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ON THE INCOMPLETENCE OF MAXWELL'S EQUATIONS.

ON THE INCOMPLETE OF MAXWELL'S EQUATIONS

Physics. Discoveries in physics.

Doctor of technical sciences, prof. Etkin V.A.

It is shown that Maxwell's equations do not take into account the displacement flows,
Caused by the displacement of the poles of electric and magnetic dipoles.
Equations of the electromagnetic field, taking into account these currents,
And the internal consistency of these equations is justified

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Introduction

It is generally assumed that the displacement currents enter the right-hand side of Maxwell's equations [1] in an absolutely equal way with transfer currents. However, "up to the present time, no one has solved these equations through displacement currents, since such solutions proved to be simply impossible" [2] . One possible reason for this is that taking into account the displacement currents in Maxwell's equations is, surprisingly, only apparent. Indeed, the notion of flow, coming from mechanics, is closely related to the concept of the flow of a fluid from a certain volume and the presence of its momentum. In particular, in the theory of irreversible processes (TNP) , combining thermodynamics with the theory of heat transfer, hydrodynamics and electrodynamics [3] , under the flow Is understood as the product of a transferable field quantity On the speed of its transfer , And under the density of this flow Is the product of the density of the indicated field quantity At this speed. Meanwhile, displacement flows, expressed in the theory of electromagnetism by partial derivatives From the vectors of electric and magnetic induction, "can not be considered the speed of something" [2] .

It is known that if any function of the radius vector of a point of space r (for example, the electric field strength , Explicitly depends on the time t , the rate of its change is determined by the expression:

Since the derivative , I.e. Determines the density of free charges , [2] , and - the velocity v of its displacement relative to a stationary observer, it is the second term (1) that expresses the displacement current of the free charge. As for the derivative , Then it characterizes only the rate of change of the electric field at a given point in space. Meanwhile, it is this derivative that appears on the right-hand side of the Maxwell equation, along with the conduction current j [1,2] :

Thus, this equation actually does not contain the bias current in its general physical sense.

Displacement currents and their analytical expression

JK Maxwell introduced the notion of displacement current on the basis of a rather particular mechanical model in which electromagnetic phenomena were modeled by vortices in an elastic vacuum connected by imaginary "wheels" [1] . Subsequently, all the "scaffolding" that Maxwell used was discarded, and the "additive" he introduced Faraday's law, called the "displacement current," lost its connection with its original model concepts. In modern electrodynamics this term is used more "by tradition" without sufficient reason. This circumstance manifests itself most clearly from the positions of thermokinetics [5] , which generalizes the TNP to the processes of the useful transformation of any kinds of energy.

As is known, the thermodynamic method consists in finding extensive parameters that characterize the specificity of the processes being studied in the system as a whole, establishing their connection with other parameters (equations of state) and using the properties of the total differential of a number of functions of this parameter. This method is also applicable to spatially inhomogeneous media (in particular, containing free or bound charges). Let the state of such a body be characterized by certain density fields Extensive thermostatic variables (Masses of k-th substances, entropy, charge, etc.), as shown in the figure. The dotted line shows the homogeneous distribution of this parameter with an average density . As follows from the figure, redistribution Between parts of the system, caused by the deviation of the system from equilibrium, is accompanied by the transfer of some of its part * From one area of ​​the system to another in the direction indicated by the arrow. This leads to a displacement of the center of this quantity, determined by its radius vector , From his position In a homogeneous system, where

Here - radius vectors of the center of elements Magnitudes Respectively, in the inhomogeneous and homogeneous state of the system. According to (3) , the deviation of the system from the homogeneous state is expressed in the displacement of the center At a distance of And in the appearance of a certain vector quantity

Which we named after L. Onsager (who first introduced the notion of a "heat displacement vector" with a similar structure) "displacement vectors" (Respectively, electric charge, entropy, k-th substance, etc.) [5] . If for the origin Accept the position of the center of magnitude In the homogeneous state (putting = 0 ), the parameters Will acquire the meaning of "moments of distribution" of this quantity, and their density - the moments of its distribution per unit volume of the system. In the particular case of conductors, where - free charge system , The value Acquires the meaning of the vector of electric displacement in an open conductor as a whole, and its density - the meaning of the electric displacement vector (induction) per unit volume of such a system D [2] . The latter is confirmed by the fact that in both cases .

This approach can be extended to the processes of polarization and magnetization in dielectrics and magnets [5] . The understanding of the unity of the processes of electric and magnetic polarization is facilitated if these processes are represented as a result of the separation of a neutral as a whole and a homogeneous material continuum (including a physical vacuum) into a number of elementary regions dV with diametrically opposite i- properties. Such, in particular, are positive and negative electric charges or unlike magnetic poles possessing certain "magnetic charges" [6] . We denote the unlike elementary "charges" Respectively one and two strokes . Then the position of their centers For the system as a whole is defined by the expressions:

Since in the processes of polarization the system as a whole remains neutral , The displacement vector of the bound charge of the system Will have a similar form (4) [5] :

Where - Dipole shoulder; Is its average value. The amount It is convenient to represent the sum of the moments of both arms of the dipole. For this we represent the shoulder of a dipole Expression . Then, taking We have . For a unit volume of a dielectric and a magnet, this parameter is equal to . This makes it expedient to introduce the concept of a "dipole charge" . Unlike free electric charge rе , unlike electric and magnetic charges generated by polarization are connected in dipoles and do not exist separately. Formally the structure of electrical And magnetic The dipole moment in unit volume of the system coincides with the structure of the polarization vectors [2] and magnetization Unit volume of a dielectric and a magnet [6], which differ from Only in that they have a dipole shoulder Counts from an arbitrary observation point taken for ) [2] . On this basis, we will continue to use, along with These more commonly used terms. Since the polarization processes are related to the displacement of electric charges, the "elec- tronic" (Faraday) state of the unit volume of the system is characterized by the electric induction vector D , and the equation of state is expressed by the well-known relation . Similarly, for Of magnets , Where - relative dielectric and magnetic permeability of the system as a function of the absolute temperature T.

Since the displacement of an elementary dipole charge The spatial coordinates (dri ≡ dr) are simultaneously and equally changed (dri ≡ dr), then for uniformly polarized media (ρiд ≠ ρiд (r)), the divergence of the displacement vector ZIV determines the magnitude of the polarization charge:

Div ZIV = ∂ (ρiдΔri) / ∂r = ρiд. (7)

Taking into account the identity D ≡ εoE + P, which follows from the dielectric state equation, and taking into account that divE = ρe / εo and divP = divZeV = ρеd, we arrive directly at the first equation of the electromagnetic (EMF) field in the form:

Div D = ρе + ρiд. ( 8 )

Similarly, in accordance with the equation of state of magnets B ≡ μоН + М, taking into account the absence of free magnetic "monopoles" (div H = 0) and the expression M = ρmΔΔrm, we arrive at the fourth EMF equation:

Div B = ρmd. ( 9 )

This equation differs from the one proposed by Maxwell (div H = 0; div B = 0) due to the presence of dipole-like opposite magnetic poles (as a result, divM does not vanish [6]).
We now explain the meaning of the derivatives (∂ZIV / ∂t). To do this, consider the total differential ZIV = ZIV (r, t) for a particular case of homogeneous polarization:

DZIV / dt = ρiд (dri / dt) + Δri (dρi / dt) = ρiд vi + Δri (dρi / dt). ( 10 )

The first of the terms on the right-hand side of (10) is determined by the transfer rate of the ith parameter θi (including the total electric and magnetic dipole charge) vi = dri / dt and determines the displacement flux density jiс = ρivi in ​​its general physical meaning. The second component characterizes the rate of local variation of ρi and in accordance with the general balance equations for some field value ρi [3]
Dρi / dt = - div ji c + σi (11)
Is determined by the divergence of the displacement streams jiс or its internal sources σi, but not by these flows themselves. In particular, for dielectrics with rigid dipoles, the density of dipole charges is unchanged (polarization is orientational in nature), however, the displacement currents and the associated effects remain. Further, according to (1) in a stationary field (∂E / ∂t = 0), the conduction currents are the only sources of the magnetic field. Meanwhile,
Some facts point to the need to take into account displacement currents in this case [6]. This confirms once again that the derivative (∂D / ∂t) in the Maxwell equation does not completely determine the displacement fluxes jiс.

Accounting for displacement currents in the equations of the electromagnetic field.
With special clarity, the need to take into account displacement currents in the equations of the electromagnetic field is manifested in the thermodynamic derivation of Maxwell's equations [7]. Let the i-th processes of redistribution of the parameters ρi (ρе, ρед and ρмд) take place in a certain system. In accordance with this, we represent the energy per unit volume of the UV system as a function of the variables ZIV (including the polarization vectors P = ρesΔrе and the magnetization M = ρmsΔrm, that is, UV = UV (ZIV) .In this case, its total differential is given by:

print version
Author: D.Sc., prof. Etkin V.A.
PS The material is protected.
Date of publication 06.09.2004гг