasma in liquid
electrolytes is considered, there is no basis for developing such methods. The methods developed are usually not an efficient
means to solve problems using an excessive complexity of the corresponding equations.
Another general method is concerned with constructing a phenomenological macroscopic theory based on the general regularities
and hypotheses obtained in experiment; this accurate method will be developed in the present work. For many applied problems, it
is sufficient to consider plasma as a conducting gas. Such an approximation is strictly justified only in the case of dense plasma
when a free path of charged particles is much smaller than a characteristic dimension of a system, and particle collisions are of
paramount importance. A velocity distribution of particles is a Maxwellian distribution function; at each point, it is fully
determined by local density, temperature, and macroscopic velocity. With these conditions satisfied, a macroscopic
phenomenological description of a gas medium, metal and dielectric plasma, liquid electrolyte plasma, or electrolyte plasma can
be made as used here.
A necessary condition for existence of the local thermodynamic equilibrium plasmastate is a high frequency of collisions of
plasma particles (Maxwelian distribution) so that the plasma state does not change markedly over the period between collisions on
the mean free path. This condition means the following: 1) electrons have time to transfer electric field energy to heavy particles;
2) ionization processes are almost completely counterbalanced by recombination (the Saha equation holds true); 3) a great number
of excited atoms release their energy at collisions; 4) particle energy exchange dominates over the processes in which plasma
energy markedly increases or decreases.
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For the local thermodynamic equilibrium plasma state, this condition means a radiation field (photons) that is out of equilibrium
with plasma, but energy loss by radiation is minor in a number of cases as compared to other mechanisms of energy transfer. With
the above assumptions taken into account, plasma electrodynamics can be described by the Maxwell continuity, motion, and
energy equations [45] (Landau, L.D. & Lifshits, E.M., 1982). For transport processes in ion conductors to be described on a
macroscopic level, irreversible thermodynamics equations will be used, not separating distinctly ion carriers.
The possibility of such a description was first suggested by L.D. Landau. This approach as justified above does not allow
molecular diffusion coefficients of cations and anions and their dissociation degree in an ion conductor to be introduced and
determined. It is due to the interaction of mass and charge fluxes. Ion fluxes are not determined. The conduction current Jq, the
mass flux Jm, and the heat flux JT for a non-equilibrium medium are of the form (Landau, L.D. & Lifshits, E.M., 1982) [45].
J q    E   gradn    T  gradT
J m   Dm gradn  DA*  E  DT gradT
(60)
J T   kgradT  I q        D gradn
*
T
*
*
whereDm is the molecular diffusion coefficient, D A is the ambipolar diffusion coefficient, DT is the coefficient allowing for heat
transfer due tothe admixture motion, β is the specific ambipolar electrical conductivity due to n , and П is Peltier’s coefficient.
3.3 Interaction of Unsteady Electric, Thermal, and Diffusion Fields with Regard to
Mass Transfer in a Layered Medium by the Example of an Electrochemical Cell
3.3.1 Introduction
To investigate the interaction of electric and thermal fields with regard to mass transfer and contact phenomena is a complex and
topical problem of theory and practice of various fields of natural science and technology.
The present work is aimed at constructing a physical and mathematical model for the unsteady electric field interaction in a
layered medium, when unsteady thermal phenomena and mass transfer are taken into account without a distinct separation of
charge carriers. The media in contact are considered to be homogeneous. For the sake of a clear representation, two-layer onedimensional models will be analyzed.
3.3.2 Interaction of Electric and Thermal Fields
In various substances, the processes of charge and energy transfer are interrelated. The amount of released heat is determined by
Joule heating and by Thomson and Peltier effects. The problem of interaction of unsteady thermal phenomena is discussed in
[46](Tsurko, 1999) having no regard of the Thomson effect.
According to [45](Landau, L.D. & Lifshits, E.M., 1982), the expressions for conduction current density and energy flux density in
the case of no external magnetic field or in the case of its slight influence are of the form, respectively,
I q    T  gradT  grad  , IT  k  T  gradT  I q  П   
Here a(T)is the specific thermoelectromotive force, П = αTis Peltier’s coefficient, φ is the potential, and k(T) is the thermal
conductivity. It should be noted that the problem is always nonlinear.
Having omitted the magnetic field strength from the system of equations (see Equations (13)–(15)), according to the method [47–
49](Shvab, 1994; Shvab, A. I. & Imo, F., 1994; Grinchik, N. N. et al., 1997), the equation for electric field strength is obtained
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I
1
  2E
 0 q   2 E
2
2
c t

t
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(61)
The heat balance equation has the form [45](Landau, L.D. & Lifshits, E.M., 1982)
cp
T
 div  k T  gradT   I q  E   T   T  T   gradT
t


(62)
For the boundary surface, the following relation [50](Tamm, 1976) is valid:
divi  I qn1  I qn 2  
(63)

t
Here ε is the permittivity, c2=l/(ε0·μ0), μ is the magnetic permeability, ε0 is the electric constant, μ0 is the magnetic constant, cpρ is
the product of specific heat of a medium and its density, and β(T)= ∂α(T)/ ∂T.
In deriving the equation (see Equation (62)), the condition of local electroneutrality of substance has been used. A differential
formulation of the problem and the method of its solution are given in more detail.
Let us investigate the contact of homogeneous media 1 and 2 with different electrophysical properties on the segment [0≤x<l] =
[0≤x<ξ;] ∪[ξ≤x≤l]. The magnitudes ε, λ, μ, E, and φ have first-kind discontinuities at the boundary surface point x= ξ. The case of
plane contact will be considered when the influence of surface currents can be disregarded and the electric double layer thickness
is much smaller than the characteristic dimension of an object. Let us put that E = –grad φ. The equations (see Equations (61)–
(62)) for the one-dimensional problem will take the form
((64)
2
 2E
 
T
 1  E







T
E


0


2
c 2 t 2
t  
x
   x
cp
T
 
T
  k T 
t x 
x
T
T 



 E   E    T   T   T  
     T 

x
x 



((65)
The conditions (see Equations (41) and (63)) will be written, respectively, in the following form
1 0 E1 x  0   2 0 E2 x  0  
((66)
x 
T
T 


 2  E2  2 T 

1  E1  1 T  

x  x  0
x  x  0
t


((67)
x 
Having differentiated the equation (see Equation (66)) with respect to time and having taken into account the equation (see
Equation (67)), at the boundary of the boundary surface the condition of equality of total currents is obtained
((68а)
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 
   E   T  x
 
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E 

0
   0
t  x

Here and below, for the arbitrary function f the notation is used
 f  x 
f
x   0
f
x   0
The equality (see Equation (66)) is a corollary of the relation
lim divD   [50] (Tamm, 1976). Having taken into
x   0
consideration the finiteness of a space electric charge and its continuity, in the one-dimensional case it is obtained
(68b)
E 

0
 0 x 
x 
The relations (see Equation (68)) reflect the laws of conservation and continuity of electric charge [45, 50] (Landau, L.D. &
Lifshits, E.M., 1982; Tamm, 1976). At the contact point, the equality of energy temperatures and fluxes (Landau, L.D. & Lifshits,
E.M., 1982) is seen:
T  x
(69a)
0
T
T




0
 k T  x     T  x  E   T  T   



 x 
(69b)
Thus, when electric and thermal fields are interacting in a layered medium, the equality of charge fluxes (see Equation (68a)), the
equality of charges (see Equation (68b)) (when the conditions of quasi-neutrality of contacting media beyond the electric double
layer are satisfied), the equality of temperatures (see Equation (69a)), and the equality of energy fluxes must be fulfilled at the
boundary surface of adjacent media. The relations for charge and energy fluxes take into account cross thermoelectric phenomena.
The process of the electric field charge is considered over the finite time interval [0≤t≤t0]. The initial conditions have the form
T  x, 0  T0  x  , E  x,0   f1  x  ,
E  x,0 
 f2  x 
t
(70)
For example, the boundary conditions are set for the value x = 0. The value of the totalcurrent density is assumed to be known
 
T
E 

     T  x  E    0 t   j1  t 

 
 x 0
(71)
and heat transfer at the boundary surface is given by the Newton relation

T
T



*
 k T  x      T  x  E   T  T      1  T  T  x 0



 x 0
(72)
where γ1 is the heat transfer coefficient and T* is the ambient temperature.The same relations in the right-hand side are
considered.
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3.3.3 Interaction of Electric and Thermal Fields with Regard to Mass Transfer
Let us consider the model of unsteady heat and mass transfer for electrochemical systems by the example of electrolysis. For this
problem, an electric double layer occurs at a place where an electrolyte contacts metal anode and cathode. Let us set aspace charge
density beyond the electric double layer at the metal–electrolyte contact boundary to be zero at the initial time moment. Hence,
according to [51](Stratton, 1938), further it remains constant while the voltage drop in electrodes and leads is small. The influence
of electrodes on the electrolyte temperature field will be taken into account in terms of heat transfer coefficient.
Two approaches to modeling diffusion and electrical phenomena have currently been developed, each of which has certain
disadvantages and advantages. The first approach [37, 38, 42](Levich, 1959; Kharkats, 1988; Sokirko, A. & Kharkats, Yu.,
1989)involves the consideration of the flows of appropriatekind in a completely or partially dissociated electrolyte. The equations
derived contain many parameters. A reliable procedure of their determination is absent in the majority of cases. Furthermore, the
proposed theory fails to take into account the interaction of cations and anions of dissolved substance with each other and with
solvent molecules. It must also be borne in mind that, apart from the dissociation reaction, simple molecules associate and, in
turn,dissociate into complex and simple ions. It is difficult to determine diffusion coefficients of complex ions; it is also difficult
to allow for the force interaction of complex cations and anions with each other and with the external electric field. The
difficulties in question become more serious when a multicomponent electrolyte is to be described. It should be noted that such
difficulties also arise in modeling, for example, certain problems of plasma physics.
The second approach [52–56](Grinchik, 1993; Grinchik, N. N. et al., 2000; Grinchik, N. N.; Muchinskii, A. N. et al., 1998;
Grinchik, N. N. et al., 1998) fails to introduce and determine molecular diffusion coefficients of cations and anions and the
dissociation degree of the electrolyte. It is based on the interaction of mass and charge fluxes. The ion flows are not determined.
The system of equations for electrodynamic processes in the electrolyte is as follows:
I
  2E
n
T
1
 0 q   2 E,
  div  I m  , c0  0
  div  I T 
2
2
c t

t
t
t
(73)
Here Iq, Im, and IT are the densities of conduction current and of mass and heat fluxes which (medium nonequilibrium state) have
*
the form (see Equation (60))[45] (Landau, L.D. & Lifshits, E.M., 1982), where Dm is the molecular diffusion coefficient, D A is
*
the ambipolar diffusion coefficient, DTis the thermal diffusion coefficient, DT is the coefficient taking into account heat transfer
due to the admixture motion, and nis the admixture concentration.
The initial
E  x,0   E0  x  ,
E  x,0  
 E  x  , T  x,0   T0  x  , n  x,0   n0  x 
t
(75)
and boundary conditions (anode-electrolyte, x= 0) are as follows:
T
E
 n

 E    T 
  0
I q     A*
x
t
 x

 1 T  T *   k
T
n
 I q       DT*
x
x
k a I q  t   DM
(76)
n
T
 DA*  E  DT
x
x
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in modeling the one-dimensional problem over the rangeG= [0≤x≤l] × [0≤t≤T]. Here kais the electrochemical equivalent of
substance deposited at the anode. The same conditions hold true for x = l as well.
The constructed system of the differential equations allowstransportprocesses in the electrochemical system to be modeled with
regard to the electric double layer influence; this influence directly determines thermoelectrical and ambipolar diffusion and
electrical phenomena at the metal–electrolyte contact boundary.
3.3.4 Numerical Modeling of Electrochemical Systems. Heating of an Electrochemical
Cell at ConstantCurrent and Pulse Electrolysis
In electrochemical systems, electrodes are as a rule metals. Electrical conductivity of metals is by a factor of hundreds higher than
specific electrical conductivity of electrolytes; therefore, the voltage drop in electrodes and leads can be disregarded. In highly
conductive metals, displacement currents and ambipolar diffusion and thermal diffusion coefficients can also be disregarded and
the system of equations (see Equations (73)–(76)) just for the electrolyte can be considered.
Let us consider the heating process of an electrochemical medium when an electric current with density Iq(t) is passing through a
solution of copper sulfate CuSO4·5H2O. Copper (99.78%) serves as an anode. Copper from the solution is deposited at the
cathode. Current efficiencies for copper are assumed to be equal to 100% and the electrochemical equivalent ke–0.6588·10-6 kg/C.
The dependence of specific electrical conductivity λ(n) of the copper-plating electrolyte on the copper sulfate concentration in
water is given in [57] (L.N. Antropov, 1989); the temperature dependence of λand kwill be considered to be insignificant. In the
*
calculations for CuSO4, DM = 5·10-10 m2 /s, D A =10-11 λ,
 A* =10-4λ, ε= 70, and μ= 1 and thermal diffusion effects are ignored.
The distance between the cathode and the anode is L = 0.05 m. Heat capacity, density, and thermalconductivity of the electrolyte
are taken to be 4.2·103 J/(kg·K), 103 kg/m3, and 0.6 V/(m·K), respectively. It should be noted that pulse action conditions have
been considered in (Kostin, N. A. &Labyak, O. V., 1995) as applied to the processes of electrode deposition of alloys. This
investigation is based on a separate description of ion transport and on the use of Kirchhoff’s law for quasi-stationary currents in a
cell. With such an approach, displacement currents are taken into account indirectly only when the concepts of electrode
capacitive currentand electrode polarization capacitance have been introduced. In [58](Kostin, N. A. & Labyak, O. V., 1995), it is
noted that the neglect of capacitive current in pulse electrolysis involves significant errors. The use of the system of equations (see
Equations (73)–(76)) to model the copper plating process permits the problems to be considered not using the concepts of
inductance and capacitance.
The obtained nonlinear system of equations was solved by the finite-difference method as in [54] (Grinchik, N. N. et al., 2000).
The process of copper plating for constant current and pulse current (Fig. 16) was modeled numerically. The constant current
density was equal to 300 A/m2. The maximum pulse current was also equal to 300 A/m2, while the current period was 0.01 s.
Figure 16 gives the modeling results of CuS04concentration and temperature distributions for different electrolysis conditions.
Measurements were carried out within 60 s after current switching on. It is seen from the figure that the concentration gradients
near the cathode surface (x=1) are different and depend on copper plating conditions. In the case of the nonstationary action, they
are substantially smaller than in the case of electrolysis withinthe constant currentregime. As shown in [58](Kostin, N. A. &
Labyak, O. V., 1995), this tendency also holds true when the same total charge is travelling through the electrochemical cell.
The analysis of the temperature field (Fig. 16b) shows that the electrolyte temperatures in the vicinity of the anode and the cathode
differ significantly. There occursnon-symmetric heating of the electrochemical cell attributed to ambipolar diffusion and
ambipolar electrical conduction. The modeling results are in agreement with the data (Dikusar, A. I. et al., 1989).
(a)
Normalized
distribution
CuSO4concentration
(b) Temperature distribution
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Fig. 16.Normalized CuSO4 concentration and temperature distributions in the electrochemical cell during 60s after the current
isswitched on. The solid curve – constant current; the dashed curve – pulse current.T, °C; x, mm
To measure the surface layer temperature of the electrolyte solution near the cathode and to comparecalculation results, an
experimental setup was designed and made. A copper anode and a cathode were placed in an electroplating bath with a sulfuricacid copper-plating electrolyte. A hole was made in the 1mm-thick cathode and a temperature sensor was mounted in it. A thermal
resistor served as this sensor. The current strength in the bath was prescribed by a variable resistor.
Figure 17 plots the modeling results and the experimental data on heating of the cathode region in relation to the current regime. It
is seen that the increase in the (constant and pulse)current density causes a temperature near the cathode surface to grow. In the
anode, no temperature increase is observed either in calculations or in experiment, which agrees with the data [59](Dikusar, A. I.
et al., 1989). The surface temperature obtained in solving the equations (see Equations(73)–(76)) in theconstant current regime is
by 10–15% more than the temperature recorded in experiments.
A more complicated situation occurs when the experimental data and the numerical modeling results are compared in the case of
pulse electrolysis. The experimental results demonstrate that in the pulse current regime the electrolyte temperature in the cathode
region increases more than in the constant current regime. At the same time, numerical calculations of pulse electrolysis show that
the heating of the cathode region here is smaller than in the constant current regime. The difference is, apparently, attributed to
convective transfer of concentration [60] (Bark, F.; Kharkats,Yu. & Vedin, R., 1998), whose influence is substantial for high
current densities at pulse electrolysis.
Fig. 17.Electrolyte heating in the cathode region as a function of polarizing current magnitudeand kind: 1) constant current; 2)
pulse current; 3, 4) appropriate experimental data. ΔT, °C; Iq, A/m2
During a longer period, natural and convective transfer of concentration as well as heat exchange with environment must be taken
into account. Nonetheless, despite the errors in determining the coefficients and the assumptions made in the model, the
calculation results are in qualitative agreement with the experimental data. In certain cases, it is possibleto use the proposed
approach for modeling of unsteady processes.
The results of Sect. 3 were published in parts[52, 53, 55, 56, 61].
4. Interaction of Transient Electric and Thermal Fields with Regard to
Relaxation Processes
Electric and thermal fields created by macroscopic charges and currents in continuous media have been investigated. Of practical
interest is the modeling of local heat releases in media exposed to a high-frequency electromagnetic field. The energy absorption
influence on the electromagnetic wave propagation should be taken into account, since transport processes are interrelated.
In an oscillatory circuit with continuously distributed parameters, energy is dissipating [62] (Kolesnikov, 2001)due to dielectric
loss as the relative permittivity ε(ω)depends on the frequency. In the general case,ε is also complex, and the relationship between
electric displacement and electric field vectors has the form D=ε(ω)E, whereε(ω)=ε'(ω) – iε''(ω); here, ε'andε'' are determined
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experimentally. As of now, in many cases, the problems of dielectric heating of a continuous medium are reduced to considering
an equivalent circuit based on lumped parameters such as capacitance, inductance, loss angle, and relativedielectricloss factor [63,
64] (Skanavi, 1949; Perre P.; Turner I. W., 1996) that are to be found experimentally.
With this approach, severe difficulties arise in determining a temperature field of equivalent circuits. Also, there occur
polarization and an electric double layer with a prescribed electric moment at contact of media with different properties.
Equivalent circuits in laminated media additionally involve empirical lumped parameters such as surface capacitance and surface
resistance [65](Jaeger, 1977). The total current can always be separated into a dissipative, or conduction current which is in phase
with an applied voltage and a displacement current shifted in time with respect to voltage. The exact physical value of these
current components greatly depends on the selection of an equivalent electric circuit. A unique equivalent circuit isa series or
parallel connection of a capacitor, a resistor, and an inductor does not exist; it is determined by a more or less adequate agreement
with experimental data.
In the case of electrolytic capacitors, the role of one plate is played by the electric double layer with specific resistance much
higher than that of metal plates. Therefore, a decrease in the capacitance with frequency is observed for such capacitors even
within the acousticfrequency range [65](Jaeger, 1977). Circuits equivalent to an electrolytic capacitor are very bulky: up to 12 R,
L, and C elements can be counted in them; it is therefore difficult to obtain, e.g., a true value of electrolyte capacitance. In
[65](Jaeger, 1977), the experimental methods for measurement of dielectric properties of electrolyte solutions at different
frequencies are presented and ε' and ε" are determined. The frequency dependence of dispersion and absorption are essentially
different consequences of one phenomenon: “dielectricpolarization inertia” [65](Jaeger, 1977). Actually, the dependence ε(ω) is
attributable to the presence of the electric double layer resistance and to the electrochemical cell in the electrolytic capacitor being
a system with continuously distributed parameters, in which the signal velocity is finite.
As a matter of fact, ε' and ε" are certain integral characteristics of substance at a prescribed constant temperature, which are
determined by the sample geometry and the electric double layer properties. It is common knowledge that in the case of an
arbitrarily time-dependent field, any reliable calculation of absorbed energy in terms of ε(ω) seems to be impossible [45](Landau,
L.D. & Lifshits, E.M., 1982). This can only be done for a specific time dependence of the field E. For a quasi-monochromatic
field, in [45] (Landau, L.D. & Lifshits, E.M., 1982) it is
E t 
1
 E0  t  e  it  E*0  t  eit 
2
(77)
H t  
1
 H 0  t  e  it  H*0  t  eit 
2
(78)
Following from [66] (Barash, Yu. & Ginzburg, V.L., 1976), the values of Е0(t) and Н0(t) must very slowly vary over the period
Т = 2π/ω. For the absorbed energy,uponaveraging of the carrier frequencyω,as in [66] (Barash, Yu.& Ginzburg, V.L., 1976),the
expression is obtained
D  t 
1 d      
 
 E0  t  E*0  t  
E t 
E0  t  E*0  t  
4 d   t
2
t
d       E0  t  *

E*0  t 
i
t

E
E0  t  

0 
4d 
t
 t

((79)
where the derivatives with respect to the frequency are taken at the carrier frequency ω.It should be noted that for an arbitrary
function Е(t), it is difficult to represent it in the form
E  t   a  t  cos   t 
(80)
since the amplitude а(t) and the phase φ (t) cannot be indicated unambiguously. The manner, in which Е(t) is decomposed into
factors andcosφ, is not clear. Even there appear greater difficulties in passing to a complex representation w(t)=u(t)+iv(t)when the
real oscillation Е(t) is supplemented with the imaginary part v(t).
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We have
( ) =
( )
( )
where the amplitude , the phase , and the instantaneous frequency
( )+
( ) =
=
( )
( ) =
( ),
,
( )
( )
=
( )
( ) ( )− ( ) ( )
( )=
̇ are determined by the expressions
=
( )
,
( 81)
.
Consequently, the vibration amplitude, phase, and frequency (81) are as arbitrary as the imaginary part. If the amplitude, phase,
and frequency are to be determined accurately, then the operator making the function ( ) corresponded to each function
( ) should be indicated.
In many cases (in particular, when a quasi-stationary approximation is used), the slowness, smoothness of the functions ( ) ,
( ) , and ( ) =
+ ̇ ( ) are needed. Such smoothness is obtained while solving many problems, particularly in the nonlinear
vibration theory when the averaging method or the associated method is used. However, the amplitude, the phase, and the
frequency determined here have been comprehended only within the framework of the averaging method and have no broader use.
Dwell on this in more detail. Using the representation of vibrations in a phase with coordinates[ , −
̇
] , the amplitude, the phase,
and the frequency are determined from the vibration representation point, i.e., from the system of the equations
( )=
( )
[
+
( )] ,−
̇( )
( )
=
This is equivalent to the general definition (81) at
=
̇( )
[
+
( )].
( 82)
. From definition (82) it follows however, that for any narrow-band
vibration, the functions ( ) and ( ) contain fast frequency components; so they are not slow (compared to
or
).
Consider a simple example. For the vibration with a slow modulation amplitude
( ) = (1 +
It is natural to set = 1 +
of , it is obtained
and
=
)
,
≪ 1,
=
. But definition (81) leads to other expressions: accurate to the terms of the order
( ) = 1+
( )=
1+
2
+
,
2
(1 +
2
).
Thus, in the amplitude and in the phase, there are fast “vibration” terms not inherent in the initial vibration and compensating each
other in the product =
. It is evident that these vibrations disappear during averaging and smooth functions and are
obtained. However vibration corrections should be allowed for even as a first (in ) approximation; they should be found
additionally and re-defined [67] (D.E. Vakman and L.A. Vanshtein, 1977).
The emerging issues have been considered in[67](Vakman, D.E. & Vanshtein, L.A., 1977) in detail. In the indicated work, it is
emphasized that certain methods using a complex representation and claiming more than the average accuracy have become trivial
when the amplitude, phase, and frequency are determined unambiguously.
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The amplitude and frequency modulation of signals is widely used in radio engineering. At present, radioengineering devices are
available, e.g., sweep generators, whose frequency varies by a certain law in an established frequency band. Today, sweep
generators are multipurpose devices; they have found wide usein automatic meters of SHF-channel parameters (damping,
impedance). Wide-band sweep generators as self-maintained devices are employed most frequently in the range 100 – 200 mHz;
here sweep generators must have various modulation forms depending on the field of application.
Automatic meters of SHF channel parameters,as a rule, must have meander modulation with a frequency of 100 kHz, rarely, of
1.10 kHz.
A monochromatic wave is infinite in space and time; it is therefore carrying no information and transfers no energy. Inreal
situations, waves represented in he form of spectral lines have amplitude and/or frequency modulation.
Spectral lines may broaden due to the interaction between the radiating atom and surrounding particles, i.e., other atoms
(molecules, ions, and electrons). Shock theory (Lorenz, Lenz, and Weiskopf) is based on the assumption of the decisive role of
spectral lines. Broadening is the violation of coherence of vibrations of an atomic oscillator, and its motion as well, i.e., the
Doppler effect; in actual practice, a laser beam then has a broadening of a spectralline and it is not strictly monochromatic. In
developing methods and measurement techniques of relative permittivity, in particular,of complex permittivity and permeability,
usually the resistance and the phase shift between real imaginary parts of resistance are to be determined; current-voltage
characteristics are also studied. However, here, a signal is assumed to be monochromatic; the influence of amplitude and
frequency modulation is disregarded as a rule. The modulation exerts an influence on measurements; therefore, under other
experimental conditions involving another “depth” of amplitude and/or frequency modulation (△ / and △ / ), electrophysical
properties of substance will be different. An electromagnetic sine monochromatic wave is a pure fiction, since energy is not
transferred; modulated signal characteristics should therefore be allowed for during measurements.
It should be emphasized that, generally speaking, equation (81) does not obey the wave equation, i.e., the coordinate dependence
is disregarded. As for ( , ̅ ) , in passing to a complex variable, for the function to be analytical, differentiable, and integrable,the
Cauchy–Riemann conditions must be satisfied. (Reference: if the function ( ) = ( , ) + ( , ) is differentiable at the point
=
+
, then the partial derivatives of the functions ( , ) and ( , ) with respect to the variables and exist at the
point ( , ); here, the Cauchy–Riemann relations are valid
(
,
)
=
(
,
)
(
,
,
)
(
= −
,
)
.
In the case of a modulated vibration of the form
( , ) = 2
(
−
)
△
2
−
△
2
,
where△ ≪ and △ ≪ , the passage to a complex variable is incorrect, since no Cauchy–Riemann conditions are satisfied
for the function ( , ) ; ( , ) is not a differentiable function and cannot be used in the Maxwell equations. In this case, a
complex modulation frequency must be introduced; a complex surface current must also be introduced, since all the functions
used in the Maxwell equations must be analytical.
Radioengineering measurements of formal coefficients are unadvisable. Complex permittivity is introduced and used in the
Maxwell equations for a monochromatic field; it is therefore difficult to allow for the “depth” of amplitude and/or frequency pulse
modulation using a complex variable.
Summing up the afore-said, it can be stated that dielectric loss calculation is mainly empirical in character. To construct
equivalent circuit and to allow for the electric double layer influence and for the field frequency dependence of electrophysical
properties are the only true conditions, under which they have been modeled; these are then fundamental difficulties in modeling
the propagation and absorption of electromagnetic energy.
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In our opinion, heat release in media exposed to nonstationary electric fields can be calculated,considering the interaction of
electromagnetic and thermal fields as a system with continuously distributed parameters from the field equation and the energy
equation that allow for the distinctive features of the boundary between adjacent media. When the electric field interacting with
substance is considered,the Maxwell equations (see Equations (6)–(7)) are used. The energy equation will be represented in the
form
Cp
dT
 div  k T  grad T    Q
dt
(83)
where Q is the electromagnetic energy dissipation.
According to (Choo, 1962), the electromagnetic energy converted to heat is determined by the expression
 d D
d  B 
Q   E    H    J q E
dt    
 dt   
(84)
In deriving this formula, the non-relativistic approximation of Minkowski’s theory is adopted. If ε, μ, and ρ = const, then heat
does not release; intrinsic dielectric loss is then linked with introducingε'(ω) and ε"(ω). The quantity Q is affected by a change in
the massdensity ρ(T).
A characteristic feature of high frequencies is a charge time lag of a polarization field in the electric field. Therefore, the
electricpolarization vector is determined by solving the equation P(t+τe)=(ε-1)ε0Е(t) with consideration of the electric relaxation
time of dipoles τe. Not going beyond the first term of the Taylor series expansion P(t+τe), from this equation it is obtained
P t    e
dP  t 
    1  0E  t 
dt
(85)
The solution (see Equation (85)) on condition that Р=0 at the initial time moment, will take the form
P
  1  0
e
t
 E   e
  t    e
d
(86)
t0
It is noteworthy that the equation (see Equation (85)) is based on the classical Debye model. According to this model,
materialparticles have a constant electric dipole moment. The indicated polarization mechanism involves a partial dipole
alignment along the electric field, which is opposed by no dipole alignment due to thermal collisions. In accordance with the
equation (see Equation (85)), the restoring “force” does not lead to electric polarization oscillations. It acts as if constant electric
dipoles possessed strong damping.
Molecules of many liquids and solids possess the Debye relaxation polarizability. Initially polarized aggregates of Debye
oscillators return to the equilibrium state P(t)=Р(0)ехр(-t/τe).
A dielectric, as a rule, is characterized by a large set of relaxation times with a characteristic distribution function, since the
potential barrier limiting the motion of weakly coupled ions may have different values [63](Skanavi, 1949); the mean relaxation
time of an ensemble of interacting dipoles should then be denoted asτe in the equation (see Equation (85)).
To eliminate the influence of initial conditions and transient processes, as usually done, it is set that t0 = -∞, Е(∞)=0, Н(∞)=0. If
the boundary regime acts for a fairly long time, the influence of initial data becomes weaker with time due to friction inherent in
each real physical system. Naturally, the problem with no initial conditions is thus obtained:
  1  0
P
e
t
 E   e
 t    e
d
(87)

Let us consider a harmonic field Е = Е0sinωt; using the equation (see Equation (87)), for the electric induction vector it is obtained
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D   0E  P 
  1  0
e
t
 E   e
  t    e
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d   0E0 sin t 

(88)
E0   1  0
sin t   e cos t    0E0 sin t
1   2 e2
The electric induction vector is essentially a sum of two absolutely different physical magnitudes: field strength and polarization
of unit volume of a medium.
If a change in the mass density is small, then for local instantaneous heat release the formula (see Equation (82)) yields
QE
2
dD E0   1  0

 sin t cos t   2 e sin 2 t 

2 2
dt
1  e
(89)
wherethe mean value of Q over a total period Т is:
2
1 E0   1  0 2
Q
  e   E2 2
2 1   2 e2
(90)
For high frequencies (ω→∞), heat release ceases to be frequency-dependent. This is consistent with the formula (see Equation
(90)) and experiment [63] (Skanavi, 1949).
Formula (90) can also be obtained in a regular manner by introducing a complex variable.The introduction and use of the
relaxation time
enable us to calculate electromagnetic energy dissipation (dielectric loss) for an amplitude and/or
electromagnetic frequency-modulated wave. The method forcalculation of static permittivity and relaxation time from the
impedance characteristics of substances is contained in Appendix 3.
Electrophysical properties,as a rule, are temperature-dependent; therefore, the system of the Maxwell equations becomes
substantially nonlinear and a complex variable cannot be used in practice.
△
The quantity
may vary as high as 10% for high-frequency ceramics, paper, and radioengineering capacitors with temperature
from −20 to + 20 . For ferroelectrics, varies by a factor of 3 to 5 [68] with temperature from −20
side of equation (12) will have the form
( )⃗ +
( )
to + 20
. The left-hand
⃗ ,
As a result, problems become nonlinear. Temperature variations cause an induced surface charge and a surface current to change;
the procedure allowing for the temperature influence on the charge and the surface current is not known.
Using the relaxation equation for the electric field, the magnetic field delay must also be taken into account, when the magnetic
polarization lags behind a change in the external magnetic field strength:
I t    i
dI  t 
 0H  t 
dt
(91)
The formula (see Equation (90)) is well known in the literature; here it has been obtained without introducing complex
parameters. In the case of “strong” heating of substance, whose electrophysical properties are temperature-dependent, the
expression (see Equation (83)) will have a more complex form and the expression for Q can be calculatedonly numerically.
Furthermore, when strong field discontinuities are present, the expression for Q,in principle, cannot be obtained because no
closing relations for induced surface charge and surface current at the boundaries of adjacent media are available; therefore, the
issue of energy relations in macroscopic electrodynamics is difficult, particularly considering absorption.
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Energy relations for a dispersive medium have been considered more than once; nonetheless, in the presence of absorption, the
issue seems to be not clearly understood (or, at least, not sufficiently known), particularly in the determination of the expression
for heat release at the boundaries of adjacent media.
Indeed, it is known from dielectrics thermodynamics that the differential of the free energy F has the form
dF   SdT  pdV  EdD
(92)
If the relative permittivity and the dielectric temperature and volume are constant quantities, then from the equation (see Equation
(92))it can be obtained
F T , D   F0  D 2 2
(93)
whereF0 is the free energy of dielectric in the case of no field.
The change in the internal energy of dielectric during its polarization at constant temperature and volume can be found from the
Gibbs–Helmholtz equation, where the external parameter D is the electric displacement. Disregarding F0, being independent of
strength field, arrives at
U T , D   F T , D   T  dF dT  D
(94)
If the relative permittivity (ε(T)) is temperature-dependent, then it can be obtained
U T , D   0 E 2 2   T  d  dT  D 
(95a)
The expression (see Equation (95)) determines the change in the internal energy of dielectric during its isothermal polarization,
but with regard to energy transfer to a thermostat, if the polarization causes a dielectric temperature to change. The expression (see
Equation (95)) will be justified in more detail. From the works on microwave heating, it is known that thisexpression is contained
in [45]. Intrinsic internal energy of unit volume of dielectric [45] is
U c T , D  


where    0    1  T

2
  2
0 
 E
T
E
.



1




2
2
T 
(95b)
 
 is the effective permittivity.
T 
A characteristic feature of high frequencies is that the polarization field lags in time behind the change in the external field;
therefore, the polarization vector is determined by solving the equation
P  t   e     1  T  d  dT  D   0E  t    *E  t 
(96)
Considering the relaxation time, i.e., not going beyond the first term of the Taylor series expansion P  t   e  , it can be obtained
P  t    e dP  t  dT    1  T  d  dT  D   0E  t    *E  t 
(97)
It may seem thatfor E  t  to be calculated correctly, the system of Maxwell equations (6)–(7) must be solved together with
equation (97). However equation (97) is the simplest, non-strict model of a polarized medium meant for calculation of dielectric
heating and requires the empirical parameter  e –the relaxation time of dipoles that can also depend on temperature. It is easy to
see that equation (97)is not in agreement with the firstMaxwell equation, which causes the emergence of the magnetic field (
rot H  0 ). It is therefore advisable to use equation (97) only for calculation of a heat source. The existence of a heat source
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causes a temperature to change and, hence, electrophysical properties as well; equation (97) must still be solved using Maxwell
equations (6)–(7).
In the existing works on microwave, heating is performedwhen complex parametersare used and the dependence ε"(T) is
disregarded. Work [69] (Antonets, I.V.; Kotov, L.N.; Shavrov, V.G. & Shcheglov, V.I., 2009) considers a one-dimensional wave
incident from a medium with arbitrary complex parameters at one or two boundaries of media, whose parameters are also
arbitrary. The amplitudes of waves reflected from and transmitted by each boundary are found. Refection, transmission, and
absorption coefficients are obtained in terms of the wave amplitudes. The well-known suggestion that, a traditional determination
of reflection, transmission, and absorption coefficients through energies (reflectivity, transmissivity, and absorptivity) in the case
of complex parameters of media comes into conflict with the energy conservation law, has been confirmed and exemplified. The
necessity to allowing for ε"(T) still further complicates the problem of calculation of electromagnetic energy dissipation when
waves are propagating through the boundaries of media with complex parameters.
The proposed calculation method of local heat release is free of the indicated drawbacks, and for the first time it has enabled one
to construct a consistent model for propagation of non-monochromatic waves in a heterogeneous medium with regard to
frequency dispersion, not introducing complex parameters.
In closing, it should be noted that a monochromatic wave is infinite in space and time, in a material medium its energy absorption
is infinitesimal, and it transfers infinitesimal energy. This is the idealization of real processes. However with these stringent
constraints, the problem of propagation of waves through the boundary is open to discussion and is far from being solved even
when complex parameters of a medium are introduced and used. In reality, the boundary between adjacent media is not infinitely
thin and has finite dimensions of electric double layers; then the approaches based on capturingmethod-based schemes for a
hyperbolic equation, when the boundary between adjacent media is not explicitly separated,seem to be promising.
5. Ponderomotive Forces in Heterogeneous Laminated Media with
Absorption
It is common knowledge that the electric field strength E is equal to the ponderomotive mechanical force per unit charge acting
upon a test charge (charged body) at a given field point. The issue of the force acting upon induced surface charges at boundaries
remains to be solved, since closing relations for σ are unavailable and a charge is not determined at the boundary surface itself.
According to [45](Landau, L.D. & Lifshits, E.M., 1982), the general form of the tension tensor of ponderomotive forces for media
with dispersion and absorption due to dipole relaxation cannot be found in a macroscopic manner. As agreed, the solving of this
problem lies in calculating both an induced surface charge at the boundaries and all tensions externally applied to the surface, not
reducing bulk ponderomotive forces to tensions, which is not necessarily possible; this issue will then be considered in more
detail.
5.1 Reduction of Bulk Ponderomotive Forces to Tensions
If f is the bulk density of ponderomotive forces, then all resultant forces applied to bodies within the volume V is
F   fdV
(99)
V
On the other hand, if bulk forces can be reduced to tensions, then a set of tensions externally applied to a closed surface S must be
equal to the same quantity. The quantity Tnis a force externally acting per unit surface alongthe external normal n; the components
of this force along the coordinate axes are denoted as Txn ,Tyn, and Tzn. The all resultant tensions externally applied to the surface
will then be equal to:
F฀
 Tn dS
(100)
S
Using the Ostrogradski–Gauss theorem and equating (see Equations (99)– 100)), it can be obtained
fx 
Txx Txy Txz


x
y
z
(101)
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fy 
fz 
T yx
x

T yy
y

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T yz
z
Tzx Tzy Tzz


x
y
z
For the bulk forces and tensions to be equivalent, not only the resultant forces applied to an arbitrary volume must remain
constant, but the moment of these forces must be invariant as well (Tamm I. , 2003). This circumstance imposes an additional
constraint on tension-tensor components. The moment of bulk forces applied to an arbitrary volume V is:
N    Rf  dV
(102)
whereR is the distance from the point O, for which the moment of forces is determined to the element dV. Consequently, the
component Nx must obeythe relation (see Equation (100)):
N x    yf z  zf y  dV  ฀
  yTzn  zTyn  dS   Tyz  Tzy  dV
V
S
(103)
V
The surface integral on the right-hand side is equal to the moment of the tension forces Tn applied to the surface S of volume V.
The moment of these tension forces will be equal to that of bulk forces only thenwhen the last integral on the right-hand side is
equal to zero. When the volume Vis arbitrary, this will be valid only when the integrand
T yz  Tzy
is equal to zero.
By repeating the same considerations for NyandNz,the following is obtained: Tzx = Txy, Txy = Tyx. Consequently, for bulk forces
and surface ponderomotive forces to be equivalent, the tension tensor must be symmetric. In anisotropic media, the tension
tensor is asymmetric; direct calculation of resultant forces in terms of tensions using formula (98) seems to be expedient.It
should be noted that Maxwell was the first who reduced field ponderomotive forces to tensions. Replacement of ponderomotive
forces by an equivalent system of tensions makes it much easier to determine forces applied to the electric field volume.
The force applied to a charged layer, given an uncompensated surface charge, is of the form
FijS    Eij1  Eij2 
(104)
If the layer is not charged, thenσ=0 and Fij  0 . The resultant force applied to the boundary of a cell is
S
F฀
 FijdS
(105)
The projections of fx, fy, and fz are determined according to the equation (see Equation (101)).In the case of a variable
electromagnetic field, it must be taken into account that fx, fy, and fz are expended in changing the density of the momentum mass
flux and the electromagnetic field. If the forces acting only upon a substance are denotedby fx , fy , fz , then according to (Tamm
I. , 2003), it can be obtained

fx  f x     1  EH x
t

fy  f y    1  EH  y
t
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
fz  f z     1 EHz
t
The expression

t
  1  EH  is known to be called the Abraham force [70](Tamm I. , 2003).
If a medium is heterogeneous and an electromagnetic wave is transmitted during its absorption and heating, then the following
scheme of calculating ponderomotive forces can be proposed:
(1) the domain under study is subdivided into N cells with constant properties;
(2) the method presented in Sect. 1 and Sect. 2 is used to calculate the surface charge σ and the surface ponderomotive force for
each cell.
This approach disregards the influence of striction forces and tensions acting only on the force distribution over the body volume,
but not influencing the value of all resultant forces and on their moment [70](Tamm I. , 2003). Allowance for ponderomotive
forces is important, e.g., for biomechanics problems.
The problem of ponderomotive forces is considered with regard to basic assumptions of classical electrodynamics [66](Barash,
Yu. & Ginzburg, V.L., 1976).
6. Maxwell Equations – Reformulation of Experiment Facts. Generalized
Wave Equation – Reformulation of Maxwell Equations
In the everyday life of researchers, it is agreed that reformulations of already found propositions and ideas can be only trivial in
essence. However the example of a transition to a concentration formulation of the electrodynamics laws in the form of Maxwell
equations argues against this opinion. This is a brilliant achievement serving as a basis for developing contemporary physics.
According to the formulation of Maxwell equations, the analysis of the electromagnetic field nature and the properties of the
system of Maxwell equations made a contributiontothe relativity theory and also to a corresponding fundamental revision of the
previous ideas of inertial reference frames of space and time.
A generalized wave equation is a reformulation of Maxwell equations. In this case, for the problem to be solved, not only the
⃗ = 0must be checked in the absence of
value of the function, but its time derivative must be prescribed. The condition
charge or
must be satisfied.
⃗ =
( ) – with regard to the charge change. At the boundary surfaceof adjacent media, 12 conditions, not six,
The conjugation conditions obtained, e.g., the equality of total currents, the equality of tangential projections of electric field rotor,
etc. hold true over each cross section of a sample under study; it is therefore promising to use capturing method-based schemes,
not separating explicitly the boundary surface of adjacent media. The use of the Dirichlet mean-value theorem for determination
of the electric field strength at a strong function discontinuity is justified.
7. Electromagnetic waves in in moving media
A successful theory of electromagnetic phenomena in moving media can be based only on Einstein’s relativity theory. However
the case of slowly moving media at / ≪ 1, i.e., with neglect of all effects proportional to the square and to high degrees of the
ratio / , will be considered. From the relativity theory, two relations [71]:
=
1
− 1−
1
μ
,
=
1
μ
− 1−
and the expression for conduction current disregarding Hall’s effect
=
1
(106)
μ
+
will be taken.
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The condition of use of relations (106) is concerned not only with a sufficiently slow motion of a medium but also with a demand
for a medium to move progressively and uniformly [50, p. 542]. More precisely, a sufficiently infinitesimal value of time and
coordinate derivatives of a medium velocity is required.
The condition for slowness of the medium motion is observed in most of the phenomena and processes of practical interest. This
is not only the condition for sufficient slowness of the medium motion but also the condition that is constant. It is the condition
of use of the above relations [71]. The effects associated with the dependence ( , ) are disregarded. In particular, Tolman’s
effect manifesting itself in a sudden breakdown of semiconductors cannot be explained on the basis of the theory presented.
Consistently consider slowly moving continuous media (continua) (V ( x, y , z , )  c ) in view of displacement currents
(dielectrics) alone and media, for which displacement and conduction currents must be taken into account at a time.
The influence of the medium motion on the propagation of waves is also seen in a simpler situation, for example, in the acoustics
problem; therefore, first the modeling of aeroacoustics problems and thereafter the distinctive features of the electrodynamics
problem will be considered.
7.1. Modeling the unsteady wave processes in moving media
In problems of acoustics, electrodynamics, and also heat conduction involving intense heat release, a
momentum transfer is usually accompanied by the medium motion; therefore, the corresponding hyperbolic
equations must be solved in relation to a moving medium.
The processes of interest are described by the wave equation of the form
( , , )
=
( , , )
,
( 107)
where the medium motion must be taken into account. As for a moving medium, the modified linear wave
equation, when  = 1/ , = 1 , was examined, with some limitations, by D.I. Blokhintsev [72, 73]. The
equations were derived and found a wider use[74-78]. But unfortunately, they were quite complex for
calculations.
The present work is devoted to the modeling of sound propagation in a moving non-uniform medium by
constructing a difference scheme that accounts for the medium motion.
It is known [71] that downstream and upstream disturbances are propagating with different velocities. This
effect will be allowed for when a difference scheme for wave equation (107) is to be constructed through the
deformation of a difference cell.
Space and time measurements in a moving medium will be analyzed here. Introduce a uniform mesh in the
), where
×
= { = ℎ , = 0, 1, 2, …, , ℎ
= },
=
= , = 0, 1, 2, …, ,
=
medium at rest (
} (Fig. l, a). In the case of the medium motion, each node means that an observer moves with velocity ;
therefore, during the time of signal propagation, within which the observer is continuously recording the
amplitude and frequency of pressure or flow velocity change, the node moves at the same additional
distance (Fig. l8, b). For a motionless observer at node isending or receiving a signal (pressure or velocity
change), signal trajectories downstream and upstream for a cell of length ℎ (ℎ is the cell length in the
Lagrange coordinate system) will be different.
Now two cases will be considered: (a) signal propagation to the right of node iand (b) signal propagation to
the left of node i. The flow velocity within the limits of the difference cell and the wave velocity are
assumed to be constant. The medium velocity V and the sound velocity С as a function of coordinate and
time will be taken into account in the difference scheme.
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xi-1
xi
xi+1
x*i-1
x*i
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x*i+1
-
h0
h
h0
+
h
Fig. 18. Difference cell deformation: a – medium at rest; b – moving medium
1. Let a motionless observer be at node i. Then a downstream signal has total velocity С+V, and for the
signal time ∆ from node i to node i+1, node i+1 is shifted to additional distance ∆ ; therefore (Fig. 18, b)
ℎ = ℎ +
where
∆ = ℎ /( +
∆ ,
ℎ = ℎ +
),
ℎ /( +
).
2. A motionless observer is at node i. Then an upstream signal has total velocity C–V, and for the signal time
∆ from node i to node i–1, the latter approaches the former at the distance ∆ ; therefore (Fig. l8, b)
ℎ = ℎ −
where
∆ =
ℎ
−
ℎ = ℎ −
,
ℎ
,
−
ℎ = ℎ
,ℎ = ℎ
1+
1−
,
,
( 108)
Hereℎ and ℎ are the cell dimensions in the Euler coordinate system. For a motionless observer at node iin
the Euler coordinate system, the initially uniform mesh undergoes deformation.
For the difference scheme, relations (108) are used. In this case, the Euler calculation region undergoes
deformation due to the medium motion.
The size of a cell (body) must not depend on the velocity sign; therefore, in constructing a difference scheme
the condition of cell size independence from the direction and the velocity sign of the moving medium will
be used. This means that a signal sent by an observer for the time j–1 downstream (or upstream) must be
received for the time j+1 at the same node iwhen the signal direction is reversed, i.e., measurements become
time-averaged. In other words, the procedure of time-averaging of measurements reflects the fact that the
same node is an emitter and a receiver of waves at a time.
Constructing the difference scheme, wave equation (107) is assumed to be valid for media at rest and in
motion in the local co-moving Lagrange coordinate system for a given individual difference cell with a
sufficiently small space step ℎ . Having solved equation (108) for ℎ , pass to Euler variables, select a space
step equal, for convenience, to ℎ = / and obtain
ℎ = ℎ/ ( 1 ±
where
/ ),
= 1+
/ ,
ℎ =
1 ℎ
2
= 1− /
+
.
ℎ
= ℎ/ ( 1 −
/
),
( 109)
The initial wave equation is then written not for a segment but for difference spatial mesh nodes
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( , , )
where the
( , , )
=
,
∗
∗
≤ ≤
,
( , ) =
( ),
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( 110)
’s are the deformed nodes built in accordance with relations (109).
The boundary and initial conditions are as follows
( 0, ) =
)=
( ,
( ),
( , )
( ),
∗(
=
0≤
),
≤ .
Denote an approximate value of the function Pat nodes
, by
, . Equation (110) will be approximated by
the scheme with allowance for non-reciprocity relation (108) and the mesh width independence from the
direction of the medium motion (time-averaging of measurements):
,
, ,
−2
,
(
=
)
,
=
,
,
2ℎ
−
where
+
,
,
−
ℎ
,
,
,
,
−
ℎ
,
,
,
=
,
, ,
,
− 1, = 1,
= 2,
,
−
ℎ
,
−
+
,
,
=
,
+ 
, ,
,
,
− 1,
,
,
,
,
, ,
−
ℎ
,
( 111)
;
2
,
,
+
.
2
The boundary conditions are
=
,
,
=
,
,
= 1,
The initial conditions are
(
=
,
∗(
)+
),
(
=
,
= 1,
),
− 1.
( 112)
,
= 0,
− 1.
( 113)
Difference scheme (111) is implemented in the following iteration process
,
, ,
,
−2
=
−
(
)
2ℎ
,
+
,
,
=
−
,
,
,
,
−
ℎ
,
,
ℎ
,
−
,
,
−
ℎ
,
+
,
,
−
ℎ
,
( 114)
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|P*|
a
b
1
3
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1
2
2
1
2
3
0
0
0.2
0.4
0.6
0
0.2
0.4
0.6
t
Fig. 19. Amplitude of acoustic pressure vs. time: = 800 s-1, a –x= 0.33: 1– V = 0; 2 – V =10 m/s; b –x=
0.66: 1– V = 0; 2 – V = 10 m/s; 3 – V = 20 m/s; dashed line – calculated magnitude of the amplitude; solid
line – time-averaged absolute magnitude of the amplitude. | ∗ | , Pa; t, s
,
=
,
=
,
,
= 0, 1, 2, …,
where
,
,
=
,
,
, ,
+
,
,
,
,
, ,
,
;
2
=
,
,
+ 
, ,
,
.
2
Iteration is being performed until the following condition is satisfied:
,
−
,
<
+
,
,
= 0, 1, 2, …
The values of each successive approximation are found using the elimination technique. It is easy to verify
that the elimination algorithm is stable for any values of hand .
Examples of numerical solution of the wave equation for moving media. A flat infinite membrane
vibrating in a gas causes its periodic compression and rarefaction around it; thus it is a source of acoustic
waves. Consider the case when a membrane performs harmonic oscillations of Asin( t) type and a pressure
at distance ℓ is equal to zero. For instance, such a problem can be observed when acoustic waves propagate
along a narrow long tube of constant cross-section, whose width is small as compared with wavelength.
Under these conditions, all quantities (velocity, density, pressure, etc.) are assumed to be constant over each
tube cross section and the wave direction coincides with that of the tube axis. The pressure difference
between the gas at the tube end and the environment is small as compared with pressure difference inside the
tube. With a sufficient degree of accuracy, the gas pressure is then assumed to be equal to zero as a
boundary condition [72].
In the presence of wind V ( x , t) along the x axis, the pressure obeys wave equation (110) with the boundary
conditions:
( 0, ) =
(
),
( 1, ) = 0,
and the initial conditions:
( , 0) = 0,
( , )
= 0,
0≤
≤ 1.
The initial conditions correspond to a medium at rest for the initial time moment. In the considered example,
equation (107) takes a linear form with the coefficients = 1/ , = 1 . In this case, a solution is sought
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0
0.05
0.1
0.15
0.2
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t
= 800 s-1. P , Pa
Fig. 20. Acoustic pressure vs. time at V = 0, x = 0.66,
3
4
2
1
0
0.2
0.4
0.6
0.8
x
Fig. 21. Acoustic pressure vs. coordinate: 1 – V = 10 m/s; 2 – V = 20 m/s at t= 0.4002 s, = 800 s-1; 3 – V =
0, and 4 – V = -3x + 5 at t = 0.3 s, = 900 s-1; curves 3, 4 – absolute magnitude of the amplitude
directly from the approximation relations of (114) type at s = 0. Calculations were carried out for different
space and time steps.
Figure 19 shows the time-dependent amplitudes of the acoustic pressure at different gas velocities. As seen,
the pressure amplitude decreases with increase in gas velocity. In this case, the most pronounced decrease in
pressure amplitude and, hence, in acoustic radiation intensity is observed at points far from the membrane
(Fig. 19, b). Figure 20 shows the pressure as a function of time at the point x= 0.66 at V= 0. When the
velocity is varied, the pressure as a function of time preserves its character. In calculations, the following
values of the parameters: A= 0.5 Pa, ℓ = 20 m, С = 320 m/s, where Сis the sound speed, were used.
Figure 21 demonstrates the acoustic pressure as a function of coordinate. The pressure amplitude and, hence,
the radiation intensity of acoustic waves decrease with increase in gas velocity since the considered medium
does not dissipate and energy dissipates only due to the gas motion. Curves 3 and 4 are obtained for the case
when the gas velocity linearly depends on the coordinate and is directed along the x axis. According to [72],
sound audibility decreases in the presence of wind. Naturally, in the real atmosphere this effect can be
associated with atmosphere turbulence, but it has also been observed in the presence of weak wind (1–2
m/s).
One more problem is considered here, i.e., the occurrence of acoustic vibrations in a resonator. The simplest
examples of such resonators are tubes with one or both open sides, Helmholtz resonators shaped as bottles,
etc. It is easy to make all resonators similar to ‘acoustic’ ones streamlined by an air flow. A tube with one
open side is identical to an absolutely soft cover when the pressure (acoustic not atmospheric) is practically
equal to zero. At the left boundary of the resonator = 0, the condition ( 0, ) = 0is valid; at the right
boundary
( , )
= 0.
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Let the tube have intrinsic noise with a set of harmonics. Then in the presence of wind, whose speed is
directed along the resonator axis, the resonance phenomenon accompanied both by a strong increase in wave
amplitude and by sound enhancement has been observed.
The calculation results for sound attenuation in the presence of wind and sound enhancement in the
resonator tube with air flowing inside it correspond to the experimental data [71].
7.2. Electromagnetic waves in a slowly moving medium. Fresnel’s drag coefficient
Assume that a considered dielectric is uniform (ε and  are constant), has no free electric charges; therefore the conduction current
I q  0 . This section is concerned with monochromatic waves propagating in uniform dielectrics. As already mentioned above, a
monochromatic wave is the idealization of a real process. In real electrodynamics, a wave is always amplitude-modulated in
character. However some of the laws of propagation of plane monochromatic waves in the dielectric will also be of use for
developing numerical methods of solution of electrodynamics equations.
Considera plane light wave with frequency  in a uniform isotropic nonmagnetic ( = 1) dielectric moving with velocity u.
Designate constant amplitudes of wave velocity vectors as Е0, Н0, D0, B0; for example, the electric field strength is then expressed
by the formula
E  E0 ei ( t knR ) ,
(115)
wheren is the unit vector in the wave direction; R is the radius-vector of point P; k is the wave number. Similar expressions will
also be valid for the remaining vectors Н, В, D.
To simplify further calculations, it should be noted that the differentiation of plane wave vectors with respect to z reduces to their
multiplication by –ik. On the other hand, since these vectors do not depend on x and y, their symbol multiplication by differential
operator  reduces to the multiplication by the ordinary vector –ikk. So when applied to these vectors the differential operator
 will be
i



 j  k  ikk
x
y
z
(not to confuse the unit vectors i and k in the coordinate axes with the imaginary unit i and the wave number k).
If the z axis of the coordinate system does not coincide with the wave direction, suffice to replace k by the unit vector n
coinciding with this direction:
  ikk .
(116)
Based on this relation, the Maxwell equations assume the form
div B   H  ik  nH  0,
div D   E  ik nE  0,
which implies that the vectors Е and Н are perpendicular to the wave direction. Thus, plane electromagnetic waves as spherical
waves are in fact transverse waves.
The time differentiation of the vectors Е and Н reduces to their multiplication by i. Bearing this in mind, the second Maxwell
equation can be presented with the use of equation (116) in the following form:
i
H   rot E   E  ik  nE .
c
Finally, one obtains
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 H   nE .
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(117)
From this equation it follows firstly that the vectors Е and Н are perpendicular to each other and secondly that the mutually
perpendicular vectors n, Е and Н form a right-handed system. Further, bearing in mind that n and Е are perpendicular, one
obtains the following relationship between the numerical values of the vectors Е and Н:
 H    nE   E .
Thus, the ratio of the numerical values of the vectors Е and Н is time-independent, i.e., these vectors have the same phases and
change synchronously.
After cancelling by the multiplier e
i ( t  k nR )
the Maxwell equations assume the form:


D0   k nH0  ,
B0  k  nE0  .
c
c
(118)
Having inserted these expressions in equation (106), one obtains:
E0  
1 u
ck
ck
 nH0    1     nE0  ,

   c

ck
ck
1 u
H0  
 nE0    1     nH0  .

   c

(119)
Thus, the induction vectors D andВ in the wave field are perpendicular to the direction of the wave n; generally speaking, the
field vectors Е and Н havenonzero components along n (if only the direction of the dielectric velocity does not coincide with that
of the wave n or with the opposite direction).
Choose the z axis in the wave direction. So equation (115) will assume the form:
E  E0 ei ( t kz )
.
For simplicity, the direction of dielectric velocity u is assumed to coincide with the wave direction or with that opposite to it:
ux  u y  0, uz   u .
In this case,
nH0   iH 0 y  jH 0 x ,
uz
uz
u

u 
 un 
 c  nE0   n  c E0   E0  c    i c E0 x  j c E0 y ,
wherei and j are the unit vectors in the х and у axes. Furthermore, taking into account that the ratio /k is equal to the wave
velocity v
v

,
k
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from (119) after multiplying these equations by /k one obtains:
vE0 x 
c
 1
H 0 y  uz  1   E0 x ,

 
c
 1
vE0 y   H 0 x  uz  1   E0 y , E0 z  0,

 
vH 0 x
vH 0 y
 1
 cE0 y  uz  1   H 0 x ,
 
 1
 cE0 x  u z  1   H 0 y , H 0 z  0.
 
(120)
As already mentioned, in our considered case, when the vectors u and n are perpendicular, not only D and В, but also Е and Н are
perpendicular to n, i.e., the light wave is transverse. Two of equations (120) contain only
only E0 y and
E0 x and H 0 y ; other two equations –
H 0 x . Consider, for example, the equations for E0 x and H 0 y :
c

 1 
 v  uz  1    E0 x  H 0 y  0,

  


 1 
cE0 x  v  u z  1    H 0 y  0.
  

From these equations it follows that if
E0 x and H 0 y are nonzero, then
2
2

 1  c
1
v

u



z


  

or
c
 1
v  uz  1   

 
(the choice of a sign is specified by the fact that at u = 0, it is necessary that v  c  ). Having taken into consideration that
is equal to the refractive index, n, of a medium, a final expression for the light velocity in the moving medium is obtained:
v
1
c 
  1  2  uz .
n  n 

(121)
Having made calculations for the case of an arbitrary angle between the medium velocity u and the wave
direction n, you can be sure that formula (121) also remains valid in the general case if in it by uzis
understood the projection of the medium velocity onto the wave direction.
Formula (121) was for the first time obtained by Fresnel in 1818 on the basis of the concept of motion of
aether, which is untenable on the modern point of view. It is a hypothetic medium where light waves are
propagating. If aether through a moving dielectric remained at rest, then according this concept, the light
velocity v would be equal to the light velocity с/п in a dielectric at rest. On the contrary, if aether was
completely entrained by the moving dielectric, then the resultant light velocity would be equal to the sum of
the light velocity с/п in aether and the aether velocity и:
v
c
u,
n
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ifu is parallel and anti-parallel to the wave direction. Assuming that aether was only partially entrained by
the medium motion, Fresnel obtained formula (121); the multiplier (1 – 1/n2) being a part of this formula is
called Fresnel’s drag coefficient.
In 1895 Lorenz showed that Fresnel’s formula had to be supplemented by some correction allowing for
medium dispersion, i.e., the refractive index as a function of wavelength. In 1851 Fresnel’s formula was
confirmed by Fizeua’s experiment and in 1914 – by Zeeman with extreme precision. The latter was also able
to confirm the validity of Lorenz’s correction.
Consider brieflylight reflection and refractionin a moving dielectric. Let a vacuum plane wave moving in the z axis and
also propagating in this direction be incident on the dielectric moving in the z direction:
E  E0ei ( t kz ) , H  H 0ei ( t kz ) .
Further, let the dielectric surface coincide with the planez = ut.
Designate values related to a wave reflected from a dielectric and to a wave refracted in a dielectric as r and
g, respectively. For example,
Er  Er0ei (r t kr z ) , E g  E0g e
i ( g t  k g z )
, etc.
In the expression for the electric field strength Er of a reflected wave, the superscript has a plus, but not a
minus since the direction of this wave is opposite to the z axis direction.
Consider any of the boundary conditions at the dielectric surface, for example, the continuity condition of
tangential components of the vector Е:
E0 t ei (t kz )  E0r t ei (rt kr z )  E0gt e
i ( g t  k g z )
atz = ut.
For this condition to be satisfied at any value of t, after zhas been replaced byut, the subscripts of all three
terms would be the same:
  ku  r  kr u  g  k g u .
(122)
Express wave vectors through frequency. For incident and vacuum reflected waves, one obtains
k


and kr  r .
c
c
As for the refracted wave in the dielectric,kg in formula (120) is multiplied byu; therefore, accurate to the
quantities of the order of u2/с2this formula can be supplemented bykg for the dielectric at rest:
kg 
g ng

.
v
c
Introducing these values into (122), one obtains
 u
 u
 nu 
  1    r  1     g  1   ,
c
c
c 



2
2
or within the accuracy to u / c ,
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 2u 
r    1  
c 

 u(n  1) 
g    1 

c 

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(a )
(123)
(b )
Thus, when light is refracted and reflected in the moving medium, the light frequency varies. Wherein the
reflected wave frequency r , unlike the refracted one g ,does not depend on the refractive index, п, of the
medium and, in general, on medium properties. So formula (123а) is also applied, for example, to metals.
The expression for r allows the following simple interpretation to be made. The light of any source S
reflecting, for example, from a mirror is represented as coming from the image S' of this light source in the
mirror. If the mirror is perpendicular to the incident light beam and moves in the direction of this beam with
velocity и, then the source image S'in the mirror moves in the same direction with doubled velocity2и.
Therefore, if the image S'of the source Sis replaced by a real light source with natural frequency, as our
source S, then due to Doppler’s effect, the frequency ' of light emitted by this moving source in the
reflected wave direction (i.e., in the direction opposite to the source motion) would appear to be equal to
 2u 
 '   1  
c 

This coincides with expression (123а) for r.
In real electrodynamics of non-uniform slow media, when an electromagnetic wave is amplitude-modulated and charges and
currents are present at the boundary of adjacent media аs well as velocity v ( x , y , z , ) depends on the time and coordinates, the
modeling of Doppler’s effect in this case is considerably complicated.
In a further, the wave velocity will be determined by Fresnel’s formula. In high-frequency electrodynamics, the equation of
‘aeroacoustics’ derived by D.I. Blokhintsev [71, 72] can be recommended for taking account of the medium motion when the flow
state changes rather slowly as compared to the frequency of the electromagnetic wave propagating through the medium, and
/ ≪ 1 . In this case, the telegraph equation will be of the form:

   2 E
E
 v

 2(v, )
 (v , )(v ,  ) E   , E   
2 
2
t
c  t
 t

(124)
 1 
1

(E  grad div E )   
  rot E
 (n )
  (n ) 
In the presence of the convective motion with constant velocity
one obtains
in the one-dimensional case
   2 E
2E   v2  2E

2
v
 1
c 2  t 2
tx   c 2  x 2
=
,μ =
, =
,
(125)
Note that equation (124) holds true if the medium velocity is time-dependent but is not coordinate-dependent. When the
medium velocity is determined by coordinates, equation (124) is not valid. In this case, for the problem to be solved, one can
recommend a numerical method based on the use of local Riemann invariants in a difference cell similar to Sect. 7.1 together with
Courant’s condition that combines time and space steps:
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 
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h0
 0

2
2
c 1  (v  c ) 1  (v 2  c 2 ) 
(126)
where is the rate of transfer of disturbance in a medium at rest and ℎ and △
step in the Lagrange coordinate system, respectively.
are the length of a difference cell and the time
A length of a difference cell and a time step are chosen so that the medium velocity during the signal time would not substantially
vary. Moreover, the wave amplitude must not change significantly. In this case, a wave can be considered to be monochromatic to
some approximation and Fresnel’s formula can be used for wave velocity calculations.
Actually, equation (125) was solved numerically for acoustics problems in the absence of conduction current. The upwind sound
‘audibility’ decreases, whereas the downwind audibility increases. These well-known experimental facts were confirmed by
numerical calculations. A comparison of the data for media at rest and in motion has shown that only the sound intensity changes,
but no phase shift is observed.
The idea of construction of a difference scheme to Eq. (118) with account of the medium’s motion is based on the works of M.
Laue [79] and A. Einstein [80] in which it is proved that a light pulse propagating in the form of an individual wave packet may be
transformed as the velocity of a material point. By virtue of the singularity of this proposition and because of the fact that it is
little-known, we cite M. Laue ([79]): “Imagine a light pulse propagating in the form of a wave packet and moving with a velocity
relative to a viewer motionless in the medium. With this packet, these moves a certain object which is in the field of a light
beam at all times and hence remains illuminated by this beam at all times. Clearly, for the other viewer moving with a certain
arbitrary velocity relative to the first one this object will also seem illuminated at all times.
For this condition to be observed, it is necessary that the velocity of the packet of light waves, in passing from the frame of
reference of the first viewer (in which he is motionless) to the frame “of the second viewer” be transformed as the velocity of a
(here “time” means the “time of a
material point”. Also, we give A. Einstein’s reasoning [80]: “Let, at an instant of time
quiescent frame” – A. Einstein), a light beam emerge from A, be reflected at B at an instant of time , and return to A at an instant
of time . Taking into account the principle of constancy of the light velocity, we find
−
=
−
,
+
=
−
,
is the length of a moving rod, measured in the “quiescent frame”. Here is the light velocity; the notation is preserved
where
according to [80]. Next: “Let, at an instant of time , a light beam be sent from the origin of the frame along the axis to a
and be reflected from it, at an instant of time , back, to the origin where it arrives at an instant of time ; then these
point
must exist the relation
1
2
or, writing the arguments of the function
have
1
2
( +  ) = 
and applying the principle of constancy of the light velocity in the quiescent frame, we
 ( 0,0,0, ) + 
0,0,0,
+
−
itx is assumed to be infinitesimal, it follows that
1
or
1
2 V−v
+
1
+
∂
V + v ∂t
+
=
= 
, 0,0, +
−
.
∂
1 ∂
+
∂x
V − v ∂t
∂
∂
v
+
= 0.
∂x
V − v ∂t
It should be noted that we could have taken, instead of the origin, any other point as the reference point of the light beam;
therefore, the equation just obtained holds true for any values of x , y and z.
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If it is taken into account then the light along the y and z axes, being viewed from quiescent frame, always propagates with the
velocity √V − v . Consider
∂
= 0,
∂y
Since t is the linear function, it follows from these equations that
= a t−
∂
= 0.
∂z
v
V −v
x ,
where is still the unknown function φ( v) and it is assumed for brevity that in the origin of the frame k , we also have t = 0 at
τ = 0.”
The necessity of giving this reasoning stems from the fact that it is the case of two moving bodies, not of a continuum, that is
proposed in [80]. For a continuum (as has been shown), the introduction of at least one more point and the use of two-layer time
difference schemes for a hyperbolic equation are necessary (see Item 3.1).
The concepts of space and time developed by A. Einstein were generated on the experimental-physical basic. This is their
strength. However the reasoning indicated above [80] is absent, in fact, from recent publications and textbooks on field theory.
Considering that the concept of simultaneity of not-single-site events is relative, A. Einstein introduced the actually novel concept
of time which is used in relativity theory.
In our case we have v/ c ≪ 1, but the principle of discrete consideration of propagation of waves [80] is assumed to be promising
for the electrodynamics of a slowly moving continuum with the results of Item 3.1 for aeroacoustics problems (the wave velocity
is determined from Fresnel formula).
The proposed method of numerical solution of the wave equation in a moving medium may, of course, be generalized to such
phenomena that propagate with a superlight velocity. Geometrically, e.g., this is quite simply done, but such phenomena cannot
serve as signals.
This means that it is impossible to arbitrarily use them to actual, e.g. a diode at a distant site. There can exist optical media in
which the “light velocity” is lager then c. But in this case by the light velocity we mean the propagation of phase in an infinite
periodic wave flow. This velocity can never be used for signaling. Conversely, the wave front, in all circumstances and for any
properties of the optical medium, propagates with a velocity c.
These features of determination of a signal were noted even by A. Sommerfeld [81] (the wave velocity is determined from the
Fresnel formula).
Boundary Conditions.At the inlet and the outlet of a continuous slowly moving medium, the conditions of equality of total
currents discussed earlier must hold. With account taken of the medium’s motion, an expression for expression for the total
current has the form
⃗ȷ
= λE⃗ +
∂D⃗
∂( εε E⃗)
1
= λE⃗ +
+ v⃗ 1 −
∇ εε E⃗
∂t
n
∂t
In a general sense, in the presence of diffusion and heat transfer, we must use a more general expression for the total current,
which is necessary to employ in electrolytes and plasma.
When classical conditions with the introduction of a surface current and a surface charge are used, it is difficult to take account of
the convective transfer of a medium not only at the interfaces of adjacent media but in the continuum as well, since the classical
Maxwell equations are local.
7.3. Electromagnetic phenomena in a slowly moving medium with regard to conduction
and displacement currents
When phenomena in dynamos, electric motors, and unipolar generators with rotating parts are being studied, the question about
properties of both electric and magnetic fields is urgent. In particular, the occurrence of an electric field during the rotation of
permanent magnets is fundamentally important for the theory of unipolar machines that have found industrial use in electrical
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engineering already since the second half of the last century. Since 1832 [84] and up to date [85-89], many works dealing with the
analysis of the operation of unipolar generators have been published. However despite the efforts of many researchers, the
rigorous theory of unipolar machines has not been developed up to now. Owing to a wide use of the following concepts in
electrical engineering such as effective voltage, total voltage, potential difference, [50, p. 545, p. 548], consider in more detail the
conditions, under which the use of the concept of electric potential is well founded.
7.3.1. Electric potential
7.3.1.1. Force work in the electrostatic field
Considering the work of electric field forces, one can talk about a difference of potentials, or electric
voltage. The elementary work performed by force F acting upon a point electric charge in the electrostatic
field of strength Е is:
dA  fdl cos(F, dl )  q ' E cos(E, dl )dl ,
where dl is the elementary motion of charge; Edl is the angle between the directions of the vectorsE and dl.
The total work at a final motion of chargeq' from pointn to pointm of the electric field is
m
a  q '  Edl cos( E, dl ) .
(127)
n
The work of repulsive forces of like charges, when they are far from each other, is positive and, when they
are close to each other, is negative. The work performed during the motion of electric charge q' in the
electric field created by q does not depend on the electrical circuit configuration, but it is determined only by
initial and final charge positions – by the property of electrostatic forces being potential. The work
performed by electric forces during the motion of a single positive charge along closed circuit L is
numerically equal to the circulation of the electric field strength (voltage):
A฀
 Edl
L
and in the electric field
฀ Edl  0 .
(128)
L
or in the differential form
rot E  0 .
(129)
7.3.1.2. Potential and potential field
According to equations (128) and (129), the work of the potential force along the arbitrary closed trajectory
of motion of its point of application is identically equal to zero. For this condition to be satisfied, the
integrand in equation (128) must be a total differential of some scalar function of the coordinates U(x, у, z),
i.e., the force function:
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E x dx  E y dy  E z dz  dU .
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(130)
Hence
Ex 
U
U
U
, Ey 
, Ez 
x
y
z
or
E
U
U
U
i
j
k  grad U .
x
y
z
(131)
Definitions (128) and (131) are considered to be equivalent. However, as shown in [90], if a potential is
chosen in the form U  arccos( kr / r ) , or in the Cartesian coordinates
x
U  arccos
x 2  y 2  z2
,
or in the cylindrical z, ,  coordinates (where   x 2  y 2 ) , U = , then the electric field potential in
equation (131) will not be potential in the sense of equation (128). Hence, for a force field to be defined as a
potential one, formula (128) must be satisfied.
Knowing the electric field strength at each point, the difference of potentials can also be calculated at any
two points of the field. If dSis the element of the charge motion andES is the projection of the field strength
vector onto the directiondS, thenESdS is the work. The difference of potentials at points1 and 2 can then be
expressed by the equation
2
U12   ES dS ,
(132)
1
where integration is made along any circuit L connecting the considered points in the direction from point 1
to point 2.
It should be noted that in an a.c. electric field condition (129) is not fulfilled; therefore formula (131) cannot
be used.
In virtue of the afore-said for dynamos, electric motors, and unipolar generators, the use of the concepts of voltage and potential
difference is of no physical sense as, generally, rot E  0 . The value of the integral (132) substantially depends on the choice of
the integration path as it is possible to talk only about the voltage U12 that exists between points 1 and 2 along this path.
7.3.2. Analysis of models for conductor motion in the magnetic field
In theoretical electrical engineering, in order to model the conductor motion in the magnetic field, usually one uses simplified
models involving lumped parameters: inductance, capacity, difference of potentials. As mentioned above, their use for modeling
purposes is difficult and they will not be considered within the scope of the present work.
In theoretical electrodynamics, an electromagnetic field moving in the magnetic field of a conductor is determined from a system
of the Maxwell equations [45, p. 319-323] and from the use of convective current, as well as of relativistic relations for
polarization and magnetization of moving media.
7.3.2.1. Convective current. Polarization and magnetization of moving media
Electric current to emerge in conductors is the motion of charged particles (electrons and ions). If a microscopic velocity of a
particle with number k is designated as


vk and a charge – as еk, then the density of the current j can be introduced as the sum ei vi
for all particles within small volume V divided by V
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
1
jq 
V

e v
i i

 jv*   e v .
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(133)
i

Here v is the macroscopic velocity of a medium. The vector
*

j is a normal current of a conductor. The vector  ev is a current
due to macroscopic charge transport and is called the density of convective or transported current. In the studies of pulsed
pyrolysis processes [73-76], the macroscopic velocity of an electrolyte solution was assumed to be equal to zero. For the influence
of the macroscopic motion of a medium (electrolyte, conductor, plasma) on transport processes to be allowed for, a time
derivative of current must be calculated



dj djq
dv  d  e
,

 e
v
d
dt
dt
dt
(134)

where dv / dt is the total acceleration of a medium


dv v
 

 ( v )v .
dt t
The charge
e
(135)
is determined from Poisson’s equation

div D   e
(136)
.
From the conservation law of an electric charge one obtains

d e
 div( jпр )  0 .
dt
(137)
In many cases, in continuous media, for example, in plasma the condition of electrical neutrality (with the exception of a double
layer at the interface of adjacent media) is satisfied. So
an excess charge with density
e
 e  0,  e / t  0 . At the same time, it should be borne in mind that if
has emerged within a conducting medium, then this charge when acted upon by an electrostatic
field caused by it will decrease with time according to the law (no medium motion)
Here
e0
The time
 e   e0 exp( t /  m ) .
(138)
 m   0 /  .
(139)
is the charge at t = 0 аnd
m
is called the Maxwell relaxation time. The quantity
m
ranges widely. In dielectrics, it can range within many
minutes, whereas in metals,  m is equal to 10-17 s. Formula (138) is obtained by solving equations (136)–(137) for constant values
of
, .
7.3.2.2. Conductor motion in the magnetic field
As known, Ohm’s law


j   E is valid, generally speaking, only for fixed conductors.
To establish a relationship between a current and a field in a moving conductor, pass from the reference
system K to another reference systemK', in which the conductor (or its separate section) is at a given time
moment. In this system, one has j   E ' whereE' is the electric field strength in K'. But according to the
known formula to transform fields, E' is expressed through the field in the system K as
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E'  E 
1
 vB ,
c
ISSN 2349 – 4425
(140)
wherev is the rate of the system K' with respect to the systemK, i.e., in this case, it is the conductor velocity
(naturally, it is assumed to be small as compared to the light velocity). Thus, one finds
1


j    E   vB   .
c


(141)
It is just the formula establishing the relationship between the current and the field in the moving conductor.
As for the derivation of this formula, the following note should be made. Having passed from one reference
system to another, the field was transformed, but the quantity j remained unchanged. The transformation of
the current density at v ฀ c would yield additional terms of high order infinitesimal. In formula (141) the
second term, which has been yielded by the field transformation, generally speaking, is not small in
comparison with the first, although it also contains the multiplier v/с. So, if the electric field itself is caused
by the electromagnetic induction of a variable magnetic field, then its order of magnitude contains an extra
multiplier 1/s as compared to a magnetic field.
According to [45, p. 321], the energy dissipation in the conductor with a given current flowing in it cannot depend on the
conductor motion. So the Joule heat release density in a moving conductor expressed in terms of the current density is yielded by
the same formula as in a fixed conductor. But instead of the product jE, now one has
1


Q  j  E   vB   .
c


Thus, in the moving conductor the sum
(142)
E   vB  / c serves as an effective electric field strength creating the conduction
current.
Joule-Lenz’s law in the local form
2

1


Q  j 2 /     E   vB  
c


(143)
is satisfied not only for a potential field but also for an eddy electric field [91, p. 59].
The electromagnetic field in the moving conductor is defined by the system of equations [45, p. 322]
1 B
,
c t
4
4
j
rot H 
c
c
div B  0.
rot E  
1


 E   vB   ,
c


(144)
Having expressed Еin terms of Н from the second equation and having substituted it into the first equation, one obtains
B
c2
 rot H 
 rot  vB   
rot 
.
t
4
  
(145)
In a uniform conductor with constant conductivity  and magnetic permeability , one has
H
c2
 rot  vH  
H, div H  0.
t
4
(146)
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In [45, p. 322], the expression rot[ vB] is disclosed and the left-hand side of equation (145) assumes the form [45, p. 323]
B
 ( v)B  ( B) v .
t
But this sum is nothing more than the time derivative of Вresponsible for changes in the В orientation relative to a rotating body.
Indeed, the sum of the first two terms is a substantial time derivative dB / dt yielding В changes at a point moving with velocity
v. The third term allows for changes in the В orientation relative to the body: it is equal to zero during the pure translational
motion of a body (v = const) and is equal to
 B at the body rotation ( v   r  , where  is the angular velocity).
In [45, p. 323], the phenomenon of unipolar induction occurring at the rotation of a magnetized conductor is also considered. It
means that if a fixed conductor is connected to a rotating magnet via two sliding contacts (А and В in Fig. 22), then current will
flow through this conductor. Pass to the coordinate system rotating together with the magnet. If  is the angular rotation velocity
of the magnet, then in a new system the conductor rotates with the angular velocity  (the magnet is fixed).
Fig. 22. To explanation of the phenomenon of unipolar induction occurring at the rotation of a magnetized conductor
Thus, it is now the case of a conductor moving in a given constant magnetic field Вcreated by the fixed magnet; the field
distortion by the conductor itself is neglected. The electromotive force acting between the conductor ends is given by the integral
E
1
1
[ vB ]dl   [B[r ]]dl

c ACB
c ACB
taken along the conductor line. According to [45, p. 323], just this formula solves the task stated.
In [45, p. 324], calculation is also made of the electromotive force of unipolar induction occurring between the pole and the
equator (Fig. 22) of a uniformly magnetized sphere that uniformly rotates about the axis coinciding with the magnetization
direction. Introducing the total magnetic moment of the sphere М = 3В0/4, in [45, p. 324] the expression for the electromotive
force between the pole and the equator is obtained
E
M
,
ac
(147)
where is the rotation frequency of the sphere; а is the sphere radius; В0 is the magnetic induction in the sphere; с is the light
velocity. When deriving formula (147) it was assumed that inside the sphere no currents are present and rot[ vB ]  0 . Formula
(147) and also the hypothesis on the existence of a static electromagnetic field are not confirmed experimentally despite
extensive experiments [91]. As a result, to replace a charge by a rotating magnetic field is impossible. An electric charge is a
cornerstone concept of electrodynamics. At the same time, a charge as such has neither mass, nor energy; in essence, it is
immaterial and is simply a source and a carrier of an electrostatic field. Owing to this, once А. Einstein said that an electron is a
stranger in electrodynamics’ [92, p. 327]. To date, it is of course impossible to refuse from an electric charge.
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Work [50, p. 549] contains the expressions for electric field potentials excited by rotating an iron magnetized sphere of radius а
with frequency  in a conducting medium (for example, in electrolyte). The magnetic induction, В0, of the electric field excited by
constant magnetization of the sphere is constant and parallel to the rotation axis According to [50, p. 549], the potential is equal to
1  

  1R 2
B0 
(3cos2   1)  a 2  R 2 
3c  21  32


1a 5
(3cos2   1)
 2   B0
3c (21  32 ) R3
at R  a ,
(148)
at R  a ,

where1, 2 is the electrical conductivity of iron and electrolyte, respectively; R is the radius-vector of the sphere;

angle between R and the rotation axis. However formula (148) is also not confirmed experimentally.
n
is the polar
As already mentioned above, the introduction of the concepts of potential and potential difference makes sense only for potential
fields. In our case, rot E  0 ; therefore, the calculation of electromotive force, for example, by formula (147) or the introduction
and calculation of
1,2
according to (148) has no physical meaning. Indeed, the electromotive force E (146а) in a closed linear
circuit С is yielded by the integral
E
1   
1 d


E
฀C  c vB   dl   c dt
(149)
that is not equal to zero. In this case, the introduction of the concept of potential is conditional enough.
In a uniform conductor with no regard to displacement current, as shown in [45, p. 322], an electromagnetic field is governed both


by the system of equations (144) or, if E is omitted, by parabolic equation (146) with respect to H . This equation is considered
to be constitutive among the equations of magnetic hydrodynamics. It should be borne in mind that in hyperbolic equation (124)
the displacement current is defined not only by


  2 E
, but also by the convective terms for a conducting medium (124). Exact
c2 t
relativistic relations for D and B are of the form [50, p. 542]:
 v  
  v 
D   H    0  E   B   ,
c 
c 

 v 
  v 
B   E   0  H   D   .
c 
c 


(a )
(150)
( b)

2
2
Solving these equations with respect to D and H and discarding the terms of the order v / c , relations (106) can be obtained.
Relations (150) are obtained, assuming that both time and coordinate derivatives of a medium are infinitesimal enough.
Expression (141) for conduction current in a moving medium is obtained under the assumption of a slight distortion of an
electromagnetic field due to a moving conductor [45, p. 323]. In many cases for a conductor, this assumption is justified as the


v  v  
electric field strength in a conductor is small; hence, the terms
 c E  ,  c D  in relation (150b) can be ignored. When a
permanent magnet rotates with a large angular velocity in a unipolar generator, relativistic relations and also Ohm’s law for a

moving conductor (141) cannot be satisfied because of large velocity gradients with respect to r ( v

 r  ).

Indeed, many of the techniques of experimental measurement of B are based on the expression for Lorentz’s total
electromagnetic force:
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


Fi  qE  q  vB  ,
 0 q  v r 
B
.
4 r 3
ISSN 2349 – 4425
(a )
(151)
( b)
Formula (151,а) uses the properties of a total field initiated by both considered charges and the applied electromagnetic field.
However, the self-electromagnetic field of a charge, as a rule, is considered to be small [46, p. 37].

Formulas (151) are valid when a charge moves with small acceleration so that its velocity would not change for the time v / c
[45, 46, 50]. If displacement and conduction currents in an unsteady non-uniform moving conductor are to be calculated at a time,
then the problem of modeling becomes even more complicated. In our opinion, to solve such a problem, it is advisable to use the

classical aeroacoustics method for solution of the hyperbolic equation with respect to E in the moving reference system K' in
combination with relativistic relations (150) and Ohm’s law for a moving conductor (141).
7.3.3. Method for modeling electromagnetic phenomena in a slowly moving medium
with regard to displacement and conduction currents involving the velocity field
assignment
The calculation method is based on the physical process splitting of a task.
1. To solve such a task, a general system of hyperbolic equations (13)–(15) with regard to the influence of the medium motion

v ฀ c is used to the aerodynamic approximation (Sect. 7.1). Relativistic relations are not allowed for.
2. Having determined the electric field strength Еij at each node i and at the time moment j, the total current is calculated as:
jij   Eij 
Dij
t
3. The magnetic field strength is calculated in terms of the total current from first Maxwell equation (6)
jполн i , j  rot H i , j
4. From the condition of independence of energy dissipation of an electromagnetic field in a medium in motion and at rest [45,
p. 32] for a finite-difference cell, one has (143)
2
1


 Ei2, j    Ei , j 1   vi , j Bijj   .
c


(152)
Solving equation (152) with respect to Ei , j 1 , one defines a new value of the electric field strength with regard to the medium
motion and the magnetic field influence.
At the interface of adjacent media the equality condition of total currents must be satisfied
jполн 1  jполн 2 .
In our opinion, if displacement currents can be neglected, then it is better to use a total system of equations and the proposed
calculation method to solve the task stated, especially in the presence of large velocity gradients of a moving medium.
7.4. Properties of electromagnetic waves
7.4.1. Properties of electromagnetic waves for a medium with ε = const,  = const,  = 0,
  x   ,   t  
For ease of analysis, consider the task in the one-dimensional statement.
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Discard the magnetic field Н from Maxwell equations (6). To do this, multiply the first of the equations by 0 and differentiate
the both its parts with respect to t:
 0 0 
2E
2 H




.
0
t 2
xt
Differentiate the second equation with respect to х:
2E
2H




.
0
x 2
xt
As the right-hand sides of these equations are identical, the left-hand sides are also identical, i.e.,
2E
1 2 E
.

t 2  00  x 2
(153)
The same equation should also be obtained for Н, if the electric field Еwas omitted from (6).
Equation (153) is the wave equation. Hence it follows that the electric Е and magnetic Н fields can propagate in space, i.e.,
electromagnetic waves can exist. It can then assumed
E   (t  x / V ), H   (t  x / V ) ,
wherev is the electromagnetic wave velocity.
The coefficient at  E / x in equation (153) is then the square of the wave velocity:
2
2
V
1
 0 0
1
c

,


(154)
whereс is the light velocity at ε =  = 1, i.e., in vacuum. Thus, the expression is obtained for the velocity of electromagnetic waves
(Maxwell’s law).
Electric and magnetic fields in the electromagnetic wave exhibit a mutual influence. Instantaneous values of Е and Н at any point
are connected by a certain relation that can also be found from the Maxwell equations. To do this, use general expressions (153)
for Е and Н (choosing in them some certain sing, say, a minus) and substitute them into one of the Maxwell equations, for
example, into first equation (6). Since
E
  ',
t
H
1
 '
x
V
(as before the prime means the differentiation with respect to the whole argument), this substitution yields
1
 0 '   ' .
V
Passing from the derivatives to the functions themselves, one arrives at
1
 0    C ,
V
whereС is the integration constant. Since electromagnetic waves are of interest here, i.e., only variable magnetic fields,
Сresponsible for an arbitrary constant field cannot be taken into account. Replacing v by its expression (154), finally one obtains
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 0 E  0  H .
(155)
This formula shows that Е and Н are proportional to each other in the propagating electromagnetic wave.
From (155) it follows that Еand Н reach a maximum at a time and vanish at a time, i.e., they are in the phase.
Consider an arbitrary area S in the electromagnetic wave (Fig. 23) and calculate energy W to be transferred by the
electromagnetic wave through this area for a small time t.
Fig. 23. Calculation of the energy flow of electromagnetic waves
To do this, build over the area S a parallelipiped, whose edges are parallel to the wave velocity v and have length Vt. The volume
of this parallelepiped is
  SV t cos  ,
where is the angle between the normal n to the area S and the velocity v. As the wave covers the distance Vt for the time t, it
is obvious that the energy W contained inside the mentioned parallelepiped will pass through our area. Therefore, if u is the
energy of unit field volume (volumetric energy density), then
W  u   uSV t cos  .
The volumetric energy density of electromagnetic wave is the sum of the electric field energy
energy
 0 E 2 / 2 and the magnetic field
0 H 2 / 2 :
u  ( 0 E 2  0 H 2 ) / 2 .
The electric field strength Е and the magnetic field strength Н in the electromagnetic wave are related by
 0 E  0  H . It
can then be written
u   0 E 2  0 H 2    0 0 EH .
Taking into account that V  1 /
  0 0 , one has
W  EHS cos   t .
Hence, the energy through the area S per unit time or, or W / t is
W / t  EHS cos  .
The results obtained can be presented in more convenient form. Introduce the electromagnetic energy flux vector defined by the
following formula:
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P   EH .
ISSN 2349 – 4425
(156)
Formula (156) for energy flux density remains valid for any variables of the electromagnetic field, including in the presence of
dispersion [45, p. 397].
In the electromagnetic wave, Е and Н are perpendicular to each other; therefore the electromagnetic energy flux Р = ЕН. The
direction of the vector Р is perpendicular to Е and Н, i.e., it coincides with the direction of the wave velocity v. Then formula
(156) can be given in the form:
W / t  Pn S .
Here
(157)
Pn  P cos  is the projection of the vector Р on the direction of the normal n to the area S.
Thus, the energy motion in the electromagnetic field can be characterized by means of the energy flux Р. Its direction yields the
energy direction. The module of the energy flux per unit time through the surface with an area equal to unity is perpendicular to
the energy direction.
The concept of energy flux vector was introduced by N.A. Umov in his works dealing with the energy motion in different media,
and expression (156) for a special case of the electromagnetic field was obtained by J. Poynting. The electromagnetic energy flux
vector Рis therefore called the Umov–Poynting vector or the Poynting vector.
If the tangents to the lines at each point coincide with the direction of the vector Р (energy flux vector lines), then these lines will
indicate the paths, along which the electromagnetic field energy propagates. On the other hand, the lines, along which the light
energy moves, are called beams in optics. As the light is also electromagnetic waves, these light beams are nothing more than the
lines of the energy flux vector of light electromagnetic waves.
The derivation of expression (157) is not rigorous since the phase wave velocity v is assumed everywhere to coincide with the
energy motion. Generally, it is certainly not the case. Nevertheless expression (157) obtained using non-rigorous considerations
often appears to be valid.
Example 1 A propagating electromagnetic wave. Let a plane electromagnetic wave propagate in vacuum along the x axis. The
field strengths Е and Н at some point х are expressed by the formulas
E  E0 sin(t  kx ), H  H 0 sin(t  kx) ,
wherek = 2/. The instantaneous value of the Umov–Poynting vector is therefore equal to
P  E0 H 0 sin 2 (t  kx ) .
However in practice, one deals not with an instantaneous value of the energy flow but with its time-averaged value P . As
sin 2   1 / 2 and, moreover, for vacuum, (ε =  = 1)  0 E0  0 H 0 , one has
P   0 / 0 E02 / 2 .
It is the average energy passing through the unit surface per unit time, or the wave intensity. The result obtained shows that the
energy to be transferred by the electromagnetic wave is proportional to the square of the fluctuation amplitude.
Example 2 A standing electromagnetic wave. Now calculate the Umov–Poynting vector for a standing wave. According to the
afore-said, the fluctuation of electric and magnetic field strengths Е and Н in a standing wave can be represented by the formulas
E  2 E0 cos(kx   E / 2) sin(t   E / 2),
H  2 H 0 cos( kx   H / 2)sin(t   H / 2).
In these expressions, Е and Н denote phase delays of the reflected wave of electric and magnetic fields:
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 E  2  2l /    ,  H  2  2l /    .
Here  and  are the phase changes under reflection equal either to , or to zero, and l is the line length (in the case of free waves,
l is the distance between an emitter and a reflecting surface). Introduce the notations
2 E0 cos(kx   E / 2)  E1 ,
2 H 0 cos(kx   H / 2)  H 1 ,
Fluctuations at any given point can then be written in shorter form:
E  E1 sin(t   E / 2),
H  H1 sin(t   H / 2),
whereE1 and H1 are time-independent. It is seen that if  = , then  = 0 and vice versa. Assuming, for example,  = , one has
E  E1 cos(t  2 l /  ), H  H1 sin(t  2 l /  )
For the electromagnetic energy flux one obtains
P  E1H1 sin(t  2 l /  ) cos(t  2 l /  )  ( E1H1 / 2)sin(2t  4 l /  ) .
In this case, the value of Р fluctuates with frequency 2 and periodically changes the sign. Therefore the time average is
Р = 0.
Hence, in the standing wave there is no energy flux (this is the explanation of the name of this-type fluctuations). Periodic changes
in the Р sign show that the direction of the energy motion changes periodically. Energy fluctuates only between the crests of
electric and those of magnetic fields. This process resembles energy fluctuations between the inductance and the capacity in a
closed oscillatory circuit.
Example 3 A conductor with direct current (DC). Consider a cylindrical conductor of radius r, where there is DC with density j
(Fig. 24).
Fig. 24. Energy motion in the case of the conductor with direct current
The electric field strength Е and the magnetic field strength Н near the conductor surface are directed, as shown in Fig. 24;
therefore the Umov–Poynting vector is directed into the conductor perpendicular to its lateral surface. This shows that energy
continuously flows into the conductor from the environment.
Calculate this energy. If  is the specific resistance of conductor material, then according to Ohm’s law one has
Ej.
The magnetic field strength near the conductor is
H  i / (2 r )  jr / 2 .
Hence
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P  EH   rj 2 / 2 .
Energy flowing through the entire lateral surface of a conductor section of length l during one second is equal to
W / t  P  2 rl   j 2 r 2l .
But according to Joule–Lenz’s law (in differential form)
 j2
is the amount of heat releasing per unit volume for unit time and
 r 2l is the conductor volume. As a result, it is found that energy flowing into the conductor is equal the amount of Joule–Lenz’s
heat, as it should be according to the energy conservation law. This example illustrates that electromagnetic energy, due to which
heat is released, flows into the conductor through its lateral surface, but not along its axis, as it seems at first glance.
7.4.2. Properties of an electromagnetic wave for an unsteady non-isothermal
conducting medium ( T ( , R),  (T ),  (T ),   const )
In this case, the expression for total current with the equation (97) is of the form:


 D

 
   
jполн   (T ) E 
  (T ) E   0    (T )  T
E .
t
t  
T  
As previously, consider the task in the one-dimensional statement. For ease, it is assumed that  (T )  0 , then equation (153) will
have the following form
0
2  
  
1  2 Ex



(
)
T
T
E

 x
 2t  
T   0 x 2
(158)
From equation (158), it is seen that in the general case, expression (154) for the electromagnetic wave velocity is not satisfied as
ε(Т()); therefore, one can talk only about a local wave velocity in the difference cell. Moreover, relations (155) between Е and Н
are violated since the self-energy of unit volume of dielectric depends on the temperature that is the time function.
Work [93] considers the relations ε(T) and also contains the temperature coefficients of dielectric permeability (ТК) TK 
4
1 d
 dT
. For rutile, for example, ТК is ( 7  10 / 1 C ); the TKsign can be both negative and positive. Due regard of ε(T) will
substantially transform formulas (88) and (90) in the course of calculation of electromagnetic energy dissipation on account of
dielectric heating.
o
The expression for the self-energy of unit volume of dielectric and its polarizability, considering the dependence ε(T), was earlier
obtained in Sect. 4 (formulas (95, b) and (97)). It should be noted that expression (89) for definition of the heating of dielectric
also remains valid with regard to ε(T), since it is introduced from thermodynamics according to relations (92)–(94). At weak
absorption, when  = const, the expression for Q can be obtained using Poynting’s theorem.
Poynting’s theorem Proceedingfrom the concept of energy localization in the field itself and bearing in mind the energy
conservation principle a conclusion should be made: if in some definite domain energy decreases, then this may occur only due to
the energy flux through the boundary of the considered domain (a medium is considered to be immobile).
At the same time, there is a formal analogy with the conservation law of charge. The meaning of the law is that a charge
decrease per unit time within a given volume is equal to the flux of the vector j through the surface bounded by this volume.
If we are talking only about the energy of the electromagnetic field, then its total energy within this volume will change
due the energy flux from this volume and due to energy transfer to charged particles, i.e., due to the electromagnetic field work.
Macroscopically, this statement can be written as

dW
฀
 PdS  Q
dt
(159)
wheredS is the surface element.
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This equation expresses Poynting’s theorem: the decrease in the energy flux per unit time within a given volume is equal to the
energy flux through the surface bounded by this volume plus the work per unit time (i.e. the power Q = QТ + Qе) done the field on
charges and dipoles within a given volume.
The energy change per unit time per unit volume of a body is calculated as

div P . Using the Maxwell equation, this equation
reduces to the form:

B 
W
 D
H
div P   HrotE  ErotH     E

t 
t
 t
(160)
7.4.3. Properties of electromagnetic waves for a medium with ε = const,  = const,  = 0
for   x   , t  0
As known, expression (115) is valid only for a monochromatic wave infinite in space and time, i.e., within the range


  x   ,   t   in the uniform medium. In this case, the vectors E and H are mutual perpendicular. For the
task with the initial conditions, when t  0 , for an infinite medium in the one-dimensional statement, expression (115)
 

E  E0 exp i t  knR  is not already the solution to the Maxwell equations. Formulate the task with the initial conditions


for an infinite medium in the one-dimensional statement in the form of equation (152):
utt  a 2u xx  0 ,
where u  E ,
utt 
(163)
2 E
2 E
u

,
, a  1 /   0 0  .
xx
t 2
x 2
u( x,0)   ( x ),
ut ( x, 0)   ( x ).
(164)
Reduce this equation to its canonic form containing the mixed derivative. The equation of characteristics
dx 2  a 2dt 2  0
is split into two equations:
dx  adt  0, dx  adt  0,
whose integrals are represented by the straight lines
x  at  C1 , x  at  C2 ,
Introducing, as usual, new variables
  x  at,   x  at,
the equation of string vibrations reduces to the form
u  0.
(165)
Find the general integral of the last equation. Obviously, for every solution of equation (165)
u ( , )  f * ( ).
where f *( ) is some continuous function of the variable  alone.
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Integrating this equality with respect to  at defined  yields
u ( , )   f * ( ) d  f1 ( )  f1 ( )  f 2 ( ) ,
(166)
wheref1 and f2 are the functions of the variables  and  alone. Vice versa, whatever the functions f1 and f2 were twice
differentiable, the function u( , ) determined by formula (166) is the solution to equation (165). As every solution of the
equation can be given in the form of equation (166) for an appropriate choice of f1 and f2, formula (166) is the general integral of
this equation. Hence, the function
u( x, t )  f1 ( x  at )  f 2 ( x  at )
(167)
is the general integral of equation (165).
The solution to the considered task is assumed to exist; it is then assigned by formula (167). Define the functions f1 and f2 so as to
obey the initial conditions:
u( x, 0)  f1 ( x)  f 2 ( x)   ( x ),
(168)
ut ( x, 0)  af1' ( x )  f 2' ( x )   ( x ).
(169)
Integrating the second equality arrives at
x
1
f1 ( x )  f 2 ( x )    ( )d   C ,
a x0
wherex0 and Сare the constants. From the equalities
f1 ( x)  f 2 ( x)   ( x ),
x
f1 ( x )  f 2 ( x ) 
1
 ( )d   C
a x0
it is found
x
1
1
C
 ( )d  ,
f1 ( x )   ( x ) 

2
2 a x0
2
x
1
1
C
 ( )d  .
f 2 ( x)   ( x) 

2
2a x0
2







(170)
Thus, the functions f1 and f2 are determined in terms of the assigned functions; at that, equalities (170) must be valid for any value
of the argument. In formula (167) the functions f1 and f2 are determined ambiguously. If the constant С is subtracted from f1 and
added to f2, then u does not vary. In formula (170) the constant С is not determined in terms of and ; however it can be
discarded not varying the value of u. When f1 and f2are added, the summands C/2 and – C/2 cancel each other. Having substituted
the found values of f1 and f2, into (167), one obtains
x  at

 ( x  at )   ( x  at ) 1  x at
u( x , t ) 

   ( )d     ( )d 
2
2a  x0
x0

or
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u( x , t ) 
 ( x  at )   ( x  at ) 1 x at

 ( )d .
2
2a x at
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(171)
Relation (171) called D’Alembert’s formula is obtained, assuming the existence of the solution to the task stated. This formula
proves the uniqueness of the solution. Indeed, if the second solution of task (163) would exist, then it could be represented by
formula (171) and coincided with the first solution.
It is easy to see that formula (171) obeys the equation and the initial conditions, assuming that the function  is twice
differentiable and the function  is once differentiable. Thus, the method as outlined here proves both the uniqueness and
existence of the solution to the task stated.
The relations obtained in Sect. 7.2.3 for refracted light reflection in the moving dielectric for a plane wave incident from vacuum
(115) should also be specified as, when a wave is incident on the interface, it is not any more monochromatic because of the
medium non-uniformity. Moreover, normal electric field components and a surface charge were always assumed to be absent.
This condition is not always satisfied.
7.4.4 Heat transfer by fluctuation electromagnetic field
Heat transfer by radiation is one of the fundamental sub-disciplines of the theory of heat transfer and is of considerable interest for
researchers in physical phenomena, and it plays an important role in modern technology. The fundamentals of the theory of this
phenomenon and their great practical value are explained by the fact that in many engineering devices heat fluxes to be transferred
by radiation should be taken into account. Usually, heat fluxes caused by heat conduction also occur in such devices. In liquids
and gases heat fluxes are possibly transferred by convection. Often radiation makes a substantial, but sometimes decisive
contribution to a total heat flux. This speaks in favor of the study of heat transfer by radiation. A number of monographs are
devoted to this class of problems. However an intensive development of modern engineering and technology puts new demands
on the theory of heat transfer by radiation. Some of these demands do not touch the fundamentals of the theory and are mainly to
improve the accuracy of the existing methods and to develop new calculation methods. Other demands are most fundamental and
touch the fundamentals of the theory of heat transfer by radiation (more precisely, by a fluctuation electromagnetic field).
Thelatteraremostimportantboththeoreticallyandpractically.
The traditional theory of heat transfer by radiation (here it will be called classical) is based on Kirchhoff’s law that establishes a
simple relationship between radiative and absorptive characteristics of bodies. However it is well known that Kirchhoff’s law is
applicable only when the approximation of geometric optics is valid. Sizes of bodies, curvature radii of solid’s surfaces and the
distance between them significantly exceed characteristic wavelengths in absorption spectra of these bodies. In the classical
theory, these conditions are always assumed to be satisfied and, as a rule, are not specifically stipulated. In this case, heat transfer
is calculated through the summation of repeatedly reflected and absorbed waves.
From the above it is clear that the classical theory of heat transfer by radiation becomes inapplicable when the mentioned
conditions for validity of the geometric approximation (in particular when distances between bodies participating in heat transfer
are small). Such conditions can arise, for example, in designing multilayered heat insulators.
It should also be noted that the classical theory does not answer the question of heat transfer between gyrotropic media in
the presence of an applied magnetic field even when the conditions of applicability of the geometric approximation are satisfied.
Meanwhile, this question is rather important in practical terms as there is an opportunity to regulate a heat flux between bodies by
changing a magnetic field, not varying temperature of these bodies and distances between them [96].
7.4.4.1 Some results of the classical theory. Difficulties in solving the thermal radiation
problem
As known, one of the fundamental results established by Kirchhoff through the application of thermodynamics laws to
equilibrium thermal radiation was the proof that the spectral energy density uωof this radiation in vacuum is a universal function of
frequency and temperature. Thestructureofthefunctionuωcanbeonlypartiallyrefinedwithintheframeworkofthermodynamics.
ThiswasdonebyU. Wien. To completely establish the universal function form became a task of a further stage of the development
of the theory of thermal radiation – the stage that is characterized by the application of statistical methods. V.А. Mekhelson was
the first, who emphasized the necessity of using molecular-statistical concepts, to find the form of the universal function, i.e., to
theoretically derive an energy distribution in the equilibrium radiation spectrum and in this way he achieved notable success.
Following this V. Mikhelson’s idea, U. Wien advanced in knowledge and for uω he obtained the formula well describing a true
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energy distribution at high frequencies. A final decision of the issue based on the quantum hypothesis and resulting in the
expression for uωvalid at any frequencies, as known, was given by M. Planck.
According to Planck’s law, the bulk energy density of equilibrium radiation in vacuum over the range of (positive)
frequencies from ω to ω + Δω is
u0    ( , T )
Z
V
where ε (ω, T) is the mean energy of an oscillator at temperature Т
 ( , T ) 


 
,   kT
2 e /  1
(172)
and ΔZ is an asymptotic value of the number of natural oscillations (oscillators) of a field within a sufficiently large volume V
over the wave number range (k, k + Δk)
Z  V
k 2 k
 2

V
2
 2c3
(173)
Thus,
u0 
 3  1
1

 
.
2 3 
 c  2 e /  1
(174)
From formula (174) it is seen that at very low temperatures, when Θ  0, the energy does not decrease infinitely as
according to (172), it is obtained that
 (, P)   / 2 [101].
At   << Θ, ε(ω, Т) = Θ, i.e., ε is equal to mean energy per one freedom degree according to the law of an equal distribution of
classical statistics. Expression (174) in this case becomes Rayleigh–Jeans’law
u0 
 2 k 2

 2c3  2c
(175)
For a further study, of interest is to emphasize validity conditions (173). For simplicity, the volume V is assumed to be a
cube with an edge ℓ. When defining the asymptotic (at ℓ → ∞) value ΔZ two conditions must be satisfied:
1) The volume of the spherical layer Δk in thickness in the first space octant of wave numbers equal to
 2
k k must be
2
larger in comparison with the volume of the grid elementary cell (π/ℓ)3 of proper wave numbers. In other words, ΔZ equal to a
doubled (to account for independent polarizations) ratio of these volumes must be sufficiently large:
 2
k k
3k 2 k
Z  2 2

฀ 1.
( / )3
2
2) The correction for ΔZ, having the order of ℓ2kΔk [98], must be small in comparison with the dominant term, which
arrives at the condition
1
฀ 1.
k
Cancelling the numerical coefficients and expressing the both equalities in terms of the wavelength (k = 2π/λ) yields [97]
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3
    
฀   , ฀ 1

 
(176)
Thus, validity conditions (173) are: first, low monochomaticity of the spectral interval and second, sufficiently large sizes
of the considered volume in comparison with λ. Of course, the representation of the volume of the spherical layer part, which is
within the first space octant of wave numbers in the differential form
 2
k k already assumes that Δk << k. This means that an
2
upper bound is set on nonmonochromaticity [97]

฀ 1

(177)
It is easy to see that the both bounds on the spectral range given by (176) and (177) as a result has second condition (176) – the
condition of applicability of the approximation of geometric optic within the considered volume.
By the definition of the radiation intensity Iω, the quantity Iω cos θdΩdσdω is the energy flux over the interval of positive
frequencies from ω to ω + dω per unit time through an area dσ in the solid angle dΩ = sin θdθdφ, whose axis forms the angle θ
with the normal N to dσ. Hence, the power transferred per unit area, whose normal is N, is equal to
S N 

I cos  d  
  /2
2
 /2
0
0
 d
I

cos  sin  d
(178)
As in the isotropic medium, equilibrium radiation is also isotropic, i.e., Iω does not depend on the direction, in this case
S N   I
(179)
In the transparent medium, the radiation intensity Iω is related to the bulk energy density uω by the relation
u  
I d 
,
w
where w 
u 
(180)

is the envelope velocity at the frequency ω (k is the medium wave number). Thus, for equilibrium radiation
k
4 I 
w
(181)
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In particular, for vacuum (w = с) from (179) and (181) it follows that
I 0 
S 0 N
c

u0 ,

4
(182)
i.e., the field of applicability of Rayleigh–Jeans’ law (175)
I 0
k 2

4 3
(183)
As known, one of Kirchhoff’s laws establishes the following relationship between the equilibrium radiation intensity Iω in
the medium, the emissivity ηω of the medium and its absorptivity αω:
I 


(184)
By the definition of absorptivity, ηωdVdΩ is the power over the frequency range (ω, ω + dω) transferred to the solid angle
dΩ by the medium volume element dV that is partitioned inside the medium itself; αω is simply the energy indicator of wave
attenuation in the considered medium.
A relation between the equilibrium radiation intensity Iω in the transparent isotropic medium and the equilibrium radiation
intensity in vacuum is established by Clausius’ law
I   I 0 n 2
(185)
where n is the refractive index of the medium at the frequency ω. If dispersion is absent, i.e., n does not depend on the frequency
ω, then ω = c/n and from (181), (182) and (185) it follows that
u  u0 n 3
(186)
Relation (185) combines the equilibrium radiation intensities within some volume of the medium and vacuum. For a plane surface
differentiating a uniform medium and vacuum, the radiation intensity of the medium in vacuum is described by Kirchchoff’s law
I  I 0 (1  R )
(187)
where R is the energy reflection coefficient dependent on incident wave direction, frequency and optical constants of the medium.
For a further consideration, it is necessary to clarify quantities Iω and ηω, аs well as the limits of applicability of laws (184)
and (187), from which the first and the last laws explicitly apply to an absorbing medium, whereas the two others are quite
understandable only in the case of full transparency of the medium.
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First of all, it should be noted that law (187) have no bounds on the absorption quantity as it is a direct consequence of the
thermal equilibrium condition. Indeed, at equilibrium the intensity of waves incident from vacuum on the flat boundary of the
medium at the angle θ is equal to I0ω. Reflected waves have the intensity I0ωR, which together with Iω – the intensity of waves
emitted by the medium in the reflection direction – must again yield I0ω. This is expressed by (187).
In the transparent medium, radiation is being created only by external bodies. In the absorbing medium, each volume
element of the medium itself moreover becomes a source of chaotic thermal radiation. In what follows, because of absorption, the
energy flux is caused both by a wave field of an emitter and by a quasi-stationary field that inversely proportional to the second
and third degree of distance. The emissivity ηω characterizes that part of the total energy flux from the medium element dV, which
moving away from dV, coincides with the distance only according to the exponential law (with αω), i.e., that part caused by the
wave field. For that part of the energy flux to be dominant even at distances r from the emitting element dV (approximately,
correlation radius), absorption must be sufficiently small.
Thus, the introduction of ηω and law (184) make sense only under this condition.
Despite the fact that the definition of intensity (178) in form is not associated with the absorption value, the condition for a
small absorption value is also extended to (185). The thing is that for all derivations of (184) the intensity at this or that solid angle
is interpreted as energy transferred by a beam of plane continuous waves, whose normals lie at this solid angle. In the uniform
isotropic medium, the direction of the energy flux vector coincides with the wave normal direction. But this usual meaning of the
intensity concept is lost in the case of absorption as the field cannot be represented by a set of pane continuous waves. Moreover,
as each volume element of the absorbing medium is a source of thermal radiation, the latter cannot be given as a set of plane
continuous waves being particular solutions to the uniform field equations. Such waves can be only “admitted” to the absorbing
medium from the outside or can be properly excited by assigned (i.e., external) regular sources located in the medium itself. That
is why the attempts to extend the above-said to the case of strongly absorbing media by considering reflection and rarefaction of
plane waves at the interface of such media (or absorbing and transparent medium) are doomed to fail. However such attempts
have been.
The first attempt in this direction was made by M. Laue. For a medium with a complex refractive index n(1-iκ) he obtained
the following result:
I   I 0 n 2 (1   2 )
In his article published in [Ann. D. Phys. 7, 63, 1950], Fragstein more carefully considered the ratio of reflection and transmission
coefficients at the interface of absorbing media. Having corrected a mistake made at this point, M. Laue obtained the following
formula:
I   I 0 n 2 / (1   2 )
(188)
However, neither this nor any other formula for the equilibrium radiation intensity inside the absorbing medium is compatible
with Kirchhoff’s law (187) for the radiation intensity of such a medium in vacuum.
Using Frenkel’s formulas for plane waves with two independent polarizations and the ratio of solid angles in medium and
in vacuum, which follows from the refraction law, the following can be emphasized. If the equilibrium radiation intensity inside
an infinite medium is isotropic and is evenly distributed between the both polarizations, then the external radiation intensity of a
finite medium cannot be reduced to (187). The demand to have radiation intensity from the outside (187) appears to be satisfied
only when the anisotropic equilibrium radiation intensity is inside the infinite medium, but it is absurd to a uniform and
isotropic medium. In the direction of the normal to the interface, the mentioned anisotropic radiation intensity exactly takes the
value (188) found by Fragstein who confined himself just to the case of a normal intensity decrease. As a result, he did not notice
that the calculated in this way equilibrium radiation intensity in the infinite medium should be different, but differently dependent
on the solid angle for different polarizations.
7.4.4.2 Electrodynamic prerequisites for definition of radiation properties and
modeling of radiant energy transfer
In [99] it is proved that the classical electromagnetic theory for natural, non-polarized light can be successfully used both for
modeling radiant energy transfer and for defining radiation properties of optically smooth finished surfaces with mirror images.
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For natural, non-polarized light the electric field has no specific orientation relative to the plane of incidence. But in this case, the
field can be represented by parallel and perpendicular components that will be equal [99]. Hence, the directional hemispherical
spectral reflectance of a mirror surface in this case is equal to the arithmetical mean value of the quantities
  ฀ (  ,  ) and
   ( ,  ) :
  ( ,  ) 
  ฀( ,  )    ( ,  ) 1 sin 2 (    )  cos2 (    ) 

1
.
2
2 sin 2 (    )  cos2 (    ) 
(189)
Equation (189) is known as Frenkel’s formula. Here β is the angle of incident wave and χ is the angle of refracted wave [99]. Just
the use of the mentioned assumption and formula (189) allowed radiant energy transfer to be modelled successfully. In [99],
however, it is noted that practically it is impossible to taken into account the influence of surface finishing. Just because of the
differences between real and ideal material, which are considered by theory, the experimental values of quantities substantially
differ from those predicted by theory. These differences are due to such factors as the presence of admixtures, surface roughness
contamination in the case of changing the crystal structure of a surface to be finished. In [99], it is emphasized that because of the
difficulties in defining surface conditions, comparison of theory and experiments is not always satisfactory even in the case of
advanced theories. At the same time, to solve modern engineering tasks dealing with high temperatures, heat insulation properties
of dielectric and metal surfaces should be known. It should be noticed that the spectrometric analysis of substances based on
Buter’s law is applicable only to transparent media [104]. Meanwhile, most of the unknown and known objects of nature are
heterogeneous, nonuniform media: powders, fibers, suspensions, polymer solutions, fabrics, etc. However the above-said applies
to objects with “dense packing” where the concept of reflection of a layer surface makes sense. The situation is different when a
“rarefied” medium is considered, for example, a cloud, a mist. Here it is difficult to talk about a surface [104; 105; 106]. It should
also be noted that in [99], аs well as in Mie’s known work [100] it is not shown how calculation results are influenced by an
induced surface current charge at the surface of spherical cloud particles; in what follows, particles must be separated from each
other (more than by 3 diameters).
The approach free from the above restrictions is described in [96, 97]. However at the interfaces of adjacent media the
known М.А. Leontovich‘s conditions [98] are used, which have been considered many times in the present work. It is impossible
to formulate boundary conditions for a set of faces (on each of the faces), drops, and particles. Technically, this procedure cannot
be simply implemented, as Sommerfeld–Malyutints’ condition obtained only for a monochromatic waves should be additionally
used for the face point. When calculating electromagnetic energy dissipation the terms T


and T
are not allowed for, i.e.,
T
T
in the assumption of weak thermal radiation absorption (see Sect. 7.4.3), when  = const and  = const.

In our opinion, to model radiant energy transfer, generalized wave equation (12) for E and heat conduction equation
involving a heat source (89) must be used. In this case, parallel and perpendicular components of the electromagnetic field must
be determined from equation (189) as thermal radiation has no specific orientation. At strong absorption in some cases, the heat
conduction equation should take into consideration a finite heat transfer rate, i.e., the wave character of heat conduction should be
noted. When the ordinary parabolic heat conduction equation is used, reasonable paradoxes arise and will be considered in more
detail.
7.4.5 Heat conduction equation considering a finite heat propagation velocity
7.4.5.1 Paradoxes of classical heat conduction theory
When establishing the concepts of temperature field, two fundamental physical quantities – temperature T and heat flux vector q
were used. Using the thermodynamic laws, the temperature is determined by the following formula:
dQ  CdT
(190)
This equality qualitatively expresses the known experimental facts when some amount of heat dQ applied to the solid changes its
thermal state. Quantitatively, the latter is expressed in increasing or decreasing a solid temperature by a value dT. To describe the
called physical process, the quantity С in (190) must be an empirical constant different for each substance and called heat capacity
of a solid.
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According to the experimental data, the temperature is represented by a scalar quantity that depends on the space (X, Y, Z)
and time (t) coordinates. If points with the same temperature in space are connected, then an isothermal surface is obtained. The
above plotting in the plane will result in a family of isothermal lines. It is obvious that the temperature does not change along the
isotherm. The largest temperature drop per unit length occurs in the direction of the normal to the isothermal surface.
Quantitatively, it is characterized by a temperature gradient grad T. The temperature gradient is the vector along the normal to the
isothermal surface in the direction of increasing values of temperature, i. e.,
 T
grad T  T  n
(191)
n

Here n is the unit vector along the normal in the direction of increasing values of temperature and ∂T/∂n is the temperature

derivative along the normal n to the isothermal surface.
From the experimental data it follows that heat is transferred from points with less temperature. If heat is assumed to be
transferred along the normal to the isothermal surface, then the presence of the temperature gradient will be a necessary condition
for heat propagation. Formula (190) shows that the temperature change at a given point in space is a consequence of the heat
amount supplied to the indicated point. As a result, along with the vector grad T, the heat flux vector is introduced

 dQ 1
q  n
dt S
(192)
This vector defines the heat amount per unit time and per unit area S of the isothermal surface. The sign “minus” in (192) means


that the vector q is opposite to the temperature gradient. Lines, whose tangents coincide with the direction of the vector q , are
called heat flux lines. Heat flux lines are perpendicular to isothermal surfaces at the points of their intersection

The relation between vectors q and grad T was established by Fourier and has the following form:

q   grad T .
(193)
Here λ is the empirical coefficient called thermal conductivity and is a thermophysical characteristic of material or substance.
To derive the heat conduction equation, it is necessary to use the relation defining a heat balance in the vicinity of a given
point of a temperature field [99]:
cV 
T

  div q
t
(194)
where cV is the specific heat capacity at constant volume V; ρ is the density of material.
The left-hand side of equation (194) defines the heat amount supplied along heat flux lines in the vicinity of a given
temperature field point and its right-hand side according to (190) yields a temperature change due to supplied heat per unit time at
a given field place.
Substitution of (193) into (195) arrives at the classical heat conduction equation
cV 
T
  T
t
(195)
Here T isLaplace’soperator.
This parabolic-type equation and its fundamental solution are well studied. To analyze the specific features of this solution,
re-write (195) for the one-dimensional case:
T
 2T
k 2
t
x
(196)
The quantity k = λ/ cV ρ is called thermal diffusivity. A fundamental solution of (196) is of the form
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T ( x, t ) 
 x2 
1
exp  
.
4 kt
 4kt 
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(197)
Find a mean-square value of temperature shift starting with its initial value x0 during the time t:
 ( x  x ) T ( x, t )dx  2kt .

 T ( x, t )dx
2
x
2
0
Define a temperature shift rate
v
d
k
 x2 
.
dt
2t
(198)
Hence it follows that lim v→∞, if t→0, i.e., at the initial time moment temperature the temperature nonuniformity spreads
instantaneously. The same paradox is seen in the theory of Brownian motion [2].
Using the above fundamental solution, the equation for a maximum surface temperature can be found. To do this,
differentiate expression (197) in time and make the result obtained to be equal to zero. As a result, x  2kt  0 . Hence
2
x  2kt and
vmax 
dx
k

dt
2t
(199)
i.e., the formula for definition of a displacement speed of a maximum temperature surface coincides with (198) and vmax has the
specific feature mentioned above.
The essence of the second paradox of the heat conduction theory lies in the following. Consider a semi-infinite solid at a
constant initial temperature T0 Let a wall temperature x = 0 be instantaneously increased to Tω Here the initial boundary
conditions will be of the form:
T = T0 at t = 0, x > 0
T = Tω at x = 0, t > 0
T  T0atx  , t > 0
Solving equation (196), the known error function is obtainable
T ( x, t )  T0
2
 x 
 erfc 


T  T0

 2 kt 


exp    d .
x
2 kt
Using it, calculate the wall heat flux q( x, t )   
 x2 
T  (T  T0 )
exp  

.
x
 kt
 4kt 
At x = 0 and within the limit, when t → 0, the heat flux becomes infinite. Hence, classical, simple heat conduction equation (195),
derived using Fourier’s law, does not describe the real physics of the process according to [107-112]. It should be noted that to
realize the 1st kind boundary conditions, i.e., to assign T(τ) at the wall, is also impossible technically; i.e., the process with an
infinitely large heat transfer coefficient cannot be implemented. Finally, equation (195) appears to correspond to an infinite
propagation velocity of thermal disturbance, when a temperature change at any one point of a solid instantaneously alters it at all
other points. Some works [107-114] are concerned with the problem of obtaining hyperbolic heat conduction equations that yield
a finite heat propagation velocity and, hence, their solutions.
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7.4.5.2 Derivation of the hyperbolic equation considering heat flux inertia
Having taken into account heat flux inertia defined in terms of the relaxation time (relaxation coefficient) τr, Maxwell–Cattaneo’s
formula [112] was obtained
q( x, t )   
T
q
 r
x

(200)
where q is the heat flux; T the temperature;λthe thermal conductivity; τr = a/w2, a the thermal diffusivity, w the displacement
velocity of thermal disturbance.
Based on formula (200) the hyperbolic heat conduction equation is derived
T ( x, t )
 2T ( x , t )
 2T ( x, t )
a
r


 2
x 2
(201)
Analytical and numerical equations for finite and semi-infinite regions under the 1st and 2nd kind boundary conditions are
obtained in [110-111], according to which their analysis allows the following conclusions to be made. A temperature distribution
is characterized by the motion of a thermal wave with a temperature jump at its front in the spatial coordinate. In the solutions for
finite solids, a backward wave appears with a temperature jump at its front. The jump sign is opposite to the one in the forward
wave. The temperature jump of the backward wave causes negative temperatures to appear in the cooling process, i.e., the solid
temperature may seem to be lower than a temperature assigned by the 1st kind boundary condition at the plate surface. According
to [110-111], the above facts evidence that locally, both the first and second laws of thermodynamics have got broken. In [111111], the introduced relaxation corrections for heat flux and temperature gradient have resulted in obtaining hyperbolic heat
conduction equations involving the third and fourth space and time derivatives (mixed derivatives). There it is also stated that this
equation is a result of the generalized system of Onsager’s equations. This issue will then be considered in more detail.
7.4.5.3 Derivation of the hyperbolic equation on the basis of the generalized system of
Onsager’s equations
The derivation is based on the equation for the substance flow J (heat flux) defined by x (x is the thermodynamic force), by the
temperature gradient ∂T/∂x obtained by A.V. Luikov [107] on the basis of the generalized system of Onsager’s equations
J  Lr
J
X
 LX  L
,


(202)
where Lr, L, L’ are the phenomenological transfer coefficients.
If the time derivative of the moving force x ( X /   0 ) is neglected, then according to [111], Cattaneo’s known
equation (201) [112] can be obtained. However if the quantity X /  is not ignored, then the formula for heat flux will be of
the form
T
 2T
q
q  
  r
 r
x
x

(203)
where L   r .
To derive the hyperbolic equation with the use of formula (203), use the heat balance equation applied to the onedimensional temperature field from [111]
c
T
q

(204)

x
where c is the heat capacity and ρ is the density.
Substitution of (203) into (204) yields
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T
 2T
 3T
 2q
  2   r 2   r

x
x 
x
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(205)
Re-write (205) as follows:
c
T
 2T
 3T

  2   r 2   r

x
x 

 q 
 
 x 
(206)
Equation (206) with regard to (204) will assume the form [111]
  2T
T
 2T
 3T 
 r 2  a  2  r 2 
c


x  
 x
(207)
In [111], it is stated that hyperbolic equation (207) is derived considering all terms of the equation that has been obtained
from A.V. Luikov’s proposed generalized system of Onsager’s differential equations applied to the one-dimensional heat transfer
process.
Also, the hyperbolic equation with higher-order derivatives is derived
  2T
T
 2T
 3T
 3T
 4T 
c
  1 2   2 3  a  2  1 2   2 2 2 



x 
x  
 x
(208)
Ibid, for equation (208) the analytical solution is found for a finite rod thermally insulated from its side surface under the
constant nonzero initial condition. One end of this rod is thermally insulated and the constant 1st kind boundary condition is
assigned on the other. The mentioned initial and boundary conditions are described by the formulas:
T ( x,0)  T0
(209)
T ( x ,0)
0

(210)
 2T ( x, 0)
0
 2
(211)
T (0, )
0
x
T ( , )  Tst
(212)
where Т0 is the initial temperature, Тst is the rod temperature at х = δ, δ is the rod length.
The studies of different analytical solutions of hyperbolic heat conduction equations obtained allow conclusions to be made
according to [111] that the solutions do not promote temperature jumps inside a solid as well as negative temperatures. Moreover,
the analysis of the results [111] makes a conclusion that heat transfer coefficient cannot exceed some value maximum for given
specific conditions (defined in terms of a thickness of a solid and its physical properties) under other heat transfer conditions.
Indeed, as mentioned above, the first-kind heat transfer condition (212) cannot be realized. In our opinion, it is more correct to
assign a generalized heat flux on the rod end [109]. Consider the problem statement for a semi-infinite rod thermally insulated
from its side surface. A time-varying heat flux enters the rod end. For equation (201) [109], we have:
T ( x,0)  T0 ,
T ( , )  Tst
T ( x ,0)
0

(213)
(214)
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T
q
(0, )   r
(0, )  q(0, ),   0
x

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(215)
Other boundary condition (215) is given in the integral form

q( )    
0
  t
T
1
(0, t ) exp  
 dt
x
r
 r 
(216)
It should be noted that the solution of (201), (213)–(215) obtained in [109] does not contradict the thermodynamics and physics of
the process without introducing mixed derivatives, i.e. not using equation (208). It should be emphasized that in the particular
case, q(0, ) can be defined by Newton’s law, i.e., heat exchange with environment occurs under the 3rd kind boundary
conditions [109]

  t 
  T
(0, t ) exp  
 dt   (T (0, )  Te )

 r 0 x
 r 
(217)
where Тe is the environmental temperature.
This problem also has the analytical solution [101] that agrees with experimental data and does not contradict the
irreversible thermodynamics of processes. Mathematical physics textbooks, as a rule, consider hyperbolic equations only for the
1st kind boundary conditions [113]. In this case, the conditions for consistency of initial conditions and boundary conditions [114]
must be satisfied. Consistency conditions (boundary-value conditions must obey the hyperbolic equation), as a rule, are not
satisfied [110-111]. Not obeying the consistency equations [114], it is impossible to obtain the only classical solution of the
problem. In [115-116], it is shown that the account for temperature time derivatives in the gradient does not consider
thermodynamic constraints [115]. They cannot be used in the form given in [111].
7.4.5.4 Theory of heat waves in the medium with heat sources
Write heat balance equations (83) in the general form for the one-dimensional case:
CP (T )  (T )
T
q
   Q (T )   (T ) E 2

x
(218)
where Q, ρ, λ, CP are temperature-dependent, and the heat flux q takes into account relaxation (equation (200)).
To obtain the heat conduction equation, formula (218) is to be re-written in the following form;
T
 
T 
2q
CP (T )  (T )
   (T )
 Q (T )   (T ) E 2
 r
 x 
x 
x
(219)
Differentiateequation (218) in time:

2q
 
T
  CP (T )(T )
x  

 Q T  2 T
E



 T  T
(220)
The last relation allows equation (219) to be re-written as follows:
T  
Q
  T

rE2

 C P (T )  (T )
   CP (T )  (T )   r

  

T  

 
T 
2
    (T )
  Q (T )   (T ) E
x 
x 
r


(221)
Here λ and λ* are the thermal conductivities.
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Large electric field strength values are described by the dependence dλ/dT. Joule heat will also influence the effective heat
capacity of material.
8. Some discussion and conclusions
Abouttotal current in the 2nd kind conductor and its extension to electrodiffusion of electrons and holes in the conductor.
In [116, 117], Nernst–Planck relations (68) for electrodiffusion of currents are extended to electrodiffusion of electrons and holes
and it is emphasized that the current density is defined not only by the effect of electrons and holes in the electric field, but also by
their diffusion. It should be noted that transport processes in the semiconductor can occur at high frequency, for example, in
Gunn’s diode. It is therefore necessary to generalize the expression for total current, considering the unsteadiness of concentration
and temperature gradients.
As known, total current can be larger and smaller than conduction current and, in the particular case, it vanishes, for example, in a
spherical condenser filled with conducting medium.
An electric bias D at a distance r from the condenser center is equal to D 
q
, the current displacement density
4 r 2
dq
 1  dq
jC  
, and the current displacement strength iC  4 r 2 jC  . When a condenser is discharged, this current is
2 
dt
 4 r  dt
dq
directed from the inner lining to the outer one, conduction current – oppositely (from plus to minus) and its strength i  
.
dt
So, the total current iполн  i  iC  0 and, despite the motion of charges between linings, the magnetic field is equal to zero. The
electric field strength in this case decreases exponentially and can be calculated from the equation
 0
E
 E  0 ,
t
(221)
 t 
E  E0 exp    ,
  
 p
 p   0  is the Maxwell relaxation time.
Let electrolyte or any 2nd kind conductor be a conducting medium. Equation (1.27) must then be of the form:
jпр  jcм   (E   n )   0
E
 0,
t
(222)
where jпр   (E   n ) .
The solution of the equation at t = t0, E = E0 is:
 t  t0   t
 t 
E  E0 exp  
  n( ) exp  

   
 
p 
p t0
p



d .

(223)
At sufficiently large times, the first term disappears; inside the condenser the electric field is set, which is defined by an entire
history of the change n(), which contradicts the physics of the process. Indeed, at t   in 2nd kind conductors E  0 or is
calculated from the conditions when the conduction current is equal to zero in 2nd kind conductors
 ( Е   n )  0 .
(224)
In virtue of the afore-said, the definition of displacement current needs some other interpretation.
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If a medium is uniform and isotropic, and serves as a 1st kind conductor, then it is convenient to represent total current in a
somewhat modified form:
jполн   E   0
jпр
E
   ( E) 
 E  0 
.
  jпр   p
  t 
t
t
(225)
Displacement current can be written as:
jc   p
jпр
t
.
(226)
In this case, if total current is equal to zero, then conduction current decreases exponentially:
 t 
jпр  jпр0 exp   
  
 p
(227)
andnocontradictionsarise. For electrolyte, electrolyte plasma, i. e., for 2nd kind conductors
jпр    Е   n  .
(228)
For convenience of further manipulations, it is assumed that T  0 . The case of T  0 will be considered separately.
Hence, define total current:
jпр    Е   n    p

  (E  n )  .
t
(229)
At present, for 2nd kind conductors,  is found using indirect methods for electromagnetic wave reflection
and propagation; direct methods for  definition in conductors are unavailable. It is clear that the
displacement current in (229) is also defined by the polarizability of dielectric; it is therefore more
convenient to transform (229):
jполн   (E   n )   0
  (E   n )   0
E
E  0 
 (  1) 0

 (n ) 
 t
t
t
E  0 

 (n ).
 t
t
(230)
The last term in (230) is new and is for the first time obtained in the present work.
E
is the most simple, but not the only manner to describe thermodynamics
t

of complex media in the presence of cross effects. While obeying the conditions for quasi-neutrality of a medium, when
 0,
t
  0 , expression (229) can be easily obtained, considering that divjполн  0 ,
The expression for the current displacement as
 0
div ( E  n )  0 .
(231)
If the usual expression for displacement current is used,
div ( 0
E
)0
t
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or
divE  0 .
(232)
However formulas (231) and (232) are contradictory; it is therefore advisable to represent the displacement current at
electrodiffusion as:
jcm   0
E  0 

 (n ) .
t
 t
(233)
Total current is determined not only by a value of transfer potentials Е and n, but also by their time derivatives. The neglect of
d/dt(n) contradicts the immutable electric charge conservation law.
Proceeding from the general concepts of electromagnetic properties of a medium, an expression can be obtained for total current
in the presence of diffusion and heat fluxes. Let a formal consideration be confined only to one physical assumption: if some solid
body is placed into an applied electromagnetic field, then an average field within the body volume is small in comparison with
intramolecular fields. In other words, average fields inside the body are assumed to be weak.
Similarly [47] it will be shown that total current depends on vectors E , B, n, T and their histories
E B 

,
, n, T ,
t t t
t
i.e.,
jполн  f (E, B, n, T ,
E B 

,
, n, T ) .
t t t
t
(234)
As fields are weak, the function jполн can be expanded into a series in powers of variables and be confined to the first power of
expansion. In essence, this expansion is made in terms of the powers of small deviation of the type E/Eвн.ат, where Eвн.ат is the
strength of the intra-atom field.
Expanding jполн into a series in powers of variables, it should be taken into account that jполн is not the polar vector. Therefore, all
terms of the series expressing the desired expansion must also be polar vectors (not scalars, not axial vectors).
It should be noted that the electric field strength Е is the polar vector and n and T are also the polar vectors. Vice versa, the
magnetic field strengths Наnd also В are the axial, pseudo-vectors. This is seen from the formula for Lorentz’s force F = q[VH].
Indeed, the behavior of Н at the vector reflection at the coordinate origin and upon substitution of (–r) for r is defined by the
behavior of polar vectors F and V and the properties of their vector product. When (–r) is substituted for r, the vectors F and V
reverse their directions; the vector product also reverses its sign. Hence, upon substitution of r for (–r) the vector of Н must
remain invariable. This property is considered to be a sign of the axial vector.
The desired expansion can involve vectors E, E/t, n, T, (/t)n, (/t)T . Thus, expanding into Taylor series (234) we
have
jполн   (E   gradn)   (T )gradT   0
E


  0  n   0 (T ) T .
t
t
t
As mentioned above, the neglect of the terms of form
(235)
(  t )n,( t )T contradicts the electric charge conservation law.
Expression (235) for total current with regard to mass and heat fluxes in the displacement current is obtained and justified by us
for the first time and within the accuracy up to the constant coefficient it coincides with formula (230).
We obtained expressions (230) and (235) for total current using independent methods. From these formulas it follows that the
surface charge, the electric induction vector D in the presence of diffusion and heat fluxes must be of the form
D   0 (E  n   (T )T ) .
(236)
In semiconductors and biological membranes concentration gradients of admixture are substantial; in the general case, it is
therefore necessary to allow for these polarization effects.
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It should be borne in mind that in semiconductors true electrons, but not formally introduced holes are real current carriers. In
reality, such holes or really existing particles are unavailable; therefore, by n should be understood only the concentration of
electrons.
About convective radiant energy transfer.In many tasks of engineering and astrophysics radiant energy is transferred by a
fluctuation electromagnetic field in a moving medium. To allow for the influence of medium motion on radiant energy transfer, it
is necessary to use the results of Sect. 7.3 for longitudinal and transverse components of the electromagnetic field, provided that
changes in the flow state are slow in comparison with a frequency of an electromagnetic propagating in the medium and v/c « 1.
The publications, which allow this phenomenon to be taken into consideration, are unknown for us.
In this article we tried to construct a consistent physical and mathematical model for electromagnetic wave propagation in layered
media having no regard of the matrices of induced surface charge impedances. This model is based on the Maxwell equations, the
electric charge conservation law, the total current continuity, and the Dirichlet theorem.
Twelve conditions at the interfaces of adjacent media were obtained and justified without using a surface charge and surface
current in explicit form. The conditions are fulfilled automatically in each section of the heterogeneous medium and are conjugate,
which made it possible to use through-counting schemes for calculations. For the first time the effect of concentration of
"medium-frequency" waves with a length of the order of hundreds of meters at the fractures and wedges of domains of size 1-3
μm has been established. Numerical calculations of the total electromagnetic energy on the wedges of domains were obtained. It is
shown that the energy density in the region of wedges is maximum and in some cases may exert an influence on the motion, sinks,
and the source of dislocations and vacancies and, in the final run, improve the near-surface layer of glass due to the
"micromagnetoplastic" effect.
Our numerical investigations show that the physical and mathematical model proposed herein can be used to advantage the
numerical modeling of high-frequency electromagnetic wave propagation in a medium consisting of layers with different
electrophysical properties.
The results of these calculations are of special importance for medicine, in particular, when microwaves are used in the therapy of
various diseases. For a small, on the average, permissible level of electromagnetic irradiation, the concentration of
electromagnetic energy in internal angular structures of a human body (cells, membranes, neurons, interlacements of vessels, etc)
is possible.
For the first time, we have constructed a consistent physicomathematical model of interaction of nonstationary electric and
thermal fields in a layered medium with allowance for mass transfer. The model is based on the methods of thermodynamics and
on the equations of an electromagnetic field and is formulated without explicit separation of the charge carriers and the charge of
an electric double layer. We have obtained the relations for the electric-field strength and the temperature, which take into account
the equality of the total currents and the energy fluxes, to describe the electric and thermal phenomena in layered media where the
thickness of the electric double layer is small compared to the dimensions of the object under study.
To describe diffusion and electrical phenomena in electrolytes, it is suggested that the irreversible thermodynamics methods be
adopted rather than the hydrodynamic theory of ion diffusion for experimentally observed thermodynamic flows and forces
without a clear understanding and identification of real flows of ions and their mobility.We have modeled numerically the heating
of an electrochemical cell with allowance for the influence of the electric double layer at the metal-electrolyte interface. The
calculation results are in satisfactory agreement with experimental data.
It should be mentioned that Eq. (8) is actually the boundary condition only in the case of electrostatic processes, when the surface
charge density  is specified at the interface. In the presence of alternating currents in the medium the density  will change with
time; therefore relation (8) is not a boundary condition for Dn. In this case, the surface charge density (t) should be calculated
from the solution of the conjugate problem for an electromagnetic field, i.e., in the case where the electric field distribution has
been already found, which we tried to do in this work and also in [82, 83].Section 1-5 of the present work have been earlier
published partially in [82, 83].
We have developed the “aeroacoustics” algorithm of solution of the problem of propagation of an electromagnetic field in a
slowly moving medium. When a signal propagates we introduce the Fresnel drag coefficient and use local Riemann invariants in a
difference cell. The calculation results show the absence of the influence of the medium’s motion on the phase shift of waves,
which is consistent with experimental data.
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References
[1] I. Monzon, T. Yonte, and L. Sanchez-Soto, “Characterizingthe reflectance of periodic lasered media,” Characterizing
thereflectance of periodic lasered media, vol. 218, pp. 43–47, 2003.
[2] 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.
[3] 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.
[4] 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.
[5] 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.
[6] 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.
[7] 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.
[8] O. B´arta, J. Piˇstora, J. Vlˇcek, F. Stanˇek, and T. Kreml, “Magneto-optics in bi-gyrotropic garnet waveguide,” OptoElectronicsReview, vol. 9, no. 3, pp. 320–325, 2001.
[9] 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.
[10] 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.
[11] 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.
[12] O. Keller, “V: Local fields in linear and nonlinear optics of mesoscopic system,” Progress in Optics, vol. 37, pp. 257–343,
1997.
[13] 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.
[14] 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.
[15] M. Born, Principles of Optics, Mir, Moscow, Russia, 1970.
[16] L. Kudryavtsev, “Mathematical Analysis,” Mir, Moscow, Russia, 1970.
[17] A. Frumkin, Electrode Processes, Nauka,Moscow, Russia, 1987.
[18] A.N. Tikhonov and A.A. Samarskii, Equations of Mathematical Physics, Nauka, Moscow, Russia, 1977.
[19] A.F. Kryachko et al., Theory of Scattering of Electromagnetic Waves in the Angular Structure, Nauka, Moscow, Russia,
2009.
[20] 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.
82| www.yloop.in
AIJ-199
American International Journal of Contemporary Scientific Research
ISSN 2349 – 4425
[21] 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.
[22] Z.P. Shul’man and V.I. Kordonskii, “Magnetorheological Effect”, Nauka i Tekhnika, Minsk,Belarus, 1982.
[23] M. Khomich, Magnetic-Abrasive Machining of the Manufactured Articles, BNTU Press, Minsk, Belarus, 2006.
[24] 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.
[25] A.M. Orlov et al., “Magnetic-stimulated alteration of the mobility of dislocations in the plastically deformed silicon of ntype,” Fiz.Tverd.Tela, vol. 43, no. 7, pp. 1207–1210, 2001.
[26] 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.
[27] A.P. Rakomsin, Strengthening and Restoration of Items in an Electromagnetic Field, Paradoks, Minsk, Belarus, 2000.
[28] 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.
[29] 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.
[30] 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.
[31] N.S. Akulov, “Dislocations and Plasticity,”ANBSSR Press, Minsk, Belarus, 1961.
[32] N.S. Akulov, Ferromagnetism,ONTI, Leningrad, Russia, 1939.
[33] I.P. Bazarov, Thermodynamics: Textbook for Higher Educational Establishments,Vysshaya Shkola, Moscow, Russia, 1991.
[34] 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.
[35] A. Einstein, Elementary Theory of Brownian Motion, Collected Papers (3), 1966.
[36] V.E. Golant et al., Fundamental Principles of Plasma Physics, Moscow, Russia, 1977.
[37] Y. Kharkats, “Dependence of the limiting diffusion-migration current on the degree of electrolyte dissociation,”
Elektrokhimiya, vol. 24, no. 4, pp. 539–541, 1988.
[38] 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.
[39] D.V. Gibbs, Thermodynamics. Statistical Mechanics, Moscow, Russia, 1982.
[40] L.N. Antropov, Theoretical Electrochemistry,Moscow, Russia, 1984.
[41] H.G. Hertz, Electrochemistry, New York, NY, USA, 1980.
[42] V. Levich et al., Physicochemical Hydrodynamics, Moscow, Russia, 1959.
[43] J. Neumann, Electrochemical Systems, Moscow, Russia, 1977.
[44] V. Skorcheletti, Theoretical Electrochemistry, Leningrad, Russia, 1969.
[45] L.D. Landau and E.M. Lifshits, Theoretical Physics, Vol. 8, Electrodynamics of Continuous Media, Moscow, Russia, 1982.
[46] V. Tsurko, Proc. Institute of Mathematics of NAS of Belarus, vol. 3, pp. 128–133, 1999.
83| www.yloop.in
AIJ-199
American International Journal of Contemporary Scientific Research
ISSN 2349 – 4425
[47] A.I. Shvab, Elektrichestvo, no. 4, pp. 59–67, 1994.
[48] A.I. Shvab and F.Imo, Elektrichestvo, no. 5, pp. 55–59, 1994.
[49] N.N. Grinchik et al., Proc. NAS of Belarus, Ser. Fiz.-Mat. Nauk, vol. 2, pp. 66–70, 1997.
[50] I.E. Tamm, Principles of the Theory of Electricity, Moscow, Russia, 1976.
[51] J.A. Stratton, The Theory of Electromagnetism, Leningrad, Russia, 1938.
[52] N.N. Grinchik, “Diffusional-electrical phenomena in electrolytes,” Journal of Engineering Physics and Thermophysics, vol.
64, no. 5, pp. 497–504, 1993.
[53] N.N. Grinchik, “Electrodiffusion phenomena in electrolytes,” Inzhenerno-Fizicheskii Zhurnal, vol. 64, no. 5, pp. 610–618,
1993.
[54] N.N. Grinchik et al., “Interaction of thermal and electric phenomena in polarized media,” Mathematical Modeling, vol. 12,
no. 11, pp. 67–76, 2000.
[55] N.N. Grinchik, Modeling of Electrical and Thermophysical Processes in Layered Medium, Belorusskaya Nauka Press,Minsk,
Belarus, 2008.
[56] 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.
[57] L.N. Antropov, Theoretical Electrochemistry, Moscow, Russia, 1989.
[58] N. A. Kostin and O. V. Labyak, Elektrokhimiya, vol. 31, no. 5, pp. 510–516, 1995.
[59] A.I. Dikusar et al., Thermokinetic Phenomena in High-Frequency Processes, Kishinev, 1989.
[60] F. Bark, Yu. Kharkats, and R. Vedin, Elektrokhimiya, vol. 34, no. 4, pp. 434–44, 1998.
[61] 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.
[62] P. Kolesnikov, Theory and Calculation of Waveguides, Light-guides, and Integral-Optoelectronics Elements.Electrodynamics
and Theory of Waveguides,’HMTI Press, Minsk, Belarus, 2001.
[63] T. Skanavi, Dielectric Physics (Region of Weak Fields), Gostekhizdat, Moscow, Russia, 1949.
[64] 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.
[65] J. Jaeger,Methods ofMeasurement in Electrochemistry, Moscow, Russia, 1977.
[66] 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.
[67] 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.
[68] B.-T. Choo, Plasma in aMagnetic Field and Direct Thermal-to-Electric Energy Conversion, Moscow, Russia, 1962.
[69] 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.
[70] I.E. Tamm, Foundations of Electricity Theory, Nauka, Moscow, Russia, 2003.
[71] V. Golant, A.P. Zhilinski, and I.E. Sakharov,Fundamentals of Plasma Physics,Moscow, Russia, p. 383, 1977.
84| www.yloop.in
AIJ-199
American International Journal of Contemporary Scientific Research
ISSN 2349 – 4425
[72] D.I. Blokhintsev, Zh. Tekh Fiz., 15, nos.1–2, pp. 72–81, 1945.
[73] D.I. Blokhintsev, Acoustics of an Inhomogeneous Moving Medium,Moscow, Russia, pp. 84–87, 1981.
[74] O.A. Godin, Acoustics of the Ocean Medium,Moscow, Russia, pp. 217–220, 1989.
[75] S.L. Odintsov, Akustich. Zh.,36, no. 2, pp. 337–339, 1990.
[76] I.A. Miniovich, A.D. Pernik, and V.S. Petrovski, Hydrodynamic Sources of Sound,Leningrad, Russia, pp. 26–28, 1971.
[77] 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.
[78] 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.
[79] M. Laue, Zs. Phys., Bd. 128, p. 387, 1950.
[80] A. Einstein, Collected Works, Nauka, Moscow, Russia, vol. 1, p. 12, 1965.
[81] A. Sommerfeld, Ann. Phys.,Bd. 44, p. 177, 1987.
[82] 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).
[83]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.
[84] Slepyan L. Unipolar induction problem.IZV Pertrograd Polytechnical Institute, 1914, Vol. XXII, Issue 1, pp. 55–175.
[85] Djurie J. J. Appl. Phys., 1975, Vol. 46, No. 2, p. 679.
[86] Bartlett D.F. et al. Phys. Rev. D, 1977, Vol. 16, No. 12, p. 3459.
[87] Rodin A.O. Unknown electromagnetic induction experiments. Electricity, 1994, No. 7 (in Russian).
[89] Luparev V.V., Kharitonov V.I. One erroneous statement associated with the uniporlar induction phenomenon. Electricity,
2012, No. 7, pp. 69–72 (in Russian).
[90] Tarnovsky A.S. Definition of the concepts ‘potential’ and ‘potential field’. Electricity, 2000, No. 1, pp. 63–64.
[92] 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.
[93] Sommerfeld А. Electrodynamics. Мoscow, 1958, 327 p.
[94] I. P. Bazarov, Thermodynamics. Textbook, 5th edition, Saint-Petersburg: Publishing House “Lan”, 2010, 384 p.
[95] B.M. Vul, Physics of dielectrics and semiconductors, Moscow: Nauka, 1988, 372 p.
[96] V. G. Polevoi, Heat transfer by fluctuation electromagnetic field, Moscow: Nauka, 1990, 90 p.
[97] S. М. Rytov, Theory of electric fluctuations and thermal radiation, Moscow: Publishing House of the USSR Academy of
Sciences, 1953, 231 p.
[98]М. А. Leontovich, Statistical Physics. # 19, # 26, Moscow-Leningrad: Publishing House “OGIZ”, 1944.
[99] R. Siegel, J. Howell, Thermal radiation heat transfer, McGraw-Hill Book Company, Toronto, 1972, p. 534.
[100] M.N. Özisik. Radiative transfer and interactions with conduction and convection, John Wiley & Sons, New York, London,
Sydney, Toronto, 1976.
85| www.yloop.in
AIJ-199
American International Journal of Contemporary Scientific Research
ISSN 2349 – 4425
[101] M. Planck, Thermal radiation laws and hypothesis on the elementary quantum of action, Collected Papers,
Thermodynamics, Poiseuille’s theory, Moscow: Nauka, 1975, 778 p.
[102] 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, .
[103]H.C. Vann de Hulst. The scattering of light by small particles, Moscow: PublishingHouse “IL”, 1961, 536 p.
[104]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.
[105]А. S. Тоporets, Light reflection by rough surfaces, in: Theoretical and Applied Problems of Light Scattering, Minsk: Nauka i
Tekhnika, 1971, 486 p.
[106] 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.
[107]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.
[108] V. A. Bubnov, Theory of heat waves, J. Eng. Phys. Thermophys., No. 3, 1982, pp. 431–437.
[109]А. G. Shashkov, V.A. Bubnov, and S.Yu. Yanovsky, Heat conduction wave phenomena, Minsk: Nauka i Tekhnika, 1993, p.
274.
[110] 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.
[111] 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.
[112]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.
[113]А. N. Tikhonov and А. А. Samarsky, Mathematical physics equations, Moscow: Nauka, 5th edition, 1977, 734 p.
[114] V. I. Коrzyuk, Mathematica physics equations, Minsk: BSU Press, 2011, 459 p.
[115] 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.
[116] V. L. Bonch-Bruevich and S. G. Kalashnikov, Physics of semiconductors, Moscow: Nauka, 1990, 622 p.
[117]W.Shockley. Theory of electronic semiconductors,Moscow: PublishingHouse “IL”, 1953, 714 p.
86| www.yloop.in