Modeling of thermal and electrical conductivities by means of a viscoelastic Cosserat continuum

Abstract

We consider a linear theory of a viscoelastic Cosserat continuum of a special type. In doing so, we associate the main variables characterizing the stress–strain state of the continuum with quantities characterizing the electrodynamic and thermal processes. Taking into account the suggested analogues, we interpret equations describing the continuum as equations of thermodynamics and electrodynamics. We identify parameters of our model by comparing the obtained equations with Maxwell’s equations and the hyperbolic heat conduction equation. As a result, we arrive at two three-dimensional telegrapher’s equations: one for temperature and the other for the electric field vector. These equations are novel. They describe electromagnetic and thermal processes and also how they affect each other more accurately compared to the classical theory. In particular, these telegrapher’s equations account for not only the skin effect described in many literature sources on electrodynamics, but also the so-called static skin effects observed in a number of experiments. In contrast to classical electrodynamics, which contains two mutually orthogonal vectors: the electric field vector and the magnetic induction vector, the proposed theory contains three mutually orthogonal vectors: the electric field vector, the magnetic induction vector and the temperature gradient. It agrees with experimental facts discovered by Ettingshausen and Nernst (the Ettingshausen effect and the Nernst–Ettingshausen effect). If thermal component is ignored, the proposed theory reduces to the system of equations, which is a generalization of Maxwell’s equations. This system of equations is novel. It is a three-dimensional analogue of Kirchhoff’s laws for electric circuits, while Maxwell’s equations are not.

1 Introduction

The idea of creating mechanical models of various physical processes has been discussed by many authors. This idea dominated in science until the late nineteenth century. Many mechanical models of thermal, electrical, magnetic and electromagnetic processes, known as models of the ether, were proposed at that time. Whittaker has provided a very good overview of these models [1]. We note that all mathematical models of the ether constructed in the nineteenth century are based on translational degrees of freedom. Originators of these models faced problems that they could not overcome [1]. At the same time, such eminent scientists of the nineteenth century as Kelvin, Fitzgerald and Maxwell came up with an idea that mechanical models of electromagnetism should be based upon rotational degrees of freedom [1]. However, the level of development of continuum mechanics did not allow scientists to create mathematical models of continua with rotational degrees of freedom until the early twentieth century, when the Cosserat brothers first developed a correct model of such a continuum. Unfortunately, by the time of the publication of Cosserat’s work, interest in the ether models had already been largely lost, and at that time Cosserat’s work practically went unnoticed. Nevertheless, some twentieth-century scientists were also interested in physical and mechanical analogues, see, e.g., [2]. There are papers published in the mid-twentieth century that draw analogies between equations describing various physical processes, including analogies between equations of continuum mechanics and equations describing processes in electrical circuits, see [3,4,5,6]. The purpose of these studies was to work out methods for solving differential equations adopting electrical circuits and electrical machines. As a result of the development of computer technology, such methods have lost their relevance. However, in modern studies, the ideas of [3,4,5,6] are revisited and applied to design the multi-physics metamaterials, see [7,8,9].

At the turn of the twentieth/twenty-first centuries, the interest in the ether models have been renewed. A large number of review papers have been published over the past 30 years, see, e.g., [10,11,12,13,14,15,16]. In addition, studies aimed at creating mechanical models of physical processes resumed in the second half of the twentieth century and in the twenty-first century. Modern theories, as well as theories of the nineteenth century, describe both processes in the free ether (in vacuum) and processes in the ether associated with ponderable matter. Discussing the research carried out in the twentieth and twenty-first centuries, we should mention both models based on translational degrees of freedom and models based on rotational degrees of freedom. We start with a number of the models based on translational degrees of freedom. In 1969, Jaswon [17] drew an analogy between Maxwell’s equations and equations describing “a gyrostatic continuum.” We note that the Jaswon gyrostatic continuum does not contain any gyrostats. This is actually a model based on translational degrees of freedom, which coincides with the one proposed by MacCullagh in the nineteenth century, see [1, 18]. We note that this model cannot be considered satisfactory, since it conflicts with Euler’s second law of dynamics. In 1976, Kelly [19] derived Maxwell’s equations in vacuum by using an ideal fluid model. The model [19] is similar to the models of the nineteenth century, but it describes some nonlinear effects and provides a mechanical interpretation of the Lorentz gauge. In 1996, Zhilin [20, 21] drew an analogy between Maxwell’s equations and linear theory of elasticity. He analyzed Maxwell’s equations in view of the comparison with incompressible solids, and proposed the generalized Maxwell’s equations. The mechanical model discussed in [20, 21] is similar to the nineteenth-century solid-type models of the ether. The cited Zhilin’s papers can be found in [22, 23]. In 1998, Larson [24] modeled electromagnetic field by means of a two-component continuum based on translational degrees of freedom. Such a complicated model allowed him to introduce the positive and negative charges. In 2001, Zareski [25] gave an elastic interpretation of electrodynamics, which is also based on translational degrees of freedom. The main feature of this work is that the author claims that his model is consistent with the special theory of relativity. In 2003 paper [26], Dmitriyev considered a solid-type model of electromagnetic field, which is similar to the one proposed by Zhilin in 1996. Later, in 2008 paper [27], Dmitriev proposed mechanical models of the Coulomb interaction and the Lorentz force. This study develops Kelly’s ideas. In 2007, 2009 and 2011 Christov published a series of works [28,29,30,31] developing Cauchy’s and Stokes’ ideas. Christov considers mechanical models of electromagnetic field, which are similar to the models of the nineteenth century, since they are also based on the concepts of the solid-type and the liquid-type ether. However, Christov’s models differ from those of the nineteenth century in that the differential equations in his works [28,29,30,31] contain the material time derivatives instead of the partial time derivatives. This enables Christov to take a fresh look at the problem of the frame indifference of Maxwell’s equations. In addition, taking into account modern cosmological theories, Christov expresses some new ideas regarding the physical interpretation of longitudinal waves present in the mechanical models. Papers [28,29,30,31] cover a wide range of topics including interpretations of Ohm’s law as a viscous effect and charged particles as localized shear waves. In 2008, Wang [32] proposed a viscoelastic continuum model of vacuum based on translational degrees of freedom. In fact, considering this model Wang develops Stokes’ ideas according to which the ether is something like a glue-water jelly. In the cited paper Wang introduces the electric charges as sources or sinks in the ether. To be exact, he defines a source as a negative charge and a sink as a positive charge. It should be noted that Wang’s and Christov’s ideas are related. In 2014, to derive Maxwell’s equations, Lin and Lin [33] used a continuum model, where both translational and rotational degrees of freedom are taken into account, but the moment stress tensor is assumed to be zero. However, in the course of reasoning, the authors made a number of additional assumptions that actually excluded rotational degrees of freedom from their model. Thus, this model of electromagnetic field is, in fact, the model based on translational degrees of freedom only.

As seen from the above review, attempts to construct models of the ether based on translational degrees of freedom are being undertaken by modern scientists despite the problems that became apparent in the nineteenth century. At the same time, in the twentieth century, various theories of the Cosserat-type continua (continua with rotational degrees of freedom) have been developed. Unfortunately, models based on the Cosserat-type continua have not yet found such widespread use in describing physical processes as we would like it to be. Nevertheless, such models are found in modern literature and now we turn to their overview. First of all, we note that the idea of describing electromechanical and magnetomechanical effects by means of continuum models with rotational degrees of freedom has been discussed by a number of authors. We can refer, e.g., to works [34,35,36,37,38,39,40,41] proposing one-component models, and works [42,43,44,45] proposing two-component models. We do not discuss these works in detail, since they are devoted to the issues that are not directly related to the topic of our work. We only note that the aforementioned works are important for developing of continuum models with rotational degrees of freedom, because they demonstrate that such models can be successfully applied to describe mutual influence of mechanical and physical processes. So, we return to the discussion of mechanical models of the ether, i.e., those mechanical models, a mathematical description of which can be reduced to Maxwell’s equations.

Russian scientist Pavel A. Zhilin was the first to pay attention to the ideas of scientists of the nineteenth century regarding the construction of ether models based on rotational degrees of freedom. He was the first to come up with the idea to use the Cosserat continuum for constructing mechanical models of physical processes and he was the first to implement this idea. In 1996, Zhilin gave a lecture at XXIII Summer School “Nonlinear Oscillations in Mechanical Systems” (St. Petersburg, Russia) titled “Reality and mechanics,” where he presented a continuum model based only on rotation degrees of freedom and showed that the mathematical description of this model can be reduced to the well-known equations of quantum mechanics: the Schrödinger equation and the Klein–Gordon equation. In 2000, Zhilin gave a lecture at XXVIII Summer School–Conference “Advanced Problems in Mechanics” (St. Petersburg, Russia) titled “The main direction of the development of mechanics for XXI century,” where he presented a linear model of electromagnetic field in vacuum. This model is also a continuum model based only on rotation degrees of freedom. Later, Zhilin developed this model and created a nonlinear theory of electromagnetic field. The exact date of the creation of this nonlinear theory is unknown. Apparently, it seems reasonable to date this theory to 2005, i.e., the year of Zhilin’s death. The lectures given at XXIII Summer School and at XXVIII Summer School–Conference were published in 2006 in the collection of Zhilin’s papers [23]; the nonlinear theory of electromagnetic field was first published in 2012 as the sixth chapter of [46]; the original version of this paper was published unedited in 2013, see [47]. A summary of the ideas of the above Zhilin’s works can be found in [18]; the biography and scientific contributions of Zhilin are presented in [48,49,50].

We are deeply convinced that modeling electromagnetic and quantum mechanical processes using the Cosserat-type continua is a very promising area of research. We are also convinced that Zhilin’s ideas on the use of the Cosserat-type continua can be applied to model other physical processes, in particular, thermal ones. Beginning in 2010, we have published a series of studies [51,52,53,54,55,56,57], in which we construct such models of thermal processes. In 2015 and 2020, we proposed two models of electromagnetic processes [58, 59], which are also based on Zhilin’s ideas. In 2019, we proposed models that take into account a mutual influence of thermal and electromagnetic processes, see [60,61,62]. All our models are based on rotational degrees of freedom. These models introduce mechanical analogues of temperature, specific entropy, the electric and magnetic field vectors, the electric and magnetic induction vectors, and also the electric current and charge densities. The study presented below develops our previous works. The main objective of this study is to model the electrical and thermal conductivities. In order to do this we construct a mechanical model of the ether associated with ponderable matter. In our opinion, the fundamental difference between modeling the free ether and modeling the ether associated with ponderable matter is as follows. The models of the free ether should be conservative. The models of the ether associated with ponderable matter should take into account dissipative processes causing the attenuation of waves. In [60,61,62], we consider models that are described by the generalized Maxwell’s equations containing a dissipative term that characterizes the external friction due to the interaction of ponderable matter and free ether. Below, in addition to the external friction [60,61,62], we introduce another type of dissipation, namely the internal friction. Then we find a match between constants characterizing the friction in the mechanical model of the ether and constants characterizing physical properties of the matter. Considering two types of dissipation we construct more detailed models of thermal and electrical conduction processes compared to the classical theory. We obtain a system of equations that can be reduced to the telegrapher’s equations for the electric field vector and temperature. We explain the meaning of these equations taking into account the following: the mechanical meaning of the dissipation mechanism; resemblance between the thermal conduction mechanism and the electrical conduction mechanism; similarity of the obtained equations to the telegrapher’s equations for current and voltage in transmission lines. We also discuss some experimental facts and the possibility to explain them by using the proposed theory. In particular, we show that the obtained telegrapher’s equations allow to account for not only the skin effect, which is well studied in electrodynamics, but also the static skin effect for electromagnetic field and the static skin effect for temperature. Finally, we show that the proposed theory without thermal component is a generalization of Maxwell’s equations and is a three-dimensional analogue of Kirchhoff’s laws for electric circuits.

2 Some known facts from electro- and thermodynamics

2.1 Maxwell’s equations for conducting media

It is well known that Maxwell’s equations can be written as

(1)

Here is the electric induction vector, is the magnetic induction vector, is the electric field vector, is the magnetic field vector, is the electric current density, is the electric charge density. We use notation ${\displaystyle \frac{\hbox {d} }{\hbox {d} t}}$ for the total time derivative [63, 64]. In most physics textbooks and monographs, Maxwell’s equations are written in terms of the partial time derivative. We follow the concepts of continuum mechanics and therefore we use the total time derivative. We note that there is a tendency in modern physics to rewrite Maxwell’s equations using the total time derivative instead of the partial time derivative. For example, in the old editions of Pohl’s book, see, e.g., [65], all equations are written in terms of partial time derivative, while in a new edition of this book, see [66], the total time derivative is used.

In the case of an isotropic conducting medium the material equations take the form

(2)

where $\varepsilon _0$ and $\mu _0$ are the permittivity and permeability of vacuum, $\varepsilon$ and $\mu$ are the relative permittivity and the relative permeability of the material. In view of Eq. (2) Maxwell’s equations (1) can be rewritten as

(3)

It is also well known that the charge conservation law

(4)

follows from Eq. (3) and, in fact, is a condition for the solvability of Eq. (3). We note that Feynman considers Eq. (4) to be the definition of the electric current density, see [67, p. 18–1].

Eliminating vector from Eq. (3) yields the differential equations for vector :

(5)

The first equation in (5) depends only on the vortex part of the electric current density and, therefore, it has the same form when and when is not equal to zero. We note that equations in (5) are not independent. Indeed, taking the divergence of the first equation in (5), we arrive at equation . Integrating this equation in time and assuming that at initial time and its time derivative also equals to zero, we obtain the second equation in (5). Thus, the second equation in (5) is a consequence of the first one.

Eliminating vector from Eq. (3) and taking into account Eq. (4), one can obtain the differential equations for vector :

(6)

We note that the second equation in (6) can be obtained as a consequence of the first one. Taking the divergence of the first equation in (6), we have . Integrating this equation in time and assuming that at initial time , we obtain the second equation in (6).

It is generally accepted that for dielectrics the electric current density in Eqs. (5) and (6) can be assumed to be equal to zero. In this case, Eqs. (5) and (6) turn into undamped wave equations, namely

(7)

In the case of conductors, Ohm’s law for the electric current density is usually used. We give the formulation of Ohm’s law using the following quotation from Feynman [67, p. 32–10]:

“It is found experimentally that an electric field in a metal produces a current with the density proportional to (for isotropic materials):

(8)

The proportionality constant is called the conductivity.”

Ohm’s law is well known from the school and university physics courses. It is used in modeling of various electrical devices, power machines, and some engineering structures. The application of Ohm’s law is not as obvious as it might seem ex facte. Its application raises questions in a number of cases described in textbooks on physics, electrodynamics and electrical engineering, see, e.g., [70,71,72]. However, for metal conductors under normal conditions, Ohm’s law is fulfilled quite accurately.

Now, we pay attention to obvious consequences of Maxwell’s equations in view of Ohm’s law. Substituting Eq. (8) into Eq. (5) yields

(9)

Next, inserting Eq. (8) into Eq. (6), we obtain

(10)

We note that, if the charge density is assumed to be equal to zero, Eq. (10) takes a simpler form

(11)

Equations (9), (10), (11) are well-known. One can find them in various literature sources, see, e.g., [68,69,70,71,72,73].

It is well known that an alternating electric current is distributed within a conductor so that the current density is the largest near the surface and decreases with depth, see, e.g., [70]. This is called the skin effect. The skin depth is usually estimated by using the approximate Maxwell’s equations, which result from Eq. (3) if we neglect the term containing the time derivative of in the last equation. If the charge density is assumed to be zero and Ohm’s law (8) is used, the approximate Maxwell’s equations yield

(12)

It is not difficult to show that from Eq. (12) it follows that the skin depth of the conductor $\delta$ is determined as

$$\begin{aligned} \delta = \frac{1}{\sqrt{\sigma \mu \mu _0 \pi f}}, \end{aligned}$$

(13)

where f is a frequency. We note that from Eq. (11) it follows that

$$\begin{aligned} \mathrm{if}\ \ f \ll \frac{\sigma }{2 \pi \varepsilon \varepsilon _0}: \quad \delta \approx \frac{1}{\sqrt{\sigma \mu \mu _0 \pi f}}, \qquad \mathrm{if}\ \ f \gg \frac{\sigma }{2 \pi \varepsilon \varepsilon _0}: \quad \delta \approx \frac{2}{\sigma }\sqrt{\frac{\varepsilon \varepsilon _0}{\mu \mu _0}}. \end{aligned}$$

(14)

The first formula for $\delta$ in (14) coincides with formula (13). This formula is well-known and widely used in engineering calculations, see, e.g., [72, p. 537]. The second formula for $\delta$ in (14) is not applied in engineering calculations. This is not surprising, since values of the relative permittivity $\varepsilon$ are not known for the conductors. We will also ignore the skin effect given by the second formula in (14). However, in addition to the skin effect described by the first formula in (14), we will consider the so-called static skin effect, which is not described in the framework of Maxwell’s electrodynamics. A definition of the static skin effect and literature references will be given below, when the corresponding differential equations are obtained.

2.2 Circuit models for conducting media

In the previous section, we discussed the approach that is based on Maxwell’s equations and is usually used in books on physics and electrodynamics. In this section, we discuss an alternative approach, which is based on Kirchhoff’s laws for electric circuits and is usually used in books on electrical engineering. After that we compare these approaches. The comparison of two approaches is important because they lead to different results, although it seems that the results should not depend on which method is used.

So now we consider a circuit model for an incremental length of the transmission line. This model is well known and can be found in many textbooks, e.g., in [74]. The circuit model includes an inductance and a capacitance, as well as a shunt conductance and a series resistance, see Fig. 1. We use the following notation: L is the inductance, C is the capacitance, G is the shunt conductance, R is the series resistance; all these parameters are specified per unit length. In order to clarify the physical meaning of G and R, we quote from [74, p. 304]: “The shunt conductance is used to model leakage current through the dielectric that may occur throughout the line length; the assumption is that the dielectric may possess conductivity, $\sigma _d$, in addition to a dielectric constant, where the latter affects the capacitance. The series resistance is associated with any finite conductivity, $\sigma _c$, in the conductors. Either one of the latter parameters, R and G, will be responsible for power loss in transmission.”

Fig. 1

An incremental length of the transmission line

Following the line of reasoning of [74], we derive differential equations describing the rates of change of voltage V and current I with respect to coordinate z along the transmission line. Applying Kirchhoff’s current law to the upper central node in the circuit shown in Fig. 1 and Kirchhoff’s voltage law to the loop that encompasses the entire section length, we obtain

$$\begin{aligned} \frac{\partial I}{\partial z} = - \left( G V + C \frac{\partial V}{\partial t} \right) , \qquad \frac{\partial V}{\partial z} = - \left( R I + L \frac{\partial I}{\partial t} \right) . \end{aligned}$$

(15)

Eliminating I from system (15) yields

$$\begin{aligned} \frac{\partial ^2 V}{\partial z^2} = L C\, \frac{\partial ^2 V}{\partial t^2} + \left( L G + R C\right) \frac{\partial V}{\partial t} + R G V. \end{aligned}$$

(16)

Eliminating V from system (15) gives exactly the same equation in terms I:

$$\begin{aligned} \frac{\partial ^2 I}{\partial z^2} = L C\, \frac{\partial ^2 I}{\partial t^2} + \left( L G + R C\right) \frac{\partial I}{\partial t} + R G I. \end{aligned}$$

(17)

Thus, we have obtained the transmission line equation, which is known as the telegrapher’s equation.

We note that the same derivation of the transmission line equation can be found in [72, 75, 76]. As shown in [72, p. 537], at high frequencies, parameters C, L, G and R are related to geometrical and material characteristics of the transmission line as follows.

In the case of the coaxial line (see Fig. 2, where the geometrical parameters are given):

$$\begin{aligned} C = \frac{2 \pi \varepsilon _d \varepsilon _0}{\ln (b/a)}, \quad L = \frac{\mu _d \mu _0}{2 \pi } \ln (b/a), \quad G = \frac{2 \pi \sigma _d}{\ln (b/a)}, \quad R = \frac{1}{2 \pi \delta \sigma _c}\left[ \frac{1}{a}+\frac{1}{b}\right] , \quad \delta \ll a, c - b. \end{aligned}$$

(18)

Here and below indices d and c refer to the quantities characterizing dielectrics and conductors, respectively, $\delta$ is the skin depth of the conductor.

In the case of the two-wire line with the geometrical parameters given in Fig. 2:

$$\begin{aligned} C = \frac{\pi \varepsilon _d \varepsilon _0}{\cosh ^{-1}(d/2a)}, \quad L = \frac{\mu _d \mu _0}{\pi } \cosh ^{-1}(d/2a), \quad G = \frac{\pi \sigma _d}{\cosh ^{-1}(d/2a)}, \quad R = \frac{1}{\pi a \delta \sigma _c}, \quad \delta \ll a. \end{aligned}$$

(19)

In the case of the planar line with the geometrical parameters presented in Fig. 2:

$$\begin{aligned} C = \frac{\varepsilon _d \varepsilon _0 w}{d}, \quad L = \frac{\mu _d \mu _0 d}{w}, \quad G = \frac{\sigma _d w}{d}, \quad R = \frac{2}{w \delta \sigma _c}, \quad \delta \ll t, \quad w \gg d. \end{aligned}$$

(20)

Fig. 2

Transmission lines: the coaxial line, the two-wire line, the planar line

As it is seen from Eqs. (18), (19), (20), for all transmission lines the following relations hold:

$$\begin{aligned} L C = \varepsilon _d \varepsilon _0 \mu _d \mu _0, \qquad \frac{G}{C} = \frac{\sigma _d}{\varepsilon _d \varepsilon _0}, \qquad \frac{R}{L} = \frac{1}{\sigma _c \mu _d \mu _0 l \delta }, \end{aligned}$$

(21)

where l is a parameter, which depends on the geometrical properties of the transmission line.

We note that some literature sources consider the simplest model of the transmission line where R and G are assumed to be equal to zero, see, e.g., [67]. At the same time, there are more complicated models of the transmission lines. However, below we discuss only the model described by Eqs. (16) and (17).

2.3 A comparison of two approaches to description of conducting media

In this section, we compare Maxwell’s equations and Kirchhoff’s laws consequences and show that there are some contradiction between them. For ease of comparison, we rewrite Eq. (11) in terms of the electric current density taking into account Ohm’s law (8). As a result we have

(22)

We also insert the parameters from Eq. (21) into Eq. (17). This yields

$$\begin{aligned} \frac{\partial ^2 I}{\partial z^2} = \varepsilon _d \varepsilon _0 \mu _d \mu _0 \frac{\partial ^2 I}{\partial t^2} + \left( \sigma _d\, \mu _d \mu _0 + \frac{\varepsilon _d \varepsilon _0}{\sigma _c l \delta }\right) \frac{\partial I}{\partial t} + \frac{\sigma _d}{\sigma _c l \delta } I. \end{aligned}$$

(23)

Comparing the coefficients at the second time derivatives of the electric current density in Eq. (22) and the electric current in Eq. (23), we see that the following correspondence takes place

$$\begin{aligned} \varepsilon _d \ \ \leftrightarrow \ \ \varepsilon , \qquad \mu _d \ \ \leftrightarrow \ \ \mu . \end{aligned}$$

(24)

Next, comparing the coefficients at the first time derivatives, we can conclude that the conductivity of dielectrics corresponds to conductivity $\sigma$ in Eq. (22):

$$\begin{aligned} \sigma _d \ \ \leftrightarrow \ \ \sigma . \end{aligned}$$

(25)

Then, under the additional assumption that the conductivity of conductors tends to infinity:

$$\begin{aligned} \sigma _c \ \ \rightarrow \ \ \infty , \end{aligned}$$

(26)

from Eq. (23) it follows that

$$\begin{aligned} \frac{\partial ^2 I}{\partial z^2} = \varepsilon _d \varepsilon _0 \mu _d \mu _0 \frac{\partial ^2 I}{\partial t^2} + \sigma _d\, \mu _d \mu _0 \frac{\partial I}{\partial t}. \end{aligned}$$

(27)

Equations (22) and (27) have exactly the same structure. However, Eq. (27) contains the conductivity of dielectrics, whereas conductivity $\sigma$ in Eq. (22) is defined as a coefficient in Ohm’s law (8), which is usually used for conductors. Then, it would seem that we should infer that the conductivity of conductors corresponds to conductivity $\sigma$ in Eq. (22):

$$\begin{aligned} \sigma _c \ \ \leftrightarrow \ \ \sigma . \end{aligned}$$

(28)

Assuming that the conductivity of dielectrics tends to zero:

$$\begin{aligned} \sigma _d \ \ \rightarrow \ \ 0, \end{aligned}$$

(29)

from Eq. (23) we obtain

$$\begin{aligned} \frac{\partial ^2 I}{\partial z^2} = \varepsilon _d \varepsilon _0 \mu _d \mu _0 \frac{\partial ^2 I}{\partial t^2} + \frac{\varepsilon _d \varepsilon _0}{\sigma _c l \delta }\, \frac{\partial I}{\partial t}. \end{aligned}$$

(30)

It is easy to see that conductivity $\sigma$ is in the numerator of the coefficient at in Eq. (22), whereas conductivity $\sigma _c$ is in the denominator of the coefficient at ${\displaystyle \frac{\partial I}{\partial t}}$ in Eq. (23). Even if we take into account Eq. (13) for the skin depth $\delta$, we do not arrive at a coefficient where conductivity $\sigma _c$ would be in the numerator. Indeed,

$$\begin{aligned} \frac{\varepsilon _d \varepsilon _0}{\sigma _c l \delta } = \frac{\varepsilon _d \varepsilon _0 \sqrt{\pi f \mu _c \mu _0}}{\sqrt{\sigma _c}\, l}. \end{aligned}$$

(31)

Moreover, if we use the last formula in Eq. (14) for $\delta$, we do not arrive at a coefficient with conductivity $\sigma _c$ in the numerator anyway. In this case, we obtain the formula

$$\begin{aligned} \frac{\varepsilon _d \varepsilon _0}{\sigma _c l \delta } = \frac{\varepsilon _d \sqrt{\mu _c \mu _0 \varepsilon _0}}{2 l \sqrt{\varepsilon _c}}, \end{aligned}$$

(32)

which does not contain conductivity $\sigma _c$ at all. Thus, we are forced to conclude that there are some contradictions between the equation for the electric current density, obtained from Maxwell’s equations, and the equation for the electric current, obtained on the basis of Kirchhoff’s laws.

2.4 Ohm’s law, Fourier’s law and the Wiedemann–Franz law

It is well known that Ohm’s law for the electric current density and Fourier’s law for the heat flux are similar to each other. Indeed, they have the following form:

(33)

where $\mathbf {h}$ is the heat flux vector, $T_\mathrm{a}$ is the absolute temperature, $\lambda$ is the thermal conductivity. Fourier’s law usually contains a minus sign. But the sign in Fourier’s law depends on how the heat flux is defined. It can be defined as entering the control volume or as leaving the control volume. We define the heat flux so that the signs in Ohm’s law and in Fourier’s law coincide. Comparing the equations in (33), we see that

(34)

In 1853, Gustav Wiedemann and Rudolph Franz discovered that ratio $\lambda / \sigma$ has approximately the same value for different metals at the same temperature. In 1872, Ludvig Lorenz has established the proportionality of $\lambda / \sigma$ with the absolute temperature. Thus, in accordance with the Wiedemann–Franz law, the ratio of thermal conductivity $\lambda$ to electrical conductivity $\sigma$ is proportional to $T_\mathrm{a}$:

$$\begin{aligned} \frac{\lambda }{\sigma } = L T_\mathrm{a}, \end{aligned}$$

(35)

where coefficient L is known as the Lorenz number. This coefficient is a constant, which is approximately the same for most metals. The Wiedemann–Franz law is confirmed by experiments at high temperatures (above room temperature) and at very low temperatures (temperatures of a few kelvins), but it may not hold at intermediate temperatures. According to the concept of modern physics, both the thermal conductivity of metals and their electrical conductivity must be determined by the properties of the electron gas. The conventional explanation of the Wiedemann–Franz law (35) is based on this concept.

In view of the aforesaid, it seems obvious that the differential equations describing the thermal conductivity process and the electrical conductivity process must be exactly the same. However, this is not quite true. Below we discuss this issue in detail.

2.5 A comparison of the equations describing the processes of thermal and electrical conductivities

We start with two heat conduction equations: the classical one and the hyperbolic one. If the volumetric heat sources are absent, the classical heat conduction equation, based on Fourier’s law, can be written in terms of the absolute temperature or in terms of the heat flux vector as

$$\begin{aligned} \Delta T_\mathrm{a} - \frac{\rho c_v}{\lambda } \frac{\hbox {d} T_\mathrm{a}}{\hbox {d} t} = 0 \quad \Rightarrow \quad \Delta \mathbf {h} - \frac{\rho c_v}{\lambda } \frac{\hbox {d} \mathbf {h}}{\hbox {d} t} = 0, \end{aligned}$$

(36)

where $\rho$ is the mass density, $c_v$ is the specific heat at constant volume. The hyperbolic type heat conduction equation is written as

$$\begin{aligned} \Delta {T}_\mathrm{a} - \frac{\rho c_v}{\lambda } \frac{\hbox {d} {T}_\mathrm{a}}{\hbox {d} t} - \frac{1}{c_r^2}\, \frac{\hbox {d}^2 {T}_\mathrm{a}}{\hbox {d} t^2} = 0 \quad \Rightarrow \quad \Delta \mathbf {h} - \frac{\rho c_v}{\lambda } \frac{\hbox {d} \mathbf {h}}{\hbox {d} t} - \frac{1}{c_r^2}\, \frac{\hbox {d}^2 \mathbf {h}}{\hbox {d} t^2} = 0, \end{aligned}$$

(37)

where $c_r$ is the heat wave velocity. It is easy to see that Eq. (37) has the same structure as Eqs. (9), (11) and (22) resulting from Maxwell’s equations.

Now, we discuss how Maxwell’s equations can be reduced to an equation that corresponds to the classical heat conduction equation. It is evident that we need to make two simplifying assumptions. Firstly, we must suppose that the electric charges are absent, and hence, and . Secondly, we must neglect the term containing the time derivative of vector in Maxwell’s first equation, i.e., in fact, we must use Ampere’s law instead of Maxwell’s first equation. In view of these assumptions, from Eq. (3) it follows that

(38)

Taking into account Ohm’s law, we can rewrite Eq. (38) in terms of vector or in terms of vector as

(39)

Comparing Eq. (39) with Eq. (36) we see that these equations have the identical structure. In addition, we note that Eq. (39) coincides with Eq. (12), which is usually used for determination of the skin depth.

Thus, in the general case, Maxwell’s equations can be reduced to the differential equations analogous to the hyperbolic type heat conduction equation, and under the simplifying assumptions, Maxwell’s equations can be reduced to the differential equations that are analogous to the classical heat conduction equation. However, if we look at these equations more closely and compare the coefficients at the first time derivatives, we will find one important difference. In the heat conduction equations (36), (37), the thermal conductivity $\lambda$ is in the denominator of the coefficient at the first time derivative, whereas in the equations of electrodynamics, the electrical conductivity $\sigma$ is in the numerator of the coefficient at the first time derivative. This looks very strange in view of what has been said above regarding the relationship between the thermal conductivity and the electrical conductivity. In addition, we note that in the circuit model for an incremental length of the transmission line the electrical conductivity of a conductor $\sigma _c$ is in the denominator of the coefficient at the first time derivative of the electric current, see Eq. (23). In other words, the coefficient at the first time derivative in this equation depends on the electrical conductivity in the same way as the corresponding coefficients in the heat conduction equations (36), (37) depend on the thermal conductivity. This is consistent with what is said above regarding the electrical conductivity and the thermal conductivity of metals.

2.6 Summary

Thus, having studied the known facts, we have found two contradictions. One of them is revealed when comparing equations resulting from Maxwell’s equations and equations resulting from Kirchhoff’s laws. The other one is revealed when comparing the equations following from Maxwell’s equations with the heat conduction equations. These contradictions lead us to the idea that it is necessary to construct a new theory.

3 The linear theory of the viscoelastic Cosserat continuum of a special type

3.1 Preliminary remarks

Below we present the basic equations of the Cosserat continuum of a special type, which we will use to model thermal and electrodynamic processes. We start with the general equations of the linear isotropic Cosserat continuum. After that we formulate a number of simplifying assumptions regarding kinematics and constitutive equations. Finally, we write down a summary of the equations, to which we will give the meaning of analogues of the basic equations of electrodynamics and thermodynamics.

3.2 Kinematics, equations of motion and strain tensors

Let vector $\mathbf {r}$ identify the position of some point of space. We use the following notation: $\mathbf {v}(\mathbf {r},t)$ is the velocity vector field; $\mathbf {u}(\mathbf {r}, t)$ is the displacement vector field; $\varvec{\theta }(\mathbf {r},t)$ is the rotation vector field, and $\varvec{\omega }(\mathbf {r},t)$ is the angular velocity vector field. In the linear approximation, the kinematic relations have the form

$$\begin{aligned} \mathbf {v} = \frac{\hbox {d} \mathbf {u}}{\hbox {d} t}, \qquad \varvec{\omega } = \frac{\hbox {d} \varvec{\theta }}{\hbox {d} t}. \end{aligned}$$

(40)

We refer the inertia characteristics to an elementary volume fixed in space and containing an ensemble of particles. We introduce the following notation: $\rho (\mathbf {r},t)$ is the mass density of the continuum at a given point of space at the current time; $\mathbf {J}(\mathbf {r},t)$ is the mass density of inertia tensors of the particles which occupy a fixed elementary volume V located in space near the point identified by the position vector $\mathbf {r}$ at the current time, and $\rho \mathbf {J}$ is the volume density of the inertia tensors. We consider an isotropic continuum, i.e., we assume that

$$\begin{aligned} \mathbf {J} = J(\mathbf {r}) \mathbf {E}, \end{aligned}$$

(41)

where $J(\mathbf {r})$ is the mass density of moments of inertia, $\mathbf {E}$ is the second order identity tensor. The kinetic energy, the linear momentum vector and the angular momentum vector constitute the dynamic structures of the continuum. The mass density of kinetic energy of the continuum has the form

(42)

The mass densities of linear momentum and angular momentum are

(43)

where is calculated with respect to the origin of the reference frame. We note that the first term in the expression for is called the moment of momentum and the second term is called the proper angular momentum or, what is the same thing, the dynamic spin.

Next, we introduce the stress vector $\varvec{\tau }_n$ and the moment stress vector $\mathbf {T}_n$ modeling the surrounding medium action on surface S of elementary volume V. By standard reasoning, we introduce the concepts of the stress tensor $\varvec{\tau }$ associated with the stress vector $\varvec{\tau }_n$ and the moment stress tensor $\mathbf {T}$ associated with the moment stress vector $\mathbf {T}_n$. These tensors are defined by the relations $\varvec{\tau }_n = \mathbf {n} \cdot \varvec{\tau }$ and $\mathbf {T}_n = \mathbf {n} \cdot \mathbf {T}$, where $\mathbf {n}$ denotes the unit outer normal vector to surface S. Now, we can write the dynamics equations as

$$\begin{aligned} \nabla \cdot \varvec{\tau } + \rho \mathbf {f} = \rho \frac{\hbox {d} \mathbf {v}}{\hbox {d} t}, \qquad \nabla \cdot \mathbf {T} + \varvec{\tau }_{\times } + \rho \mathbf {L} = \rho \, \frac{\hbox {d} (J \varvec{\omega })}{\hbox {d} t}\,, \end{aligned}$$

(44)

where $\mathbf {f}$ and $\mathbf {L}$ are the mass densities of external forces and moments, respectively, $(\ )_{\times }$ denotes the vector invariant of a tensor that is defined for an arbitrary dyad as $(\mathbf {a} \mathbf {b})_{\times } = \mathbf {a} \times \mathbf {b}$. We note that, in Eq. (44) the mass density $\rho$ is assumed to be equal to its reference value.

Under the assumption that the energy supply from external sources is absent, the energy balance equation takes the form

(45)

where is the internal energy per unit mass, the double contraction is defined as $\mathbf {a} \mathbf {b} \cdot \cdot \, \mathbf {c} \mathbf {d} = (\mathbf {b} \cdot \mathbf {c})(\mathbf {a} \cdot \mathbf {d})$, the vector product of a second rank tensor and a vector is defined as follows: if $\,\mathbf {A} = \mathbf {a} \mathbf {b}$ then $\mathbf {A} \times \mathbf {c} = \mathbf {a} \mathbf {b} \times \mathbf {c} = \mathbf {a} (\mathbf {b} \times \mathbf {c})$. Substituting Eq. (40) into Eq. (45) and introducing the notation

$$\begin{aligned} \varvec{\varepsilon } = \nabla \mathbf {u} + \mathbf {E} \times \varvec{\theta }, \qquad \varvec{\varTheta } = \nabla \varvec{\theta }, \end{aligned}$$

(46)

we obtain

(47)

where tensors $\varvec{\varepsilon }$ and $\varvec{\varTheta }$, defined by Eq. (46), are called the strain tensors.

We note that from Eqs. (40), (46) it follows that

$$\begin{aligned} \frac{\hbox {d} \varvec{\varepsilon }}{\hbox {d} t} = \nabla \mathbf {v} + \mathbf {E} \times \varvec{\omega }, \qquad \frac{\hbox {d} \varvec{\varTheta }}{\hbox {d} t} = \nabla \varvec{\omega }. \end{aligned}$$

(48)

Equations that directly relate velocities and strain tensors are more convenient for our purposes. Therefore, below we will use Eq. (48) instead of Eqs. (40) and (46).

3.3 Simplifying assumptions and the constitutive equations

Now, we make a few simplifying assumptions that allow us to construct a mathematical model of the Cosserat continuum of a special type. In Sect. 4, we will introduce thermodynamic and electrodynamic analogues and give a physical interpretation of the model.

Hypothesis 1

The velocity vector $\mathbf {v}$, the external body force $\mathbf {f}$ and the stress tensor $\varvec{\tau }$ are equal to zero:

$$\begin{aligned} \mathbf {v} = 0, \qquad \mathbf {f} = 0, \qquad \varvec{\tau } = 0. \end{aligned}$$

(49)

Thus, below we deal with the Cosserat continuum, the particles of which possess only rotational degrees of freedom and are able to interact with each other only through the moments.

Hypothesis 2

The moment stress tensor $\mathbf {T}$ has the following structure:

$$\begin{aligned} \mathbf {T} = T \mathbf {E} - \mathbf {M} \times \mathbf {E}, \end{aligned}$$

(50)

where the scalar quantity T characterizes the spherical part of tensor $\mathbf {T}$ and the vector quantity $\mathbf {M}$ characterizes the antisymmetric part of tensor $\mathbf {T}$.

Assuming this structure of the moment stress tensor, we exactly follow the ideas of our previous works, see [54, 55, 60, 62]. The physical meaning of representation (50) will be clarified below.

Hypothesis 3

The external moment $\mathbf {L}$ is the moment of linear viscous damping proportional to the proper angular momentum:

$$\begin{aligned} \mathbf {L} = - \beta J \varvec{\omega }. \end{aligned}$$

(51)

Here $\beta$ is the coefficient of damping. In the linear theory this quantity is assumed to be constant.

In fact, moment $\mathbf {L}$ models the dissipation of energy of the continuum caused by its interaction with some other continuum, which is ignored in the model. The structure of moment $\mathbf {L}$ is chosen in accordance with the results obtained by solving two model problems, see [52, 53].

Hypothesis 4

The constitutive equation for the spherical part of tensor $\mathbf {T}$ has the following form:

$$\begin{aligned} \frac{1}{C_\varTheta }\, \frac{\mathrm{d} T}{\mathrm{d} t} + \frac{1}{\eta _\varTheta }\,(T - T_*) = \frac{\mathrm{d} \varTheta _\rho }{\mathrm{d} t}, \qquad \varTheta _\rho = \frac{\varTheta }{\rho }, \qquad \varTheta = \mathrm{tr}\,\varvec{\varTheta }, \end{aligned}$$

(52)

where $C_\varTheta$ is the stiffness parameter, $\eta _\varTheta$ is the viscosity parameter, $T_*$ is the reference value of T. Both parameters $C_\varTheta$ and $\eta _\varTheta$ are assumed to be constant.

Hypothesis 5

The constitutive equation for vector $\mathbf {M}$ characterizing the antisymmetric part of tensor $\mathbf {T}$ has the following form:

$$\begin{aligned} \frac{1}{C_\varPsi }\, \frac{\mathrm{d} \mathbf {M}}{\mathrm{d} t} + \frac{1}{\eta _\varPsi }\, \mathbf {M} = \frac{\mathrm{d} \varvec{\varPsi }_\rho }{\mathrm{d} t}, \qquad \varvec{\varPsi }_\rho = \frac{\varvec{\varPsi }}{\rho }, \qquad \varvec{\varPsi } = \varvec{\varTheta }_\times , \end{aligned}$$

(53)

where $C_\varPsi$ and $\eta _\varPsi$ are the stiffness and viscosity parameters, respectively. They do not depend on time and space coordinates.

According to constitutive equations (52), (53), the rheology of the continuum under consideration is similar to the classical Maxwell’s model of a viscoelastic material. The only difference is that the considered continuum is based on rotational degrees of freedom.

3.4 The system of the basic equations and its consequences

Here, we present a set of equations that we are going to discuss in the next sections. First of all, we note that the system of the basic equations can be divided into two independent systems. The first system includes as follows: three constitutive equations, trace and vector invariant of the relation between the strain measure and the angular velocity, and the angular momentum balance equation. The second system consists of the deviatoric part of the relation between the strain measure and the angular velocity.

The constitutive equations can be rewritten as

$$\begin{aligned} \frac{1}{C_\varTheta }\, \frac{\hbox {d} T}{\hbox {d} t} + \frac{1}{\eta _\varTheta }\,(T - T_*) = \frac{\hbox {d} \varTheta _\rho }{\hbox {d} t}, \qquad \frac{\rho }{C_\varPsi }\, \frac{\hbox {d} \mathbf {M}}{\hbox {d} t} + \frac{\rho }{\eta _\varPsi }\, \mathbf {M} = \frac{\hbox {d} \varvec{\varPsi }}{\hbox {d} t}, \qquad \varvec{{\mathcal {K}}} = \rho J \varvec{\omega }, \end{aligned}$$

(54)

where $\varvec{{\mathcal {K}}}$ is the proper angular momentum per unit volume.

Trace and vector invariant of the relation between the strain measure and the angular velocity are

$$\begin{aligned} \nabla \cdot \varvec{\omega } = \rho \frac{\hbox {d} \varTheta _\rho }{\hbox {d} t}, \qquad \nabla \times \varvec{\omega } = \frac{\hbox {d} \varvec{\varPsi }}{\hbox {d} t}. \end{aligned}$$

(55)

In view of the definition of $\varvec{{\mathcal {K}}}$, the angular momentum balance equation takes the form

$$\begin{aligned} \nabla T - \nabla \times \mathbf {M} - \beta \varvec{{\mathcal {K}}} = \frac{\hbox {d} \varvec{{\mathcal {K}}}}{d t}. \end{aligned}$$

(56)

The second system of equations allows us to find deviator of the symmetric part of tensor $\varvec{\varTheta }$ when vector $\varvec{\omega }$ is already known. This system of equations can be represented in tensor form as

$$\begin{aligned} \frac{\hbox {d} (\mathrm{dev}\,\varvec{\varTheta }^s)}{\hbox {d} t} = \mathrm{dev}\,(\nabla \varvec{\omega })^s, \end{aligned}$$

(57)

where upper index s denotes the symmetric part of a tensor. Below we discuss only the first system. Discussion of the second system is beyond the scope of this work.

Now, we transform Eqs. (54–56). We note that from the second equation in (55) it follows that ${\displaystyle \frac{\hbox {d} (\nabla \cdot \varvec{\varPsi })}{d t}} = 0$. Integrating this equation in time under the zero initial condition, provides

$$\begin{aligned} \nabla \cdot \varvec{\varPsi } = 0. \end{aligned}$$

(58)

Taking into account Eq. (58), from the second equation in (54) we obtain ${\displaystyle \frac{\hbox {d} (\nabla \cdot \mathbf {M})}{\hbox {d} t} + \frac{C_\varPsi }{\eta _\varPsi }\, \nabla \cdot \mathbf {M} = 0}$. Solving this equation for $\nabla \cdot \mathbf {M}$ under the zero initial condition, we have

$$\begin{aligned} \nabla \cdot \mathbf {M} = 0. \end{aligned}$$

(59)

Next, we consider the second and the third equations in (54) and the second equation in (58). Eliminating vectors $\varvec{\varPsi }$ and $\varvec{\omega }$ from these equations, we obtain

$$\begin{aligned} \nabla \times \varvec{{\mathcal {K}}} = \frac{\rho ^2 J}{C_\varPsi }\, \frac{\hbox {d} \mathbf {M}}{\hbox {d} t} + \frac{\rho ^2 J}{\eta _\varPsi }\, \mathbf {M}. \end{aligned}$$

(60)

Taking the curl of Eq. (56) in view of Eq. (59) yields

$$\begin{aligned} \Delta \mathbf {M} = \beta \nabla \times \varvec{{\mathcal {K}}} + \frac{\hbox {d} (\nabla \times \varvec{{\mathcal {K}}})}{\hbox {d} t}. \end{aligned}$$

(61)

Now, we can eliminate $\nabla \times \varvec{{\mathcal {K}}}$ from Eqs. (60), (61). As a result, we obtain the telegrapher’s equation

$$\begin{aligned} \Delta \mathbf {M} = \frac{\rho ^2 J}{C_\varPsi }\, \frac{\hbox {d}^2 \mathbf {M}}{\hbox {d} t^2} + \left( \frac{\rho ^2 J}{\eta _\varPsi } + \frac{\beta \rho ^2 J}{C_\varPsi }\right) \frac{\hbox {d} \mathbf {M}}{\hbox {d} t} + \frac{\beta \rho ^2 J}{\eta _\varPsi }\, \mathbf {M}. \end{aligned}$$

(62)

Eliminating vector $\mathbf {M}$ from Eqs. (60), (61), we arrive at exactly the same equation in terms of $\nabla \times \varvec{{\mathcal {K}}}$:

$$\begin{aligned} \Delta (\nabla \times \varvec{{\mathcal {K}}}) = \frac{\rho ^2 J}{C_\varPsi }\, \frac{\hbox {d}^2 (\nabla \times \varvec{{\mathcal {K}}})}{d t^2} + \left( \frac{\rho ^2 J}{\eta _\varPsi } + \frac{\beta \rho ^2 J}{C_\varPsi }\right) \frac{\hbox {d} (\nabla \times \varvec{{\mathcal {K}}})}{\hbox {d} t} + \frac{\beta \rho ^2 J}{\eta _\varPsi }\, \nabla \times \varvec{{\mathcal {K}}}. \end{aligned}$$

(63)

It is not difficult to show that, in view of the first and the third equations in (54), the first equation in (55) takes the form

$$\begin{aligned} \nabla \cdot \varvec{{\mathcal {K}}} = \frac{\rho ^2 J}{C_\varTheta }\, \frac{\hbox {d} T}{\hbox {d} t} + \frac{\rho ^2 J}{\eta _\varTheta }\,(T - T_*). \end{aligned}$$

(64)

Taking the divergence of Eq. (56) with regard to Eq. (64), we obtain

$$\begin{aligned} \Delta T = \frac{\rho ^2 J}{C_\varTheta }\, \frac{\hbox {d}^2 T}{\hbox {d} t^2} + \left( \frac{\rho ^2 J}{\eta _\varTheta } + \frac{\beta \rho ^2 J}{C_\varTheta }\right) \frac{\hbox {d} T}{\hbox {d} t} + \frac{\beta \rho ^2 J}{\eta _\varTheta }\, (T - T_*). \end{aligned}$$

(65)

Eliminating T from Eqs. (64), (65) yields the telegrapher’s equation in terms of $\nabla \cdot \varvec{{\mathcal {K}}}$:

$$\begin{aligned} \Delta (\nabla \cdot \varvec{{\mathcal {K}}}) = \frac{\rho ^2 J}{C_\varTheta }\, \frac{\hbox {d}^2 (\nabla \cdot \varvec{{\mathcal {K}}})}{\hbox {d} t^2} + \left( \frac{\rho ^2 J}{\eta _\varTheta } + \frac{\beta \rho ^2 J}{C_\varTheta }\right) \frac{\hbox {d} (\nabla \cdot \varvec{{\mathcal {K}}})}{\hbox {d} t} + \frac{\beta \rho ^2 J}{\eta _\varTheta }\, \nabla \cdot \varvec{{\mathcal {K}}}, \end{aligned}$$

(66)

which has exactly the same coefficients as Eq. (65).

Thus, we have reduced Eqs. (54–56) to two telegrapher’s equations. One of them can be written in terms of vector $\mathbf {M}$ or in terms of vector $\nabla \times \varvec{{\mathcal {K}}}$. It describes the bending wave propagation. Another one can be written in terms of scalar T or scalar $\nabla \cdot \varvec{{\mathcal {K}}}$. It describes the torsion wave propagation. We use terms “bending wave” and “torsion wave” following the terminology generally accepted in the rod theory (the one-dimensional Cosserat continuum) and the shell theory (the two-dimensional Cosserat continuum).

4 A physical interpretation of the constructed theory

4.1 Mechanical analogues of thermodynamic and electromagnetic quantities

Following the ideas of [51,52,53,54], we interpret quantity T as a mechanical analogue of temperature and quantity $\varTheta _\rho$ as a mechanical analogue of the specific entropy. Thus, we suppose that quantities T and $\varTheta _\rho$ are related to the absolute temperature $T_\mathrm{a}$ and the specific entropy $\varTheta _\mathrm{a}$ by the formulas

$$\begin{aligned} T = a T_\mathrm{a}, \qquad \varTheta _\rho = \frac{1}{a}\, \varTheta _\mathrm{a}, \end{aligned}$$

(67)

where a is the normalization factor. We are forced to introduce the normalization factor since the units of measurement of the temperature analogue and the entropy analogue are different from the standard units of measurement of temperature and entropy.

Following the ideas of [58,59,60], we introduce the analogues of electromagnetic quantities as follows: the moment stress vector $\mathbf {M}$ is the analogue of the electric field vector ; the volume density of proper angular momentum $\varvec{{\mathcal {K}}}$ is the analogue of the magnetic induction vector ; the strain measure $\varvec{\varPsi }$ is the analogue of the electric induction vector ; the angular velocity vector $\varvec{\omega }$ is the analogue of the magnetic field vector . Thus, we suppose that

(68)

where $\chi$ is the normalization factor. We note that the normalization factor a in Eq. (67) and the normalization factor $\chi$ in Eq. (68) play the same role as the Boltzmann constant (the proportionality factor between the average relative kinetic energy and temperature) in the kinetic theory.

4.2 The elastic continuum without the external friction

We start with the simplest theory, where the external body moment $\mathbf {L}$ is assumed to be equal to zero, or what is the same thing, the coefficient of damping $\beta$ is assumed to be zero, and the viscous terms in constitutive equations (54) are ignored. The latter means that we replace Eq. (54) by the following constitutive equations

$$\begin{aligned} T = T_* + C_\varTheta \, \varTheta _\rho , \qquad \mathbf {M} = \frac{C_\varPsi }{\rho }\, \varvec{\varPsi }, \qquad \varvec{{\mathcal {K}}} = \rho J \varvec{\omega }, \end{aligned}$$

(69)

which describe elastic behavior of the continuum. Of course, later we return to the general theory. Inserting Eqs. (67), (68) into Eqs. (55), (56), (69), we obtain the following results. Constitutive equations (54) take the form

(70)

In view of Eq. (70) kinematic relations (55) is written as

(71)

For vanishing $\beta$ the angular momentum balance equation (56) assumes the form

(72)

Now, we can express the mechanical parameters in Eqs. (70), (71) in terms of the corresponding physical parameters.

First of all, we turn to electrodynamics. Comparing the second and the third equations in (70) with the well-known constitutive equations and , we obtain

$$\begin{aligned} C_\varPsi = \frac{\chi ^2 \rho }{\varepsilon \varepsilon _0}, \qquad \rho J = \chi ^2 \mu \mu _0. \end{aligned}$$

(73)

Next, we consider the expression for thermal energy per unit mass $U_T$ that corresponds to the first equation in (70), namely

$$\begin{aligned} T_\mathrm{a} = T_\mathrm{a}^* + \frac{C_\varTheta }{a^2}\, \varTheta _\mathrm{a}, \quad T_\mathrm{a} = \frac{\partial U_T}{\partial \varTheta _\mathrm{a}} \quad \Rightarrow \quad U_T = T_\mathrm{a}^* \varTheta _\mathrm{a} + \frac{1}{2}\,\frac{C_\varTheta }{a^2}\, \varTheta _\mathrm{a}^2. \end{aligned}$$

(74)

From Eq. (74) it follows that

$$\begin{aligned} U_T = \frac{1}{2}\,\frac{a^2}{C_\varTheta }\left( T_\mathrm{a}^2 - (T_\mathrm{a}^*)^2\right) . \end{aligned}$$

(75)

By the definition, the specific heat at constant volume $c_v$ is calculated as

$$\begin{aligned} c_v = \left. \frac{\partial U_T}{\partial T_\mathrm{a}} \right| _{T_\mathrm{a} = T_\mathrm{a}^*} \quad \Rightarrow \quad c_v = \frac{a^2 T_\mathrm{a}^*}{C_\varTheta }. \end{aligned}$$

(76)

As a result, we arrive at the following expression for stiffness $C_\varTheta$:

$$\begin{aligned} C_\varTheta = \frac{a^2 T_\mathrm{a}^*}{c_v}. \end{aligned}$$

(77)

Thus, we have obtained the expressions for the inertia and stiffness parameters of the model. Taking into account Eqs. (73), (77), we can rewrite Eq. (71) as

(78)

Now, we have the closed system of Eqs. (72), (78). Comparing Eq. (72) with the Maxwell–Faraday equation, we see that the only difference is the presence of the term containing $\nabla T_\mathrm{a}$ in Eq. (72). Thus, we treat Eq. (72) as the generalized Maxwell–Faraday equation. The first equation in (78) gives the relation between the potential part of vector and the rate of change of the absolute temperature. The second equation in (78) is Maxwell’s first equation, but without the electric current density. Thus, we conclude that, in fact, only the vortex part of vector can be interpreted as magnetic induction. The potential part of vector has the thermodynamic meaning. This is an important difference between the proposed theory and classical electrodynamics, where the potential part of vector is assumed to be zero.

Taking the curl of Eq. (72) and eliminating vector from the obtained equation in view of the second equation in (78), we obtain the following wave equation for :

(79)

Taking the divergence of Eq. (72) and eliminating by using the first equation in (78), we arrive at the wave equation for $T_\mathrm{a}$:

$$\begin{aligned} \Delta T_\mathrm{a} = \frac{\chi ^2}{a^2}\,\frac{\rho c_v \mu \mu _0}{T_\mathrm{a}^*}\, \frac{\hbox {d}^2 T_\mathrm{a}}{\hbox {d} t^2}. \end{aligned}$$

(80)

Next, eliminating and $T_\mathrm{a}$ from Eqs. (72), (78) yields

(81)

From Eq. (81) one can obtain two wave equations for :

(82)

It is easy to see that the first equation in (82) coincides with Eq. (79) and the second one coincides with Eq. (80). Thus, there are two wave propagation velocities in the considered system, namely, the bending wave velocity c and the torsional wave velocity $c_r$, which are calculated as

$$\begin{aligned} c^2 = \frac{1}{\varepsilon \varepsilon _0 \mu \mu _0}, \qquad c_r^2 = \frac{a^2}{\chi ^2} \frac{T_\mathrm{a}^*}{\rho c_v \mu \mu _0}. \end{aligned}$$

(83)

The bending wave velocity c is associated with the electromagnetic waves, whereas the torsional wave velocity $c_r$ is associated with the heat waves.

Thus, considering the elastic continuum without the external friction, we have determined the inertia and stiffness parameters of our model. It is important to note that this simplest theory describes a medium that does not possess the properties of electrical conductivity and thermal conductivity. To develop a theory of a conductive medium we should take into account either the external friction modeled by the body moment in the equation of motion or the internal friction introduced by means of the constitutive equations. Below, we discuss these two theories and the theory taking into account both types of friction. We emphasize that, in all theories, we use the inertia and stiffness parameters identified in this section.

4.3 The mechanical interpretations of two types of thermal conductivity and electrical conductivity

Using the above mechanical analogues we can obtain the telegrapher’s equations for temperature and the electric field vector under the condition that we take into account both types of friction. If we take into account only one type of friction we can obtain the hyperbolic type heat conduction equation and the analogous equation for the electric field vector. Let us stress that for any of two types of friction we will arrive at the equations having the identical structure, but coefficients in these equations will be different. Certainly, these coefficients will depend on the thermal and electrical conductivities in any case, but their mechanical interpretation will be different. In order to clarify this issue, below we discuss the physical meaning of friction in our model.

So, our mechanical model includes two completely different types of friction. One of them is presented by the external body moment in the angular momentum balance equation. We believe, that this type of friction is caused by the interaction between the ether associated with the ponderable matter and the free ether occupying the entire infinite space. It is due to such an interaction that some part of the ether energy associated with the ponderable matter is carried away to infinity. The energy does not return because in the free ether at the infinity there are no energy sources. This is the way how the first type of friction leads to the dissipation of the energy of the ether associated with the ponderable matter. The model problems illustrating this type of friction can be found in [52, 53, 55]. Another type of friction is the internal friction in the ether associated with the ponderable matter. We introduce this friction by the constitutive equations. In our opinion, this type of friction occurs due to the fact that waves emitted by particles of the ponderable matter are scattered on other particles of the ponderable matter. According to the proposed mechanical model, the thermal and the electrical conduction processes are caused by both the aforementioned dissipative processes in the ether. Each type of dissipation in the mechanical model can be matched to the corresponding coefficients of thermal conductivity and electrical conductivity.

Let us return to the hyperbolic type equations for temperature and the electric field vector. It is difficult to assume a priori, whether one of the considered types of friction will be preferable for obtaining these equations. Perhaps the use of any of them allows us to obtain physically reasonable results. Perhaps none of them will provide a satisfactory result. In the latter case, it will be necessary to use the model with two types of friction not only to arrive at the telegrapher’s equations, but even to obtain the hyperbolic type equations. It is clear that more research is needed to answer the question of what type of friction should be used to obtain adequate hyperbolic-type models describing the thermal and electrical conductivities. That is why, below we consider three different models: one with the external friction, another one with the internal friction, and the model incorporating both types of friction.

4.4 The elastic continuum with the external friction

Now, we consider the simplified theory, where the viscous terms in the constitutive equations (54) are ignored, but external body moment $\mathbf {L}$ is not zero and determined by Eq. (51). In this case, the angular momentum balance equation (56) takes the form

(84)

The constitutive equations and the kinematic relations have the form of Eqs. (70) and (71), respectively.

In order to arrive at the heat conduction equation, we take the divergence of Eq. (84) and eliminate vector from the obtained equation by using the first equation in (71). As a result, taking into account expressions for parameters (73), (77), (83), we obtain

$$\begin{aligned} \Delta T_\mathrm{a} = \frac{\beta }{c_r^2}\, \frac{\hbox {d} T_\mathrm{a}}{\hbox {d} t} + \frac{1}{c_r^2}\, \frac{\hbox {d}^2 T_\mathrm{a}}{\hbox {d} t^2}. \end{aligned}$$

(85)

Comparing Eq. (85) with the hyperbolic type heat conduction equation (37), we conclude that

$$\begin{aligned} \frac{\beta }{c_r^2} = \frac{\rho c_v}{\lambda }. \end{aligned}$$

(86)

In view of Eq. (86), the hyperbolic type heat conduction equation (85) takes the form

$$\begin{aligned} \Delta T_\mathrm{a} = \frac{\rho c_v}{\lambda } \frac{\hbox {d} T_\mathrm{a}}{\hbox {d} t} + \frac{\chi ^2}{a^2} \frac{\rho c_v \mu \mu _0}{T_\mathrm{a}^*} \frac{\hbox {d}^2 T_\mathrm{a}}{d t^2}. \end{aligned}$$

(87)

We note that from Eqs. (83), (86) it follows that

$$\begin{aligned} \beta = \frac{a^2}{\chi ^2} \frac{T_\mathrm{a}^*}{\lambda \mu \mu _0}. \end{aligned}$$

(88)

Inserting expression (88) for parameter $\beta$ into Eq. (84) yields

(89)

Equation (89) is a generalization of Eq. (72) and it can also be treated as the generalized Maxwell–Faraday equation. Taking the curl of Eq. (89) and eliminating vector from the obtained equation by means of the second equation in (78), we have

(90)

This equation can be rewritten in view of the Wiedemann–Franz law (35) as

(91)

Thus, in the framework of our simplified model we have obtained the hyperbolic type heat conduction equation (87) and equation (91) describing the electrodynamic processes in conducting matter. We note that the term containing the first time derivative of vector appears in Eq. (91) due to the presence of the term proportional to vector in the generalized Maxwell–Faraday equation (89), not due to the presence of the electric current density in Maxwell’s first equation. Indeed, the second equation in (78), which has the meaning of Maxwell’s first equation, does not contain any terms that can be interpreted as the electric current density.

It is easy to see that Eq. (91) has the same structure as Eq. (11), but the coefficient at the first time derivative of vector in Eq. (91) differs from the corresponding coefficient in Eq. (11). At the same time, the coefficient in Eq. (91) is similar to that coefficient in telegrapher’s equation (23), which contains the electrical conductivity of conductor $\sigma _c$, namely coefficient ${\displaystyle \frac{\varepsilon _d \varepsilon _0}{\sigma _c l \delta }}$ in Eq. (23). However, in order to obtain the correct formula (13) for the skin depth, we are forced to require that the coefficient at the first time derivative of vector in Eq. (91) is equal to the corresponding coefficient in Eq. (11):

$$\begin{aligned} \frac{\varepsilon \varepsilon _0}{\sigma (L \chi ^2/a^2)} = \sigma \mu \mu _0. \end{aligned}$$

(92)

This relation between the parameters seems to be inadequate. Therefore, the simplified theory considered in this section should be recognized as unacceptable.

4.5 The viscoelastic continuum without the external friction

We return to the assumption $\beta = 0$ and use the simplified form of the angular momentum balance equation (72). However, now we take into account the viscous terms in Eq. (54). In view of Eqs. (67), (68), constitutive equations (54) take the form

(93)

In view of Eqs. (67), (68), (93), kinematic relations (55) can be rewritten as

(94)

Thus, we deal with the system of Eqs. (72), (93), (94). Certainly, in the case of the viscoelastic continuum, we can use the expressions for the parameters obtained in the case of the elastic continuum. We mean Eqs. (73), (77), and (83). However, in the case of the viscoelastic continuum, two additional parameters appear, namely, the viscosity parameters $\eta _\varTheta$ and $\eta _\varPsi$. Below we associate these parameters with the thermal and electrical conductivities.

Taking the curl of Eq. (72) and eliminating vector by means of the second equation in (94), we obtain

(95)

Taking into account Eq. (73) and specifying parameter $\eta _\varPsi$ as

$$\begin{aligned} \eta _\varPsi = \frac{\chi ^2 \rho }{\sigma }, \end{aligned}$$

(96)

we reduce Eq. (95) to the form

(97)

It is easy to see that the first equation in (97) coincides with the first equation in (11). Solving the second equation in (97) for in view of the initial condition , we obtain the second equation in (11).

Taking the divergence of Eq. (72) and eliminating by means of the first equation in (94), we obtain

$$\begin{aligned} \Delta T_\mathrm{a} = \frac{\rho ^2 J}{\eta _\varTheta } \frac{\hbox {d} T_\mathrm{a}}{\hbox {d} t} + \frac{\rho ^2 J}{C_\varTheta } \frac{\hbox {d}^2 T_\mathrm{a}}{\hbox {d} t^2}. \end{aligned}$$

(98)

Taking into account Eqs. (73), (77) and specifying parameter $\eta _\varTheta$ as

$$\begin{aligned} \eta _\varTheta = \frac{\chi ^2 \lambda \mu \mu _0}{c_v}, \end{aligned}$$

(99)

we reduce Eq. (98) to the hyperbolic type heat conduction equation (87). This equation contains the same coefficient at the first time derivative of temperature as in the classical heat conduction equation. As for the coefficient at the second time derivative, it is difficult to assess its correctness, since the experimental values of the propagation velocity of heat waves differ significantly (sometimes by several orders of magnitude).

The approximate theory considered in this section contains two independent viscosity parameters. Thanks to this, it turns out to be possible to obtain quite adequate parameters both in the heat conduction equation and in the equation for the electric field vector. However, in view of the Wiedemann–Franz law (35), from Eqs. (96), (99) it follows that

$$\begin{aligned} \eta _\varTheta \eta _\varPsi = \frac{L \chi ^4 \rho T_\mathrm{a}^* \mu \mu _0}{c_v}. \end{aligned}$$

(100)

It seems logical that the Wiedemann–Franz law should impose some restriction on the ratio of viscosity coefficients $\eta _\varPsi / \eta _\varTheta$, but not on the product $\eta _\varTheta \eta _\varPsi$. Therefore, relation (100) seems strange.

Let us reject Eq. (99) and try to reason differently. Let us assume that

$$\begin{aligned} \eta _\varTheta = \eta _\varPsi . \end{aligned}$$

(101)

Then, in view of Eqs. (96), (101) and the Wiedemann–Franz law (35), we have

$$\begin{aligned} \eta _\varTheta = \frac{L \chi ^2 \rho T_\mathrm{a}^*}{\lambda }. \end{aligned}$$

(102)

Inserting Eqs. (73), (77), (102) into Eq. (98) yields

$$\begin{aligned} \Delta T_\mathrm{a} = \frac{\lambda \mu \mu _0}{L T_\mathrm{a}^*} \frac{\hbox {d} T_\mathrm{a}}{\hbox {d} t} + \frac{\chi ^2}{a^2} \frac{\rho c_v \mu \mu _0}{T_\mathrm{a}^*} \frac{\hbox {d}^2 T_\mathrm{a}}{d t^2}. \end{aligned}$$

(103)

Thus, assuming that $\eta _\varTheta = \eta _\varPsi$, we obtain almost the same results as in the previous section. The only difference is that, in this section, we arrive at the equations where $\lambda$ and $\sigma$ are in the numerators of the coefficients at the first time derivatives of temperature and the electric field vector, respectively, whereas in the previous section we obtain the equations where $\lambda$ and $\sigma$ are in the denominators of the coefficients at the corresponding first time derivatives. An attempt to obtain both the equations with the correct coefficients results in relation (100), the physical meaning of which is unclear. That is why, we conclude that this simplified theory, as well as the simplified theory considered in the previous section, is unacceptable.

4.6 The viscoelastic continuum with the external friction

None of the simplified theories discussed above give us satisfactory results. Therefore, now we turn to the general theory. Thus, we deal with the system of Eqs. (84), (93), (94).

Taking the curl of Eq. (84) and eliminating vector by means of the second equation in (94), we obtain

(104)

Taking the divergence of Eq. (84) and eliminating by means of the first equation in (94), we have

$$\begin{aligned} \Delta T_\mathrm{a} = \frac{\rho ^2 J}{C_\varTheta } \frac{\hbox {d}^2 T_\mathrm{a}}{\hbox {d} t^2} + \left( \frac{\rho ^2 J}{\eta _\varTheta } + \frac{\beta \rho ^2 J}{C_\varTheta }\right) \frac{\hbox {d} T_\mathrm{a}}{\hbox {d} t} + \frac{\beta \rho ^2 J}{\eta _\varTheta }\, (T_\mathrm{a} - T_\mathrm{a}^*). \end{aligned}$$

(105)

We note that the equation for is exactly the same as Eq. (104) and the equation for is exactly the same as Eq. (105). The last statement follows from Eqs. (63), (66) and relation .

Above we have identified all the parameters in Eqs. (104), (105). In order to avoid ambiguity, we will change some notation used in the previous sections. To be precise, we keep the notation for parameters given by Eqs. (73), (77), (83) unchanged and change the notations for the thermal and electrical conductivities. We replace $\sigma$ and $\lambda$ by $\sigma _c$ and $\lambda _c$, respectively, in the formulas obtained in the framework of the model with the external friction. As a result, expression (88) for parameter $\beta$ takes the form

$$\begin{aligned} \beta = \frac{a^2}{\chi ^2} \frac{T_\mathrm{a}^*}{\lambda _c \mu \mu _0}. \end{aligned}$$

(106)

We replace $\sigma$ and $\lambda$ by $\sigma _d$ and $\lambda _d$, respectively, in the formulas obtained in the framework of the model with the internal friction. Thus, we rewrite Eqs. (96) and (102) as follows:

$$\begin{aligned} \eta _\varPsi = \frac{\chi ^2 \rho }{\sigma _d}, \qquad \eta _\varTheta = \frac{L \chi ^2 \rho T_\mathrm{a}^*}{\lambda _d}. \end{aligned}$$

(107)

In the general theory, we suppose that $\lambda _d$ and $\sigma _d$ are the independent parameters, whereas $\lambda _c$ and $\sigma _c$ are related by the Wiedemann–Franz law:

$$\begin{aligned} \frac{\lambda _c}{\sigma _c} = L T_\mathrm{a}^*. \end{aligned}$$

(108)

It seems to be logical, because the internal friction is characterized by two parameters, namely $\eta _\varPsi$ and $\eta _\varTheta$, whereas the external friction is characterized by only one parameter $\beta$.

Thus, taking into account Eqs. (73), (77), (106), (107), (108), we can rewrite Eq. (104) as

(109)

and Eq. (105) in the form

$$\begin{aligned} \Delta T_\mathrm{a} = \frac{\chi ^2}{a^2} \frac{\rho c_v \mu \mu _0}{T_\mathrm{a}^*} \frac{\hbox {d}^2 T_\mathrm{a}}{\hbox {d} t^2} + \left( \frac{\lambda _d \mu \mu _0}{L T_\mathrm{a}^*} + \frac{\rho c_v}{\lambda _c}\right) \frac{\hbox {d} T_\mathrm{a}}{\hbox {d} t} + \frac{\lambda _d}{\lambda _c (L \chi ^2/a^2)}\, (T_\mathrm{a} - T_\mathrm{a}^*). \end{aligned}$$

(110)

We note that in order to pass from the general equations (109), (110) describing the propagation of electromagnetic and heat waves with losses to the equations describing the propagation of these waves without losses, we must set the thermal and electrical conductivities corresponding to the internal friction equal to zero, and simultaneously let the thermal and electrical conductivities corresponding to the external friction tend to infinity.

In view of Eqs. (67), (68) and expressions for parameters (73), (77), (106), (107), (108) we can reduce Eq. (60) to the form

(111)

and Eq. (64) to the form

(112)

We note that vector satisfies the telegrapher’s equation with exactly the same coefficients as in Eq. (109):

(113)

and the scalar function satisfies the telegrapher’s equation with exactly the same coefficients as in Eq. (110):

(114)

In fact, we have introduced two different quantities, $\sigma _d$ and $\sigma _c$, characterizing electrical conductivity and two different quantities, $\lambda _d$ and $\lambda _c$, characterizing thermal conductivity. Precisely because of this, we arrive at Eqs. (109), (110) that have exactly the same structure as Eq. (23) obtained for the transmission line. We note that we use notations $\sigma _d$, $\sigma _c$ for the electrical conductivities and $\lambda _d$, $\lambda _c$ for the thermal conductivities only to make it easier to compare Eqs. (109), (110) with Eq. (23). We do not attribute the meaning of the electrical and thermal conductivities of a dielectric to $\sigma _d$ and $\lambda _d$, as well as we do not attribute the meaning of the electrical and thermal conductivities of a conductor to $\sigma _c$ and $\lambda _c$. We emphasize that Eqs. (106), (107), (108) do not determine the viscosity parameters. These equations allow us to replace the unknown viscosity parameters with other unknown parameters, which have the dimensions of electrical conductivity and thermal conductivity.

4.7 The parameters of the viscoelastic continuum with the external friction

In order to determine parameters $\sigma _d$, $\sigma _c$ and $\lambda _d$, $\lambda _c$, we should express them in terms of the known values of electrical conductivity $\sigma$, thermal conductivity $\lambda$, and probably other some physical quantities that can be found experimentally.

We begin an analysis of Eqs. (109), (110) by considering the following particular case:

$$\begin{aligned} \sigma _d = \sigma _c = \sigma , \qquad \lambda _d = \lambda _c = \lambda . \end{aligned}$$

(115)

Taking into account assumption (115) and relations between parameters (83) and (108), we can rewrite Eq. (109) as

(116)

and Eq. (110) as

$$\begin{aligned} \Delta T_\mathrm{a} = \frac{1}{c_r^2} \frac{\hbox {d}^2 T_\mathrm{a}}{\hbox {d} t^2} + \left( \sigma \mu \mu _0 + \frac{\rho c_v}{\lambda }\right) \frac{\hbox {d} T_\mathrm{a}}{\hbox {d} t} + \frac{1}{L \chi ^2/a^2}\, (T_\mathrm{a} - T_\mathrm{a}^*). \end{aligned}$$

(117)

We note that the analysis of the physical constants data allows us to conclude that

$$\begin{aligned} \sigma \mu \mu _0 \ll \frac{\rho c_v}{\lambda }, \qquad \frac{c_r^2}{c^2} \ll 1, \qquad \sigma \mu \mu _0 \gg \frac{c_r^2}{c^2} \frac{\rho c_v}{\lambda }. \end{aligned}$$

(118)

The second inequality in (118) expresses the well-known fact that the heat wave velocity in matter is much less than the light velocity. In view of (118) we can use the following approximate form of Eq. (116):

(119)

and the following approximate form of Eq. (117):

$$\begin{aligned} \Delta T_\mathrm{a} = \frac{1}{c_r^2} \frac{\hbox {d}^2 T_\mathrm{a}}{\hbox {d} t^2} + \frac{\rho c_v}{\lambda } \frac{\hbox {d} T_\mathrm{a}}{\hbox {d} t} + \frac{1}{L \chi ^2/a^2}\, (T_\mathrm{a} - T_\mathrm{a}^*). \end{aligned}$$

(120)

Thus, we have obtained Eqs. (119), (120), which contain the correct coefficients at the first time derivatives of and $T_\mathrm{a}$. It is a matter of concern that these equations contain the same coefficients in the last terms, and these coefficients do not depend on the properties of the material. Therefore, we return to Eqs. (109), (110).

Equations (109), (110) contain four unknown parameters, $\sigma _c$, $\sigma _d$, $\lambda _c$, $\lambda _d$ and a fundamental constant $\sqrt{L \chi ^2/a^2}$. Now, we have only one relation, namely the Wiedemann–Franz law (108). Hence, we need four more relations. Certainly, we should use the electrical conductivity $\sigma$ and the thermal conductivity $\lambda$, the values of which can be found in handbooks. We suppose that the coefficient at the first time derivative of vector in Eq. (109) equals to $\sigma \mu \mu _0$ and the coefficient at the first time derivative of temperature in Eq. (110) equals to ${\displaystyle \frac{\rho c_v}{\lambda }}$. As a result, we arrive at the relations

$$\begin{aligned} \sigma \mu \mu _0 = \sigma _d \mu \mu _0 + \frac{\varepsilon \varepsilon _0}{\sigma _c (L \chi ^2/a^2)}, \qquad \frac{\rho c_v}{\lambda } = \frac{\lambda _d \mu \mu _0}{L T_\mathrm{a}^*} + \frac{\rho c_v}{\lambda _c}. \end{aligned}$$

(121)

We emphasize that the Wiedemann–Franz law (35), which relates the electrical and thermal conductivities, is associated only with the external friction, see Eq. (108). We believe that this is logical, since the external friction is determined by only one parameter, whereas the internal friction is characterized by two parameters. Certainly, this assumption leads to the fact that the Wiedemann–Franz law for $\sigma$ and $\lambda$ is fulfilled approximately. But this is in agreement with experimental data, according to which the Wiedemann–Franz law is fulfilled quite accurately above room temperature, whereas at lower temperatures significant deviations of ratio $\lambda / \sigma$ from this law can be observed. We note that, in particular case (115), the first term in the coefficient at the first time derivative of , see Eq. (116), is much greater than the second one, and the first term in the coefficient at the first time derivative of $T_\mathrm{a}$, see Eq. (117), is much smaller than the second one. We can suppose that the same relationships take place also in the general case. This means that $\sigma \approx \sigma _d$ and $\lambda \approx \lambda _c$. Since we formulate the Wiedemann–Franz law for ratio $\lambda _c / \sigma _c$, we can suppose that $\sigma _d \approx \sigma _c$.

In order to obtain two additional relations we consider ultra-low frequency electromagnetic and thermal processes. In this case, Eqs. (109), (110) turn into static equations

(122)

It is not difficult to show that from Eq. (122) it follows that the static skin depth for the electromagnetic field $\delta _{\displaystyle \varepsilon }$ and the static skin depth for the temperature field $\delta _T$ are determined as

$$\begin{aligned} \delta _{\displaystyle \varepsilon } = \sqrt{\frac{\sigma _c (L \chi ^2/a^2)}{\sigma _d}}, \qquad \delta _T = \sqrt{\frac{\lambda _c (L \chi ^2/a^2)}{\lambda _d}}. \end{aligned}$$

(123)

We do note that the static skin effect has nothing to do with the skin effect discussed in Sect. 2.1, see Eqs. (13), (14).

The static skin depths $\delta _{\displaystyle \varepsilon }$ and $\delta _T$ can be found experimentally. The static skin effect for the electromagnetic field was discovered a long time ago [77]; the experimental studies has been carried out at different times by many authors [78,79,80,81] and continue up to the present [82, 83]. The static skin effect for the temperature field was discovered quite recently; we know only a few experimental works, namely [84, 85].

Thus, Eqs. (108), (121), (123) allow us to find parameters $\sigma _c$, $\sigma _d$, $\lambda _c$, $\lambda _d$ and the fundamental constant $\sqrt{L \chi ^2/a^2}$. Certainly, we can consider the constructed theory correct if $\sqrt{L \chi ^2/a^2}$ turns out to be actually independent of the material properties. The fundamental constant can be calculated differently. One can use formula for the heat wave velocity (83), or, what is the same, use formula for quantity $\beta$ characterizing the external friction (88). Quantity $\beta ^{-1}$ coincides with the so-called relaxation time of the heat flux, i.e., it is the quantity that is usually given in works on the hyperbolic heat conduction. Certainly, the values of $\sqrt{L \chi ^2/a^2}$, obtained by the two above methods must be exactly the same. Unfortunately, the currently available experimental data do not allow us to determine the value of this fundamental constant. The point is that the generally accepted explanation of the static skin effect is based on the use of concepts of bulk conductivity and surface conductivity. Because of this, the experimental studies provide data on the bulk and surface conductivities instead of data on the skin depth. As for the experimental data on the relaxation time of the heat flux, its values given by different authors sometimes differ by several orders of magnitude. For example, in accordance with [86, 87], the experimentally determined relaxation time of the heat flux of metals is of the order of tens of nanoseconds, whereas according to [88,89,90] it is of the order of 0.01 nanoseconds. Apparently, the values of the relaxation time depend significantly on the measurement technique. For example, [91] shows that, depending on the type of scattering (phonon–phonon and electron–phonon), the relaxation time of the heat flux of gold is 12 and 1.52 picoseconds, respectively. We believe that such experimental data do not allow to estimate even the order of magnitude of $\sqrt{L \chi ^2/a^2}$ correctly. But we hope that in the future, when the experimental technique is improved, it will be possible to determine this fundamental constant.

4.8 The skin effects and modeling of various surface effects

First of all, we note that the static skin depths depend on the relative contribution of two different types of friction in the ether associated with the ponderable matter. Indeed, from Eqs. (73), (106), (107), (108), (123) it follows that

$$\begin{aligned} \delta _{\displaystyle \varepsilon } = \frac{1}{\rho } \sqrt{\frac{\eta _\varPsi }{ \beta J}}, \qquad \delta _T = \frac{1}{\rho } \sqrt{\frac{\eta _\varTheta }{ \beta J}}. \end{aligned}$$

(124)

As seen from Eq. (124), the static skin depths increase with increasing the internal friction and decrease with increasing the external friction.

In addition, we note that the dynamic skin depths also increase with increasing the internal friction and decrease with increasing the external friction. The above applies to both skin effects that occur at not too high frequencies, and the skin effects occurring at very high frequencies. It seems to us that such dependence can be explained as follows. The internal friction impedes wave processes, but promotes diffusion processes, since this friction is due to the scattering of the energy of the ether associated with the ponderable matter on particles of the matter. The external friction impedes both wave and diffusion processes, since this friction is caused by the radiation of the energy of the ether associated with the ponderable matter into the free ether.

Finally, we suggest that the proposed model describing the static skin effects will help to take a fresh look at the nature of various surface effects, and perhaps to find new approaches to their modeling. We mean both the well-known electric and thermoelectric effects that occur at the interface between two conductive materials (the Volta potential, the Seebeck effect, the Peltier effect), and the equally well-known thermomechanical and purely mechanical surface effects (the Joule–Thomson effect, surface tension, capillary action). We believe that the proposed equations can provide a more accurate description of

• the Volta potential: the electrostatic potential difference between two metals that are in contact and are at the same temperature;

• the Seebeck effect: a phenomenon in which a temperature difference between two dissimilar electrical conductors or semiconductors produces a voltage difference between the two substances;

• the Peltier effect: the cooling of one junction and the heating of the other one when electric current is maintained in a circuit consisted of two dissimilar conductors or semiconductors;

• the Joule–Thomson effect: the temperature change of a real gas or liquid when it is forced through a porous plug.

We also believe that models similar to the proposed one but based on translational degrees of freedom can be applied to describe surface tension and capillary action. Although it is possible that the proposed equations themselves may also be useful for describing these effects.

4.9 Asymptotic estimations for the telegrapher’s equations

Above, we have established how the measurable quantities $\delta _{\displaystyle \varepsilon }$ and $\delta _T$ relate to the model parameters. In this section, we discuss some asymptotic estimations for the telegrapher’s equations. We show that in different frequency ranges, the boundaries of which depend on quantities $\delta _{\displaystyle \varepsilon }$ and $\delta _T$, we can use different approximate equations. Some of these approximate equations coincide with the well-known equations for the electric field vector and temperature. The other approximate equations are novel.

In view of Eqs. (83), (121), (123) equation (109) for the electric field vector takes the form

(125)

and the heat conduction equation (110) writes

$$\begin{aligned} \Delta T_\mathrm{a} = \frac{1}{c_r^2}\frac{\hbox {d}^2 T_\mathrm{a}}{\hbox {d} t^2} + \frac{\rho c_v}{\lambda }\, \frac{\hbox {d} T_\mathrm{a}}{\hbox {d} t} + \frac{1}{\delta _T^2}\, (T_\mathrm{a} - T_\mathrm{a}^*). \end{aligned}$$

(126)

Taking into account Eqs. (121), (123), it is not difficult to show that the following estimation takes place for the parameters of Eq. (125):

$$\begin{aligned} \sigma \mu \mu _0 = \frac{1}{c\, \delta _{\displaystyle \varepsilon }} \left( \sigma _d \mu \mu _0\, c\, \delta _{\displaystyle \varepsilon } + \frac{1}{\sigma _d \mu \mu _0\, c\, \delta _{\displaystyle \varepsilon }}\right) \ge \frac{2}{c\, \delta _{\displaystyle \varepsilon }}, \end{aligned}$$

(127)

and the following estimation is valid for the parameters of Eq. (126):

$$\begin{aligned} \frac{\rho c_v}{\lambda } = \frac{1}{c_r \delta _T} \left( \frac{\lambda _c}{\rho c_v c_r \delta _T} + \frac{\rho c_v c_r \delta _T}{\lambda _c}\right) \ge \frac{2}{c_r \delta _T}. \end{aligned}$$

(128)

We note that estimations (127), (128) can be rewritten as

$$\begin{aligned} \frac{1}{\delta _{\displaystyle \varepsilon }^2 \sigma \mu \mu _0} \le c^2 \sigma \mu \mu _0, \qquad \frac{1}{\delta _T^2}\frac{\lambda }{\rho c_v} \le c_r^2\, \frac{\rho c_v}{\lambda }. \end{aligned}$$

(129)

Asymptotic analysis of Eqs. (125), (126) shows that at different signal frequencies $\omega$ we can use different approximate equations. For vector we have

(130)

The approximate equations for temperature are

$$\begin{aligned} \begin{array}{l} {\displaystyle \omega \gg c_r^2\, \frac{\rho c_v}{\lambda }: \,\,\,\qquad \qquad \qquad \Delta T_\mathrm{a} = \frac{1}{c_r^2} \frac{\hbox {d}^2 T_\mathrm{a}}{\hbox {d} t^2},} \\ {\displaystyle \frac{1}{\delta _T^2}\frac{\lambda }{\rho c_v} \ll \omega \sim c_r^2\, \frac{\rho c_v}{\lambda }: \qquad \Delta T_\mathrm{a} = \frac{1}{c_r^2} \frac{\hbox {d}^2 T_\mathrm{a}}{\hbox {d} t^2} + \frac{\rho c_v}{\lambda } \frac{\hbox {d} T_\mathrm{a}}{\hbox {d} t},} \\ {\displaystyle \frac{1}{\delta _T^2}\frac{\lambda }{\rho c_v} \ll \omega \ll c_r^2\, \frac{\rho c_v}{\lambda }: \qquad \Delta T_\mathrm{a} = \frac{\rho c_v}{\lambda } \frac{\hbox {d} T_\mathrm{a}}{\hbox {d} t},} \\ {\displaystyle \frac{1}{\delta _T^2}\frac{\lambda }{\rho c_v} \sim \omega \ll c_r^2\, \frac{\rho c_v}{\lambda }: \qquad \Delta T_\mathrm{a} = \frac{\rho c_v}{\lambda } \frac{\hbox {d} T_\mathrm{a}}{\hbox {d} t} + \frac{1}{\delta _T^2}\, (T_\mathrm{a} - T_\mathrm{a}^*),} \\ {\displaystyle \omega \ll \frac{1}{\delta _T^2}\frac{\lambda }{\rho c_v}: \qquad \qquad \qquad \Delta T_\mathrm{a} = \frac{1}{\delta _T^2}\, (T_\mathrm{a} - T_\mathrm{a}^*).} \end{array} \end{aligned}$$

(131)

We can be sure that for most conductive materials under normal conditions

$$\begin{aligned} \frac{1}{\delta _{\displaystyle \varepsilon }^2 \sigma \mu \mu _0} \ll c^2 \sigma \mu \mu _0, \qquad \frac{1}{\delta _T^2}\frac{\lambda }{\rho c_v} \ll c_r^2\, \frac{\rho c_v}{\lambda }, \end{aligned}$$

(132)

since validity of the third equations in (130) and (131) in the fairly wide frequency ranges confirmed by numerous experimental data. In this case, we do not need to use exact equations (125), (126), because in any frequency range we can find approximate equations, which have sufficient accuracy. In general, there may be materials with the parameters satisfying one of the following relations or both:

$$\begin{aligned} \frac{1}{\delta _{\displaystyle \varepsilon }^2 \sigma \mu \mu _0} \sim c^2 \sigma \mu \mu _0, \qquad \frac{1}{\delta _T^2}\frac{\lambda }{\rho c_v} \sim c_r^2\, \frac{\rho c_v}{\lambda }. \end{aligned}$$

(133)

In such a case, at certain frequencies, it is necessary to take into account all terms in the right-hand sides of the corresponding equations. We emphasize that, in accordance with estimations (129), there are no such parameters and such frequencies, under which we could asymptotically obtain an approximate equation with the second time derivative of the function and the function itself, without its first time derivative.

Now, we return to the approximate equations (130), (131). We note that the first three equations in both (130) and (131) are well-known: the first equations are the wave equations without damping; the second equation in (130) is the consequence of Maxwell’s equations and the second equation in (131) is the hyperbolic heat conduction equation (the Cattaneo-type heat conduction equation); the third equation in (130) is the approximate equation that is usually used to describe the skin effect in metals and the third equation in (131) is the classical heat conduction equation. The rest two equations in (130) and (131) are novel. Exact equations (125) and (126) are also novel.

5 A comparison of the obtained equations with the known physical laws

5.1 Electric current, Ohm’s law and the charge conservation law

In the proposed model, Ohm’s law describes the electrical conduction process that is associated with the internal friction in the ether. Indeed, the second equation in (94) can be rewritten as

(134)

Taking into account Eqs. (73), (107), we can reduce Eq. (134) to the form

(135)

The first equation in (135) is called Maxwell’s first equation. Vector on the right-hand side of this equation can be treated as an electric current density vector. Then it is logical to interpret the second equation in (135) as Ohm’s law. Thus, in the proposed model, Maxwell’s first equation written in terms of vectors and contains the term that can be treated as the electric current density. However, it is important to note that there are a number of differences between the equations obtained above and classical Maxwell’s equations.

Firstly, Ohm’s law obtained in the framework of the proposed theory contains the electrical conductivity $\sigma _d$. This is because the dissipative term in Maxwell’s first equation appears only because of the internal friction. The electrical conductivity $\sigma _c$ has nothing in common with the electric current density in Eq. (135), because the dissipative term containing $\sigma _c$ emerges due to the external friction and it appears in the Maxwell–Faraday equation, not in Maxwell’s first equation.

Secondly, if Maxwell’s first equation obtained in the framework of the proposed theory is written in terms of vectors and , then the electric current density is absent in it. Indeed, substituting Eq. (68) into the second equation in (55), we arrive at Maxwell’s first equation without the electric current density:

(136)

The electric current density appears only when we rewrite Eq. (136) in terms of vectors and taking into account the constitutive equations and assuming that the electric current density is defined by Ohm’s law written as . Indeed, in view of the expressions for parameters given by Eqs. (73), (107) and Ohm’s law , the last two constitutive equations in (93) take the following form:

(137)

It is easy to see, that substituting Eq. (137) into Eq. (136) we arrive at the first equation in (135). Certainly, the absence of the electric current density in Eq. (136) should be considered as a shortcoming of the proposed theory. In order to solve this problem, it may make sense to add the source term to the equation for strains, as is done in [92]. But this is the subject of future research.

Thirdly, the electric current introduced above has nothing to do with the electric charge. In the model under consideration, no electric charge is introduced at all, but nevertheless, the electric current is present. We note that the charge conservation law is not violated in this model since from Eq. (135) it follows that

(138)

If equals to zero at initial time, solving Eq. (138) we obtain that . Hence, the charge conservation law (4) is satisfied provided that the electric charge density is zero. Thus, in the proposed model, we deal only with the eddy currents, which are also called Foucault’s currents.

We note that, in the previous papers [58, 59], we have developed the models, where the mechanical analogues of electric current are introduced in different ways. We do not see any contradiction in this situation, since, in our opinion, electric currents can be of different nature. There are the electric currents associated with the translation motion of charged particles. At the same time, we are deeply convinced that there are electric currents associated with rotational degrees of freedom. This is the type of the electric current that we consider in the present paper. Electric currents associated with rotational degrees of freedom are also considered in [23, 46, 47]. We are convinced that there can be electric currents related to electric charges by the differential equation that expresses the charge conservation law and there can be electric currents not associated with the electric charges. The former is presented in, e.g., the mechanical models of electrodynamic processes [59]. The latter is introduced in the model discussed in this paper and in the models proposed in [23, 46, 47, 58]. We are convinced that there can be electric currents that make a system of equations non-conservative [23, 46, 47], and vice versa, there can be electric currents that keep it conservative [58, 59]. To clarify the last statement, we draw analogy with friction. There are different types of friction and most of them make a dynamic system non-conservative. But at the same time, the static friction can keep a dynamic system conservative. Physics textbooks present the view that electric current is the movement of charged particles. Therefore, it is not surprising that this point of view is widespread. However, the physical nature of the electric current is not as obvious as it might seem at first glance. We refer to a very interesting discussion on the subject, which took place almost a hundred years ago [93,94,95]. In our opinion, the issues that were raised in this discussion remain controversial at the present time.

5.2 The generalized Maxwell–Faraday equation and the heat conduction equation

Now, we discuss two features that make the generalized Maxwell–Faraday equation (84) different from the classical one. First of all, let us rewrite Eq. (84) as

(139)

Comparing Eq. (139) with Eq. (135), we see that these equations have the same structure and vector plays the same role in Eq. (139) as vector in Eq. (135). The measurement units of vector coincide with those of the electric voltage per unit area. That is why we call it the electric voltage density. Thus, the first feature that distinguishes Eq. (139) from the classical Maxwell–Faraday equation is the presence of the electric voltage density . We note that in the previous work [60], we introduced the Maxwell–Faraday equation with additional term , but we used an expression for the electric voltage density different from . In the cited work, we discussed the physical meaning of vector in detail. Here we only emphasize that, in our opinion, electric voltages, as well as electric currents, can be of different nature.

We note that a number of papers, e.g., [96,97,98] consider the so-called symmetric Maxwell’s equations. These equations differ from classical Maxwell’s equations by an additional term in the Maxwell–Faraday equation. This term is usually called the magnetic current density and denoted by . From the mathematical point of view, the magnetic current density introduced in [96,97,98] is similar to the electric voltage density in Eq. (139). However, there is a significant difference in physical interpretations. In the theories developed in [96,97,98], due to the presence of the additional term in the Maxwell–Faraday equation, an additional term $Q_m$ appears in the Gauss law for the magnetic induction vector: . The authors of [96,97,98] interpret this additional term $Q_m$ as a magnetic charge per unit volume. We suggest a completely different physical interpretation. In the proposed model, only the vortex part of vector  is related to magnetic field, whereas the potential part of vector  is associated with thermal quantities, see Eqs. (111), (112). In fact, Eq. (112), which expresses in terms of temperature and its time derivative, replaces the Gauss law [96,97,98]. That is why, there are no magnetic charges in the proposed model, despite the fact that the Maxwell–Faraday equation contains the term that is in some sense similar to the magnetic current density.

The second feature that distinguishes Eq. (139) from the classical Maxwell–Faraday equation is the presence of a temperature gradient. At first glance, this may seem strange. However, the close relationship between thermal and electromagnetic processes has been known for long time. In classical electrodynamics, we can take into account the influence of thermal processes only assuming the dependence of the electric and magnetic constants on temperature. In this approach, the temperature plays the role of nothing more than a parameter. In classical thermodynamics, we can take into account the influence of electromagnetic processes on the temperature field only considering Joule heat as a source term in the heat conduction equation. But thermoelectromagnetic effects are not limited to Joule heat and the dependence of electrodynamic parameters on temperature. We can refer to the aforementioned Seebeck effect and Peltier effect that take place at the interface between two conductive materials, see Sect. 4.8. We believe that the proposed theory can describe these effects more precisely because the generalized Maxwell–Faraday equation contains the temperature gradient. As shown above, a number of transformations make it possible to reduce the original system of equations to independent equations for temperature and electrodynamic quantities. However, the corresponding boundary value problems are not independent, since the boundary conditions can relate thermodynamic and electrodynamic quantities [61, 62]. Certainly, the mutual influence of thermodynamic and electrodynamic processes is noticeable mostly near the boundaries. In fact, the situation is the same as in the case of the classical linear theories of elasticity, thermoelasticity, viscoelasticity, etc. These theories can be formulated as two independent equations describing volumetric and shear deformations, but the corresponding boundary value problems remain coupled through the boundary conditions.

Fig. 3

The Ettingshausen effect (a) and the Nernst–Ettingshausen effect (b)

Let us note the following effects:

• the Ettingshausen effect: if there is a current in a conductor and a magnetic field normal to it, one observes a temperature gradient normal to the current and magnetic field, see Fig. 3 a;

• the Nernst–Ettingshausen effect: if a conductive sample undergoes the influence of a magnetic field and a temperature gradient normal to the magnetic field, one observes an electric field directed perpendicular to both the magnetic field and the temperature gradient, see Fig. 3 b.

These effects are nonlinear and the proposed theory cannot describe them, since it is the linear theory. But the equations of the proposed theory contain three mutually orthogonal vectors: the electric field vector, the magnetic field vector and the temperature gradient. This gives us a reason to believe that by generalizing this theory to nonlinear case, we will be able to describe these phenomena.

5.3 The generalized Maxwell’s equations and Kirchhoff’s laws

Now, we exclude the thermal processes. First of all, we rewrite Maxwell’s first equation (135) and the generalized Maxwell–Faraday equation (139) in terms of vectors , , , :

(140)

To close system (140) we should supplement it with the equations relating to and to . Taking into account expression (106) for parameter $\beta$ and the Wiedemann–Franz law (108), we have

(141)

Eliminating and from Eqs. (140), (141), we obtain

(142)

For simplicity sake, we consider a plane wave that propagates in the direction of axis z of a fixed coordinate system x, y, z:

(143)

Inserting Eq. (143) into Eq. (142) yields

(144)

Comparing Eq. (144) with Kirchhoff’s laws (15) we see that they coincide if we accept the following match for the variables:

(145)

and the following match for the parameters:

$$\begin{aligned} \varepsilon \varepsilon _0 \ \leftrightarrow \ C, \qquad \mu \mu _0 \ \leftrightarrow \ L, \qquad \sigma _d \ \leftrightarrow \ G, \qquad \frac{1}{\sigma _c (L \chi ^2/a^2)} \ \leftrightarrow \ R. \end{aligned}$$

(146)

Comparing Eq. (146) with the expressions for the parameters of transmission lines given by Eqs. (18), (19), (20) we find that the match (146) is quite logical. Thus, we can conclude that both scalar equations (144) and vector equations (142) are in agreement with Kirchhoff’s laws (15).

We note that rewriting Eq. (142) in terms of vectors and we arrive at the equations, which are similar to equations (142), but having different coefficients:

(147)

Therefore, we cannot claim that equations (147) are in agreement with Kirchhoff’s laws (15).

It seems to be logical to suppose that matches I and matches V. Let us try to eliminate and from Eqs. (140), (141) and to obtain Maxwell’s first equation with the generalized Maxwell–Faraday equation in terms of vectors and :

(148)

Equations (148) are similar to equations (142), but not exactly the same since the coefficients are different. Hence, equations (148) are not in agreement with Kirchhoff’s laws (15). This means that we cannot match to I and to V.

Thus, we conclude that in order to obtain three-dimensional analogues of Kirchhoff’s laws the following is necessary: firstly, to take into account the additional term in the Maxwell–Faraday equation; secondly, to formulate Maxwell’s first equation and the Maxwell–Faraday equation in terms of vectors and ; thirdly, to suppose that the magnetic field vector matches the current I and the electric field vector matches the voltage V. We note that, before Maxwell’s equations were obtained, experiments in the field of electricity and magnetism were, in fact, experiments with electric circuits. In these experiments four quantities, namely, the voltage V, the current I, the voltage pulse $\int V \mathrm{d}t$ and the current pulse $\int I \mathrm{d}t$ were determined by the direct measurement with the help of an ammeter and a voltmeter. As a result, quantitative relationships were established between these quantities, see [65, 66]. Then, the scalar quantities characterizing the processes in electrical circuits were associated with the vector quantities characterizing the processes in the ether surrounding the electrical circuits as follows:

(149)

It is easy to see that our assumptions that the magnetic field vector matches the current I and the electric field vector matches the voltage V are in agreement with the ideas of nineteenth-century scientists.

6 Outlook

We have proposed the linear mathematical model of the ether associated with the ponderable matter. It describes thermal and electromagnetic processes on the basis of the same approach and the same principles. This theory can be reduced to two three-dimensional telegrapher’s equations: one for temperature and another for the electric field vector. We are convinced that the telegrapher’s equation describes electromagnetic processes in conducting media more accurately than classical Maxwell’s equations. We give two arguments in favor of the latter statement. First, this equation generalizes the one-dimensional telegrapher’s equation, which is widely used in electrical engineering. Second, this equation, in contrast to classical Maxwell’s equations, allows one to describe the static skin effect observed in many experiments. The telegrapher’s equation for temperature may be useful in describing thermal and thermoelectric surface effects. It is important to note that the proposed model includes three mutually orthogonal vectors: the electric field vector , the magnetic induction vector  and the temperature gradient $\nabla T_\mathrm{a}$. This feature of our theory allows us to hope that with generalization to the nonlinear case, it will be able to describe the nonlinear thermoelectromagnetic effects, namely the Ettingshausen effect and the Nernst–Ettingshausen effect. If thermal effects are ignored, the proposed theory can be considered as a generalization of Maxwell’s equations and the three-dimensional analogue of Kirchhoff’s laws for electric circuits. Certainly, the proposed theory is not free from shortcomings and it describes not all the observed effects. But further development of this theory will be the subject of future research. Below we briefly describe the unresolved issues and the main directions of development of our theory.

Firstly, in the theory presented above, Maxwell’s first equation contains the electric current density only if this equation is written in terms of vectors and or and . If this equation is written in terms of vectors and or and , the electric current density is absent. This is a shortcoming of this theory, which should be eliminated. We see a way to solve this problem in a modification of the equation that relates the angular strain tensor to the gradient of the angular velocity vector. We mean the modification analogous to the one made in [92] for the relation between the strain measure associated with translational degrees of freedom and the gradient of the velocity vector. The idea of the modification is to add a source term into the equation for the angular strain tensor and to achieve the desired properties of the model by choosing a constitutive equation for the source term.

Secondly, the theory presented above, as well as the theories published in our previous papers [60,61,62], does not describe the conversion of electrical energy into thermal energy due to Joule heat. This is undoubtedly a shortcoming of the model, which should be eliminated in the very near future. It will be possible to say that our model describes Joule heat only when a term proportional to appears in the heat conduction equation. We think that, by changing the constitutive equations in the framework of the model of viscoelastic material, we cannot arrive at the heat conduction equation with such a term. But we hope that we will be able to derive the heat conduction equation with the term proportional to if we use the aforementioned modified relation between the angular strain and the gradient of the angular velocity vector.

Thirdly, the theory presented above, does not contain terms characterizing electric charges and terms characterizing permanent magnets. In our previous work [59] we have introduced an analogue of the electric charge density as a nonlinear effect inherent in the Cosserat continuum. We are convinced, in the framework of the model based on the Cosserat continuum, it is possible to introduce a quantity characterizing the permanent magnet distributed over space. In order to do this, it is necessary to take into account the symmetric part of the angular strain tensor and nonlinear effects. We emphasize that speaking about the quantity characterizing permanent magnets, we do not mean magnetic monopoles, which were discussed by many authors. We mean a vector characteristic of permanent magnets, which we call a magnetic charge density vector. We believe that this vector is a source of electromagnetic field just like an electric charge is a source of electric field. We note that Maxwell’s equations do not contain a term which is associated with permanent magnets. In fact, solving Maxwell’s equations, one can take into account the effect of permanent magnets only by introducing the equivalent electric currents. At the same time, it is clear that permanent magnets and equivalent coils are not the same things. Therefore, the problem of including permanent magnets in Maxwell’s equations seems to be important.

Fourthly, the theory presented above, is linear. At the same time, various nonlinear effects (electric, magnetic, electromagnetic, thermoelectric, thermomagnetic and thermoelectromagnetic) have been experimentally discovered. In order to describe these effects we need to generalize our theory, but it is not yet clear how to do this.

Fifthly, the theory presented above, is valid for immobile media. Obviously, it needs to be generalized to the case of moving media. However, at this stage, we see different ways to generalize the analogies between mechanical and electrodynamic quantities for the case of moving media, and it is not yet clear to us which of these ways is preferable.

Finally, it is important to account for other types of interactions, namely gravitational, strong and weak interactions. But this can be done only after translational degrees of freedom are added to the model.