on details, logics, and modeling physical and mathematical techniques used in the previous ~110 years sinse Lorentz found the ways to connect, loosely communicate and justify the continuum scale Maxwell-Heaviside-Lorentz electrodynamics equations to just discovered electron presence in an atom, atomic scale substances, materials.

So, there was the need to develop and connect electromagnetism phenomena and equations to those are at the atomic scale and backward to the continuum scale for the Maxwell-Heaviside-Lorentz equations.

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Introduction to Scaled Electrodynamics Description of Matter, Materials, Substances at Scales Below Continuum - an Atomic, Sub-atomic Ones

The main pathway of establishing the validity of Maxwell-Heaviside-Lorentz electromagnetism modeling equations has been to prove that at any scale (level) of description - one has the same type of model, modeling equations!

That the modeling electromagnetism equations are invariant regarding the scale of physical phenomena considered. The only changed are the coefficients created for that purpose.

Almost completely the talk here is about the sub-atomic and atomic scales equations (often mentioned as microscale) and of continuum scale modeling equations.

We do not concern at this point any smaller - as of particle physics scales, or larger as of any Heterogeneous matter, for example, astrophysics scales electrodynamics modeling statements.

As long as it is considered as the strong true statement in conventional homogeneous physics - that the Maxwell-Heaviside-Lorentz (MHL) modeling electrodynamics equations are exact, precisely valid for the atomic scale phenomena, we would accept tentatively the idea so far, that the most reliable and fundamentally proven electromagnetism governing equations are of the atomic scale.

We have paid attention in 90s to provide the scaled description based on correct mathematics of heterogeneous, scaled media MHL's electrodynamics phenomena with some hard copy publications - Travkin et al. (1999a,b,c; 2000a,b; 2002), Travkin and Catton (1999, 2001a,b,c), Travkin and Ponomarenko (2004; 2005a,b,c), Ponomarenko et al., (2001), and other; as well as with the texts on Heterogeneous Electrodynamics (HtEd) in this and other websites.

The very notion that at the atomic and close to this scale phenomena in media of different plasmas are the same ideas of Scaled description for plasma fields - as long as these fields are of particulate substances, dominated our developments at the beginning of 2000s regarding the Heterogeneous Scaled Electrodynamics and Momentum Transport for plasma theory along with Continuum Mechanics that found some presence at this website in -

The set of the Upper scale governing equations developed with HSP-VAT tools, for those have been used the Lower scale conventional solid state plasma modeling equations - that supposed to be already "averaged" - unfortunately as we point out they are not averaged -

Conventional Maxwell-Lorentz' Continuum Lower Scale Electrodynamics and the Upper Scale Heterogeneous Electrodynamics concepts, fundamentals obtained first in the second half of 1990s given in -

Lorentz' averaging theory type of application toward the dielectric materials fundamentals we have shown in -

Then in other disciplines, but completely of the same origin the problems that have been blocking advancements in considering our surrounding matters, substances in a polyscale, polyphase nature as they are, found some depiction in:

The important message sent by Lorentz was at the time -

1) The aether medium was accepted as unquestionable existing phenomenon which was an inseparable part of electrodynamics and Maxwell-Heaviside-Lorentz governing equations. This text by Lorentz (1906) shows definitely that Lorentz himself was totally in agreement with experiments by D.Miller (that finally lasted for more then 30 years) that confirmed the ether-earth drift effect while disproving the conventionally adopted experiment by Michelson-Morley's of 1887, which ran over only for 6 hours in four days (see publications in the References and other materials all over the internet).

More on that - there are studies and publications showing that the Michelson-Morley's 1887 experiment was erroneously analyzed and that the same experiment is also in a favor of the ether-drift effect existence.

2) Maxwell had no idea (or better to say he did not recognize the outstanding Weber's contribution into the electrodynamics field (the discussion is known and published) and nonexistent yet atomic physics) about electron(s), its charge, role, and all controversy around it. But much earlier in 1840s W.Weber had, while he had developed at that time and later the whole theory regarding the atomic structure of matter and its relation to electrodynamics of that matter.

Many decades later at the beginning of 1900s Lorentz knew already about an electron existence (its Weber's suggested discovery was confirmed via experiment) and wrote the book on that published in (1906). Lorentz' ideas and development were that the Maxwell's equations created for unspecified "matter" that makes transport and properties of "electricity" noticeable and meaningful, created as of continuum medium(a) are quite suitable and for an aether itself, and for an aether with electrons (the nucleus was not discovered yet, but actually did not change something), as well as for the any matter in the mathematical forms that almost undistinguishable one from another.

3) Electrons are the volumetric, of extended size particles.

We are not discussing or assuming any invitation to argue on that now, in this text.

However good they are, these three postulates were lately abandoned - while most of the students are never told why? Well, to some have been told firstly the unrelated reasons. We are turning at this point here to the multi-"phase" treatment of atomic scale Maxwell-Heaviside-Lorentz electrodynamics equations, as well as in few other places of this website.

Postulates Introduced in Physics During the Beginning of Quantum Mechanics and Particle Physics Times:

1) All particles are accepted and assumed to be viewed and treated as the "point-like," point-mass (with the mass is being Attached to this object?) or having no extended body in space. Which was done because there were tremendous difficulties in treatment of many-body problems in atomic physics. There were no mathematics for that treatment and even now these problems having approximate simulating solutions.

2) All particles and surrounding media as, for example, an air or the vacuum were treated as the one-phase Homogeneous medium with either the space indistinguishable locations of point-particles or with use of the Dirac's -function.

3) Because at that time there were also no concepts and theories of multi-phase media and their methods of interaction, modeling and solution. Those were developed much later in continuum mechanics of fluids and gases (with later on the wide applications to other sciences and engineering) while supported by the corresponding newly developed mathematical tools.

4) There were no boundaries, bounding surfaces for point-particles. No need to have those. That postulate brought in the great problems.

5) So, with introduction by Dirac the theory of electron simultaneously with the theory of -function in 1929 there were solved two and even three tasks in atomic physics:

a) physical and mathematical fields could be continually considered as the homogeneous ones and all powerful existing at the time mathematical mechanisms and tools for solution of those homogeneous problems could be applied;

b) the location and movements of particles and atoms could be assigned as for the Homogeneous fields source functions in mathematical formulation of the tasks;

c) the powerful methods of statistics were free to apply to the point-particles in space behavior, advantageous for their assigned interaction collective movements and characteristics. That was the beginning of statistical mechanics shining.

That also created the problem of near particle field's description, that became the statistical qualities of a field.

And nobody since then is willing to explain to students - that all of this vision of particles as point-mass particles and fields as the Homogeneous ones, was accepted in 1920s - 1930s because they couldn't solve correctly the particulate problems as is. With particles as small, but still physically particles and the physical fields as Polyphase, yes, of different phases nature distributed spatial fields. Now we can do this.

Since then, the electron and photon are the Point-mass objects with no volume and volumetric characteristics.

Since then, the huge body of mathematics, let alone physics, has been developed just to support this artificial picture. While physics became a metaphysical science, because of false point-particles, MHL electrodynamics, SR and GR following from this short-hand electrodynamics, and QM that became the compounding original theory for everything small enough to not study it within.

6) With these above assumptions (1-4) that was not surprising that Quantum Mechanics was introduced and advanced in this fantastic mathematical formulation as we know it now. Little later Dirac's -function added greatly to the functionality of the hard solid rock of QM that is laying on the road to further progress in physics for many decades.

These words above is not the critics of QM, the critics is in the other places of this website. Also, people started to find inconsistencies, flaws in QM in the earlier years of QM appearance. And that was not too hard.

Unfortunately, QM is being taught to students and used for research up to now as the primary theory for micro-world, sub-atomic physics. Teachers do not tell that QM is just one of the approximate theories that was developed due to insolvency of mathematics and physics regarding mentioned issues at that time.

But not at present situation, when those statements of 1920-1930s can be solved, because the theories for that solution during the last 20-30 years have been advanced far enough.

Maxwell's Ideas of Electromagnetism and Governing Mathematical Equations

Maxwell (2002a; 3rd ed.) in introductory paragraphs admitted the level of contemporary to his time knowledge of electromagnetism (electricity) phenomena.

In p. 37 one can read that:

"When we come to describe electrometers and multipliers we shall find that there are still more delicate methods of detecting electrification and of testing the accuracy of our theorems, but at present we shall suppose the testing to be made by connecting the hollow vessel with a gold leaf electroscope. This method was used by Faraday in his very admirable demonstration of the laws of electrical phenomena."

Our comments:

Maxwell straight in denoting that he uses few theorems for explanation of electromagnetic phenomena. Gauss-Ostrogradsky theorem was among of them.

In p. 38 one can read:

"35. While admitting electricity, as we have now done, to the rank of a physical quantity, we must not too hastily assume that it is, or is not, a substance, or that it is, or is not, a form of energy, or that it belongs to any known category of physical quantities. All that we have hitherto proved is that it cannot be created or annihilated, so that if the total quantity of electricity within a closed surface is increased or diminished, the increase or diminution must have passed in or out through the closed surface.

This is true of matter, and is expressed by the equation known as the Equation of Continuity in Hydrodynamics.

It is not true of heat, for heat may be increased or diminished within a closed surface, without passing in or out through the surface, by the transformation of some other form of energy into heat, or of heat into some other form of energy."

In p. 39 one can read:

"There is, however, another reason which warrants us in asserting that electricity, as a physical quantity, synonymous with the total electrification of a body, is not, like heat, a form of energy. An electrified system has a certain amount or energy, and this energy can be calculated by multiplying the quantity of electricity in each of its parts by another physical quantity, called the Potential, of that part, and taking half the sum of the products. The quantities 'Electricity ' and 'Potential,' when multiplied together, produce the quantity 'Energy.' It is impossible, therefore, that electricity and energy should be quantities of the same category, for electricity is only one of the factors of energy, the other factor being 'Potential.'"

Our comments:

Maxwell is vague about connection "electricity" with energy.

In p. 40 one can read:

"In the theory called that of Two Fluids, all bodies, in their un-electrified state, are supposed to be charged with equal quantities of positive and negative electricity. These quantities are supposed to be so great that no process of electrification has ever yet deprived a body of all the electricity of either kind. The process of electrification, according to this theory, consists in taking a certain quantity P of positive electricity from the body A and communicating it to B, or in taking a quantity N of negative electricity from B and communicating it to A, or in some combination or these processes.

The result will be that A will have P+N units of negative electricity over and above its remaining positive electricity, which is supposed to be in a state of combination with an equal quantity of negative electricity. This quantity P+N is called the Free electricity, the rest is called the Combined, Latent, or Fixed electricity."

"We shall see that the mathematical treatment of the subject has been greatly developed by writers who express themselves in terms of the 'Two Fluids' theory. Their results, however, have been deduced entirely from data which can be proved by experiment, and which must therefore be true, whether we adopt the theory of two fluids or not. The experimental verification of the mathematical results therefore is no evidence for or against the peculiar doctrines of this theory.

The introduction of two fluids permits us to consider the negative electrification of A and the positive electrification of B as the effect of any one of three different processes which would lead to the same result. "

"37. In the theory of One Fluid everything is the same as in the theory of Two Fluids except that, instead of supposing the two substances equal and opposite in all respects, one of them, generally the negative one, has been endowed with the properties and name of Ordinary Matter, while the other retains the name of The Electric Fluid. The particles of the fluid are supposed to repel one another according to the law of the inverse square of the distance, and to attract those of matter according to the same law. Those of matter are supposed to repel each other and attract those of electricity."

"This theory does not, like the Two-Field theory, explain too much. It requires us, however, to suppose the mass of the electric fluid so small that no attainable positive or negative electrification has yet perceptibly increased or diminished either the mass or the weight of a body, and it has not yet been able to assign sufficient reasons why the vitreous rather than the resinous electrification should be supposed due to an excess of electricity."

In p. 43 Maxwell summarizing his intent in the manuscript:

"In the present treatise I propose, at different stages of the investigation, to test the different theories in the light of additional classes of phenomena. For my own part, I look for additional light on the nature of electricity from a study of what takes place in the space intervening between the electrified bodies. Such is the essential character of the mode of investigation pursued by Faraday in his Experimental Researches, and as we go on I intend to exhibit the results, as developed by Faraday, W. Thomson, &c., in a connected and mathematical form, so that we may perceive what phenomena are explained equally well by all the theories, and what phenomena indicate the peculiar difficulties of each theory."

Our comments:

We need to understand that Maxwell saw the nature of electromagnetic phenomena as a kind of ones happening in and with participation of aether and by some unknown at that time (to all seems, but W.Weber and some collaborators) "portions of matter" that he forecasted would give a possibility to construct the matter related "dynamical theory of electricity" - see in our CONCLUSIONS citations on that. While he was not sure (he disputed Weber's electrodynamics) at the time of what kind of media provides for electromagnetic phenomena - even could be a fluid like.

Wilhelm Weber, Carl Friedrich Gauss and Continental School of Thought Accomplishments Regarding Electromagnetism Phenomena, Electrodynamics and even Atomic Physics

In the review by Hecht (2001) we might read that:

"It is not too difficult to see that Weber's Fundamental Electrical Law, almost unknown today, is a statement of a relativistic law of physics, long predating the statement of relativity we are accustomed to^8. Here it is the force, rather than the mass, which varies with the relative motion.

But, not only does it predate the Einstein formulation, it is methodologically far superior. One can, in various ways, attempt to show an equivalence of the two statements, but the usefulness of such efforts is doubtful. The problem lies elsewhere. The two statements lie in two entirely different domains.

One is a continuation of the Leibnizian current of physics; the other, whatever the intentions, serves to hide errors embedded in the assumptions underlying the Maxwell equations. "

Our comments:

Yes, this author had been brave enough to publish the statement on inadequacy of Maxwell-Heaviside-Lorentz electrodynamics. And only well organized political support has been preventing the issue from studying, from the ground and up systematic research, and improvement or advancement.

We have obtained the theory for collective (averaged) action of the Weber's and other known force interaction between the charged particles formulae suggested by J.Klyushin first for the two-particles interaction. This theory by Klyushin actually embraced the whole set of others well grounded fundamental interactions (including of Weber's) between the two charged particles that are still, as in the Coulomb formula, and/or are in motion and well known today. The parts of this theory are placed in this and other our sites, in hard copy publications, and will be disseminated to more applications.

We will be exploring this and many other notices, and works to go in that same developmental direction regarding the Ampère-Gauss-Weber, others electrodynamics base and features as that includes at least the two physical spatial scale physical models and what are making them if all that to be taken combined for the closed in some aspects the strictly connected and scaleported physical system effected and controlled by its external input either from Upper or at the Lower scale spatial and temporal boundaries.

We will be adding to this part of historical observations regarding the electrodynamics of continental school of thought - Ampère-Gauss-Weber, and contemporary electrodynamics working scientists, that are actually support our current time (1987 - 2010s) electrodynamics Polyphase-Polyscale-Polyphysics (3P) theory. The theory that is of scaled base from the sub-atomic particles while induces the laws of electrical particles interaction. Structural particles - not a point-mass ones.

Lorentz formulated Electrodynamics Governing Equations for the Ether, Ether and Electrons Medium, and for a Matter, Material of Continuum Mechanics

Lorentz (1906, 1952) in his lectures of 1906, when the discovery of electron had already impacted the physics world, gives the following sets of equations for an ether, for the quantity of electrons in an ether, and for a bulk formulated continuum mechanics material.

Lorentz in the "The Theory of Electrons and Its Applications to the Phenomena of Light and Radiant Heat : a Course of Lectures Delivered in Columbia University in 1906" (1906) writes in page 5: ( equations in an aether) -

"3. We are now prepared to write down the fundamentals equations for the electromagnetic field in the form which they take for the ether. We shall denote by the electric force, the same symbol serving for the dielectric displacement, because in the ether this has the same direction and, on account of the choice of our units, the same numerical magnitude as the electric force. We shall further represent by the magnetic force and by a constant depending on the properties of the ether.

A third vector is the current , which now consists only of the displacement current of Maxwell. It exists wherever the dielectric displacement is a function of the time, and is given by the formula

In the form of differential equations, the formulae of the electromagnetic field may now be written as follows:

here used as a partial derivative with respect to the time.

Our comments:

Note that only two functions used here and , where in an ether (or "vacuum") the displacement because the electric dipole moment and where because magnetic dipole moment is

Note that the in Lorentz's physical statement is the "constant depending on the properties..."

But not a light speed or something as that kind.

Further in this page Lorentz explains the equations:

"The third equation, conjointly with the second, determines the magnetic field that is produced by a given distribution of the current . As to the last equation, it expresses the law according to which electric forces are called into play in a system with a variable magnetic field, i.e. the law of what is ordinary called electromagnetic induction. The formulae (), () and (5) are vector equations and may each be replaced by three scalar equations relating to the separate axes of coordinates."

Lorentz writes interesting things about electromagnetism in an ether, p. 6:

"The formulae for the ether constitute the part of electromagnetic theory that is most firmly established. Though perhaps the way in which they are deduced will be changed in future years, it is hardly conceivable that the equations themselves will have to be altered. "

Now we write the Lorentz' electrodynamics equations for a volume in an aether with number of electrons - p. 11-12:

"7. I have now to make you acquainted with the equations forming the foundation of the mathematical theory of electrons. Permit me to introduce them by some preliminary remarks.

In the first place, we shall ascribe to each electron certain finite dimensions, however small they may be, and we shall fix our attention not only on the exterior field, but also on the interior space, in which there is room for many elements of volume and in which the state of things may vary from one point to another.

As to this state, we shall suppose it to be of the same kind as at outside points. Indeed, one of the most important of our fundamental assumptions must be that the ether not only occupies all space between molecules, atoms or electrons, but that it pervades all these particles.

We shall add the hypothesis that, though the particles may move, the ether always remains at rest.

We can reconcile ourselves with this, at first sight, somewhat startling idea, by thinking of the particles of matter ad of some local modifications in the state of the ether. These modifications may of course very well travel onward while the volume-elements of the medium in which they exist remain at rest.

Now, if within an electron there is ether, there can also be an electromagnetic field, and all we have got to do is to establish a system of equations that may be applied as well to the parts of the ether where there is an electric charge, i.e. to the electrons, as to those where there is none. As to the distribution of the charge, we are free to make any assumption we like.

For the sake of convenience we shall suppose it to be distributed over a certain space, say over the whole volume occupied by the electron, and we shall consider the volume-density as a continuous function of the coordinates, so that the charged particle has no sharp boundary, but is surrounded by a thin layer in which the density gradually sinks from the value it has within the electron to 0.

Thanks to this hypothesis of the continuity of , which we have never to trouble ourselves about surfaces of discontinuity, nor to encumber the theory by separate equations relating to these.

Morever, if we suppose the difference between the ether within and without the electrons to be caused, at least so far as we are concerned with it, only by the existence of the volume-density in the interior, the equations for the external field must be got from those for the internal one by simply putting so that we have only to write down one system of differential equations.

Of course, these must be obtained by a suitable modification, in which the influence of the charge is expressed, of the equations (2)-(5) which we have established for the free, i.e. for the uncharged ether. It has been found that we can attain our object by the slightest modification imaginable, and that we can assume the following system

in which the first and the third formula are the only ones that have been altered.

In order to justify these modifications, I must in the first place recall to your minds the general relation existing in Maxwell's theory between the dielectric displacement across a closed surface and the amount of charge contained within it. It is expressed by the equation

in which the integral relates to the closed surface.... "

Here in (19) " is the velocity of the charge".

Our comments:

Contrary to Lorentz' simplified assumption that allowed him to write the same kind of equations with the space-dependable Lower scale homogeneous equations governing the electromagnetic fields, we would use the "phase"-dependable media interacting within our region with particles - just some of the kinds, and the "vacuum" which is in reality not the vacuum at the lower scales physics.

These equations (17)-(20) are supposed to be the averaged equations - for the aether and plus electrons as a total mixed medium! Because there is less understanding and function performed due to mutual dependencies by these equations, the equations responsible for the volumetrically mutual interactive actions, but not for dimensionless point interface when the fields determined by space-dependable functions as and - unless we consider the collective, averaged fields in an area.

That means these equations (17)-(20) in the scalable notations should be written for averaging as

and these equations if averaged correctly are not the ones written in textbooks as (in other dimensions systems as SI looking slightly different) the following

**************

Well, the EM equations by Lorentz (1906) in a matter are (p. 7):

"4. There is one way of treating these phenomena that is comparatively safe and, for many purposes, very satisfactory. In following it, we simply start from certain relations that may be considered as expressing, in a condensed form, the more important results of electromagnetic experiments. We have now to fix our attention on four vectors, the electric force , the magnetic force , the current of electricity and the magnetic induction . These are connected by the following fundamental equations:

presenting the same form as the formulae we have used for the ether.

In the present case however, we have to add the relation between and on the one hand, and that between and on the other. Confining ourselves to isotropic bodies, we can often describe the phenomena with sufficient accuracy by writing for the dielectric displacement

a vector equation which expresses that the displacement has the same direction as the electric force and is proportional to it. The current in this case is again Maxwell's displacement current

In conducting bodies on the other hand, we have to do with a current of conduction, given by

where is a new constant. This vector is the only current and therefore identical to what we have called , if the body has only the properties of a conductor. In some cases however, one has been led to consider bodies endowed with the properties of bothconductors and dielectrics.

If, in a substance of this kind, an electric force is supposed to produce a dielectric displacement as well as a current of conduction, we may apply at the same time (12) and (14), writing for the total current

Finally, the simplest assumption we can make as to the relation between the magnetic force and the magnetic induction is expressed by the formula

in which is a new constant. "

Our comments:

Note here, that there were no averaging operators at all by the Lorentz time. There were no atomic physics as a discipline and the nucleus will be discovered only in 1911.

So, the mathematical formulation of the continuum electrodynamics were found based on general considerations of experiments and with the homogeneous Volume-Surface theorems by mostly as the Gauss-Ostrogradsky and Green's theorems.

But not with any kind of averaging techniques at atomic scale used for Maxwell-Lorentz equations.

In p. 8 further Lorentz gives few outstanding sentences regarding the role of modeling in electromagnetism:

"If we want to understand the way in which electric and magnetic properties depend on the temperature, the density, the chemical constitution or the crystalline state of substances, we cannot be satisfied with simply introducing for each substance these coefficients, whose values are to be determined by experiment; we shall be obliged to have recourse to some hypothesis about the mechanism that is at the bottom of the phenomena."

In pgs. 8-9 Lorentz writes:

"In a ponderable dielectric there can likewise be a motion of the electrons. Indeed, though we shall think of each of them as having a definite position of equilibrium, we shall not suppose them to be wholly immovable. They can be displaced by an electric force exerted by the ether, which we conceive to penetrate all ponderable matter, a point to which we shall soon have to revert."

Our comments:

Lorentz straightly supports here the ether as the medium for transduction of electromagnetic phenomena!

In p. 9 Lorentz continues:

"Now, however, the displacement will immediately give rise to a new force by which the particle is pulled back towards its original position, and which we may therefore appropriately distinguish by the name of elastic force.

The motion of the electrons in non-conducting bodies, such as glass and sulphur, kept by the elastic force within certain bounds, together with the change of the dielectric displacement in the ether itself, now constitutes what Maxwell called the displacement current. A substance in which the electrons are shifted to new positions is said to be electrically polarized.

Again, under the influence of the elastic forces, the electrons can vibrate about their positions of equilibrium. In doing so, and perhaps also on account of other more irregular motions, they become the centers of waves that travel outwards in the surrounding ether and can be observed as light if the frequency is high enough. In this manner we can account for the emission of light and heat."

Our comments:

Note here - that there is no mentioning of photons. Photons were just suggested recently, still did not come into the system of Lorentz's physical world, probably. The light is explained in this text as through the "surrounding ether and can be observed as light" in a vibrating ether!

What kind of averaging theory Lorentz developed and applied toward the dielectric materials fundamentals we have shown in -

Note, that these incorrect for heterogeneous, polyscale media averaging procedures dominated the first half of XX century.

Atomic Scale (Microscale) Maxwell-Heaviside-Lorentz Equations (with Partial Time Derivatives) and Their Fields and Media in SI and CGSE Systems

Lorentz style of EM equations in the vacuo for SI system

Finally we can arrive to the Lorentz style of EM equations in the vacuo for SI system (as the vacuum we will use the medium as it is the "vacuum", at least consisting as of one of its componets the cosmic relict radiation field -- Cosmic Microwave Background (CMB) radiation and more on that. With photons of it in each and with the influx of those photons as of ) when of course

or for vacuum while medium can be taken as polarization free

or for vacuum while medium can be taken as magnetization free

from

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Now the set of Maxwell GE in SI with for vacuum and point-like as for when :

Gauss' law equation

or for vacuum but with only AFTER like averaging. So, before averaging we need to take . Well, so far following the methods to connect atomic scale electromagnetism and macroscale Maxwell-Lorentz equations we do take and .

or for vacuum while also with only AFTER like averaging.

where in the Lorentz' form for properties fields we have

conservation of magnetic flux equation

where means for vacuum conditions, while the Lorentz' equation is

Maxwell's-Ampere's law equation: at the macroscales

or it can be re-written as to go closer to of the Lorentz's equation when because before averaging and with and finally

well, but in textbooks authors like or see advantages in presenting this equation via the microscale magnetic induction field. Why is that? Apparently because - in this way or reasoning they do the jump straight to the macroscale magnetic induction, as does Jackson (1999) in eq. (6.64), where already at the microscale appeared magnetic induction (the following is the Maxwell's-Ampere's law equation from the set of microscopic equations for electromagnetism in (6.64) in Jackson (1999) )

while Lorentz in his system with got to

Faraday's law of induction equation

for vacuum with homogenized particles and we are able to take in the right hand side as as it looks like reasonable.

******* system in the vacuum with **********************

Now in system of variables this set of Lorentz' style GE of Jackson for "active" vacuum will be as (following (with )

or for vacuum

or for vacuum

in "active"vacuum

while

well, these GE exactly as eqs. (2)-(5) in p. 5 in Lorentz (1906, 1952).

****************************************************

Now the set of equations in the Gaussian system (CGSE) for a substance

with

of variables for vacuum and

point-like as for

when :

Gauss' law equation

or for vacuum with charges-particles

or for vacuum

before averaging

where in the Lorentz' form for the we have

the next one is the conservation of magnetic flux equation or Gauss law for magnetic field equation which is before averaging

where means for vacuum conditions, while the Lorentz' equation is

Maxwell's-Ampere's law equation before averaging

or it can be re-written as to go closer to one of the Lorentz's equation when because for vacuum with separately included and not averaged charges-particles would be

for vacuum finally

well, but in textbooks authors like or see advantages in presenting this equation via the microscale magnetic induction field. Why is that? I know why - in this way or reasoning they do jump straight to the macroscale magnetic induction, as does Jackson (1999) in eq. (6.64), where already at the microscale appeared magnetic induction

for vacuum with charges-particles

for vacuum

while Lorentz in his system with got to

Faraday's law of induction equation

before averaging

If we take the full form of constitutive equations for system of vacuum plus the charges with

we would have

or (also before Averaging)

the same as of Lorentz (1906, 1959).

Schwinger et al. (1998) Pseudo-Averaging Techniques for Unspecified Volume Atomic Scale Fields

In the textbook by Schwinger et al. (1998) given the good example of reasoning, methods used in homogeneous physics to derive (and by this to validate, justify) the Maxwell-Heaviside-Lorentz electrodynamics equations for a continuum. Everyone was able to analyze these rather complicated mathematically ideas and methods used for "homogenized" mixing of atomic scale particles and of the "vacuum" to get Up to the continuum media electromagnetism governing equations.

We would extend our consideration of this theory in attempts to contribute for the multi"phase" Scaleportation of electrodynamics equations, models between atomic and continuum descriptions.

In Homogeneous physics - like in the tested below monograph by Schwinger et al. (1998) and others, there are no specifics, theorems to do averaging over these REVs, and even there are no definitions or wrong definitions (as in Continuum Homogeneous Mechanics) of a REV.

In p. 33 one can read that:

"4.1 Force on an Atom

"The Maxwell-Heaviside-Lorentz system of equations, (1.65) and (1.68), provides a microscopic description of electromagnetic phenomena, at the classical level, ranging from the simplest two-particle system to the detailed behavior of all particles in a macroscopic system.

However, for the latter case, we usually do not require such a complete description, since our measurements involve macroscopic quantities which are only indirectly related to the microscopic behavior of individual atoms.

We must develop a theory that is directly applicable to the macroscopic situation with only an implicit reference back to the detailed characterization of the system."

Our comments:

These authors did not believe in the possibility of true scaleportation - purely because of the lack of their knowledge, partially the reasons were about the HSP-VAT tools and advancements up to that time - the mid-90s. At the same time, it is obvious that very established in traditional Homogeneous physics authors would not be so naive to break with the ground base concepts of absence of Heterogeneous instruments in Atomic physics as it was continued for the 90-something years.

Also it is true that this book with the so straight interest and methods to justify the conventional homogeneous derivation procedures, mathematics used for "averaging" in Atomic physics serving to the needs in orthodox physics for justification of that sort of thinking and the status quo - as for a homogeneous matter even when the tools and the solid base exist for treating heterogeneous microscale electrodynamics phenomena with the heterogeneous physics and math tools.

Further down the p. 33:

"The resulting macroscopic electrodynamics is a phenomenological theory, by which is meant a theory that operates at the level of the phenomena being correlated and predicted, while maintaining the possibility of contact with a more fundamental theory - here, microscopic electrodynamics - that operates at a deeper level."

Our comments:

What an egocentric declaration - the results of statistical operations over the point-objects of conventional atomic physics are taken as of higher superiority over the exact analytical methods and procedures for modeling and simulation of relevant scaled problems.

This kind of up and down points of view regarding the statistical physics and deterministic physics cooperation registered for existing since the XVIII. It is not new completely.

Further down the p. 33-34:

"That contact exists to the extent that the macroscopic measurements can be considered to be averages, over very many atoms, of the results of hypothetical microscopic measurements.

To begin, we consider an atom, an electrically neutral assembly of point charges (outlined by us),

that are bound in a small region. We want to study the response of such a system to external electric and magnetic fields that vary only slightly over the spatial extent of that system.

We will first concentrate our attention on the net force on the system at a given time, the sum of the forces on its constituents, (1.68), (force law by Lorentz, H.A.)

Since the system is small, all the charges are near the center of mass of the charge distribution, which lies at the position . (For the purposes of the following expansion we could let represent an arbitrary point inside the charge distribution; the use of the center of mass allows us to separate intrinsic properties from those due to the motion of the atom as a whole.)

We can then expand the electric and magnetic fields about this reference point,

and like wise for , in which the subsequent terms are considered negligible."

Our comments:

Well, this is the expansion named the one that is up to the first order of variable's (function usually) magnitude. And considered in mathematics of lower order of accuracy.

In physics it is often, especially in complicated areas, taken as the final and most often used tool to get to some results out of despair in search for any kind of solution.

Here we talk on that because students in physics and techs usually can not assess the motifs for this or that direction in taught material.

Further down the p. 34:

"Here means the gradient with respect to . Now, the total force on the atom, (4.2), can be rewritten in terms of this expansion as

In p. 38 one can read that:

"4.2 Force on a Macroscopic Body

..... Macroscopic materials are made up of large numbers of atoms. What is the total force on such a piece of materials? We must sum up all the forces on the individual atoms. To the extent that the forces on the atoms vary but slightly from one atom to another, the simulation can be replaced by a volume integration, weighted by the atomic density, , the number of atoms per unit volume at the macroscopic point :

Notice that we have rewritten (4.22) with the aid of the identities

First a word about and in these expressions. In the single atom formula (4.22), the derivatives act only on and which is reflected in (4.32). For a many-atom system, the dipole moments could well vary from one location to another and so have macroscopic spatial dependence. Accordingly, and are the average dipole moments at the point .

We now define the electric polarization, , and the magnetization, , by

respectively. The resulting macroscopic form of the total force at time is

(Here the distinction between and has been dropped, because the "

Our comments:

The identities (4.33), (4.34) are not complete expressions.

Those should look like (4.33f), (4.34f)

that is the difference.

Also, authors introduced the fields and as if they are already averaged over the REV. More on that - they use in further derivations until the macroscopic equations these and as those already being averaged!

In p. 39 one can continue reading the simplifications of the expression for an averaged macroscopic force in a body:

"We proceed to simplify this in various ways. First, we use one of Maxwell's equations to obtain

and then we use the identity

which is a generalization of (4.34)."

Our comments:

Well, here in (4.39) used the full formula for gradient of scalar product of vectors. Why is that taken in full swing unlike before in (4.33), (4.34)?

But how can we use the "one of Maxwell's equations" if all those are still for the atomic (microscale) level - but here we have the mathematical expressions for fields "averaged" so far over a single separate atom in and ?

In p. 39 one can continue:

"All subsequent steps involve the statement that the integral is extended over a volume that includes the whole body, so that, on the bounding surface of that volume, . This means that in performing partial integrations through the use of the divergence theorem (Gauss-Ostrogradsky homogeneous theorem; outlined by us) the surface integrals vanish. In effect, then,

and similarly, using yields

Our comments:

The important notice can be made on these transformations is that the use of Homogeneous Gauss-Ostrogradsky theorem for the all body of interest being bounded by REV surface - means bringing the problem to the bulk, but still as for the point-like assessment of the physics. And actually not worth for further consideration as of the averaged at the Upper scale but space dependent problem.

In this case - the problem of internal electrodynamics governing equations for the piece of material, piece of matter, but not of a dot-mass weighted task.

And again - How we can use the atomic scale Maxwell equations for the not averaged or only one atom averaged expression in (4.41)?

Further in p. 39 one can continue:

"The immediate result is

The comparison of this with the microscopic description of the force on charge and current densities, (3.8) for zero magnetic charge, suggests the definition of an effective charge density, , and an effective current density, , as

Notice, that these effective densities satisfy the equation of charge conservation,

Our comments:

Notice first that in the fields and are not the averaged over the REV or over the another non-identified volume of averaging; not, they are the fields averaged over a separate single atom at a location within a REV.

Note also, that in all these the fields and are actually the electric polarization and magnetization of a separate atom or molecule, that's it. These are not the averaged functions, indeed.

These are given in an expression as BEFORE averaging determined.

But later on, we see that they will be taken as the already averaged over the volume of the REV functions.

******************************************************************

Schwinger at el. (1998) wrote further on the Macroscopic form of Maxwell-Lorentz set of equations:

In p. 40-41 one can read that:

" The microscopic regime is characterized by rapid space-time variations unlike the macroscopic one, which is characterized by scales large compared to those of atoms. Laboratory instruments, being large, directly measure average quantities. Macroscopic fields are thus defined in terms of averages over space and time intervals, V and T, large on the atomic scale but small compared to typical macroscopic intervals. We adopt the convention that lower-case letters, like f(r ,t), represent microscopic quantities while capital letters, like F(r ,t), represent the corresponding macroscopic quantities. The connection between the two is

This is a linear relation, in the sense that

where is a constant. From this follows the connection between derivatives of microscopic and macroscopic quantities, that is, that the averaged derivative of a function is the derivative of the average:

Our comments:

Here it is the very important to note and observe the following two issues:

1) The first is that the real averaging volume - Representative Elementary Volume (REV) has never been described or even briefly illustrated in homogeneous physics.

Even there is no figure that students can locate in textbooks (if anyone find and point me out the place of publication, I will comment on that).

And this is common in orthodox homogeneous physics - researcher, reader or student can be easily driven into a variable false constructions in need at any moment. Mostly in atomic and particle physics this is the frivolous REV - most often the REV that is so large that it embraces the whole subject, piece of matter, material that is under investigation. This is done to declare many surficial integrals equal to zero.

In Homogeneous physics - like in the tested in this section monograph by Schwinger et al. (1998) and others, there are no mathematical methods, specifics, theorems to do averaging over these kinds of real form REVs, and even there are no definitions or wrong definitions (as in Continuum Homogeneous Mechanics) of a REV.

That is the recent design of coarse-graining sequence algorithms of "Superatoms" in chemistry of polymers is taken from the hierarchical scaled methodology developed in Hierarchical Scaled Fluid Mechanics and Thermal physics in 80s-90s, while using only chemical specific homogeneous techniques - which is the incorrect averaging.

The very important reason to hide and not specify the REV for this kind of averaging is obvious from the averaging construction (by Lorentz) of Atomic scale (microscale) Maxwell-Lorentz equations when for the key averaged terms and at the end needs to be fixed (fitted) the mathematical expressions for surficial integrals in a way that nullify the very important terms that we need to retain for correct integration.

In this way the equations are brought to the homogeneous continuum media form obtained by Lorentz and that form is incorrect.

And they have been falsified anyway, because the correct physical and mathematical methods of HSP-VAT have been already proven for many problems that homogeneous methods just bring approximate and often misleading incorrect results..

See in our -

and in

plus achievements updated up to 2004-2005 (we do more on that - not all achievements can be stood plainly up-front):

and other pages of our websites.

Meanwhile, all other conventional textbooks on electrodynamics and materials science show the same type of mathematical procedures for heterogeneous averaging. And this falsified electrodynamics lies in the very core of the Contemporary Orthodox Homogeneous physics (COHP) - from particle and atomic physics up to astrophysics of "Big Bangs" and "Black Holes," etc.

One can read further in p. 41 that:

"What is the macroscopic role of the bound charge distributions? It must be related to the effective charge and current densities given in terms of the polarization and magnetization by (4.43) and (4.44)

Microscale "Equations of Ampere-Maxwell for induced magnetic field and Gauss law for electric field

Faraday law of induction equation and Gauss law for magnetic field equation

Our comments:

Here authors instantly substituted the magnetic field for the magnetic induction field in the equations 1), 3) and 4); which is an innocent procedure only in the "vacuum" when with magnetic dipole moment is and because the electric dipole moment and there is no still averaging performed over the REV volume!!

But they can not do changing field for BEFORE averaging over the media with other than vacuum components, and we provide the averaging arguments against this move.

One of these arguments is that authors need to embed the field in other equations also like and where is not present just at the beginning before averaging will start. All these three equations are

Also, in this way of formulating the microscopic EM equations authors avoid appearance of convective term as in the equation of Ampere-Maxwell of induced magnetic field.

Then authors immediately obtained the "averaged" equations which in reality are the adjusted to look like "averaged" by Lorentz the known equations of macroscopic electrodynamics:

"These are averaged to yield the macroscopic equations,

With the capital letters for averaged fields.

Our comments:

The important thing here is that the fields and that were not averaged within the previous set of microscopic equations became INSTANTLY averaged (How come?) in this set .

Which is the magic action indeed.

We followed here all the actions authors of point-like particles obtained for justifying the conventional continuum Maxwell-Heaviside-Lorentz EM equations throughout the last ~100 years since Lorentz' pseudo-averaging of microscale EM equations, and of ~80 years since Dirac's theory of electron and quantum mechanics induced for the point-like particulate media in particle physics.

In p. 42 finally authors get the Maxwell-Lorentz equations written in traditional Lorentz' form:

" The final form of the historical, macroscopic Maxwell-Heaviside-Lorentz equations is

Our comments:

Returning back the notation of Ampere-Maxwell for induced magnetic field equation to the original form with that was at microscale artificially turned into just at the beginning of averaging procedures. And with no explanation, note!

Two-Scale Bottom-Up Averaging of Maxwell-Heaviside-Lorentz Equations (with Partial Time Derivatives) From the Sub-Atomic Scale (~10^(-15)m) Up to Polyatomic, Polymolecular Few Phase Medium (Material) with the Upper Scale ~10^(-(7-6))m [Sc]

There have been discussions and disputes throughout the most of the time of the Maxwell-Heaviside-Lorentz governing equations existance about what is the ground and a sense for these equations form. Mainly for the two most important and used everywhere and for most often situations - at the atomic scale for the atomic, sub-atomic applications and at the scale of homogeneous medium of perfect (not only as perfect) crystal constructed (or other materials) while for start adopted as a homogeneous at the nearest to the atomic scale - the scale of crystalline structure of solid state matter.

It might be as of [m] and of larger scale of consideration.

In all known up to now averaging procedures used in atomic, particle physics, electrodynamics applied the postulates specified above plus the medium of vacuum. This medium, as appeared through the many decades of research, is not an absolute empty space. It has characteristics from which we are interested in now that are common electromagnetism properties as the speed of light dielectric permittivity , and magnetic permeability

Media and Constitutive Elements (Particles)

We will use the atomic scale (microscale) as the ground base for Lower scale electromagnetism mathematical formulation in form of Maxwell-Lorentz equations.

We won't take right now into account other known particles apart of the photons, electrons, nuclei; plus atoms and simple molecules. Adding these kinds of media to a "vacuum" we mean for this development that we are assuming still that all phenomena of the lower than scale are taken here as of a material's components properties, and the real vacuum (empty space) does not exist as appeared throughout the findings of others.

That means the fields we are describing with the scale resolutions of are considered as "mean" and are averaged of the fields that are of the Lower scales.

Particles of atomic world are of extended volume no matter how small they are, we consider the point-like particles assumption that prevailed in the first half of XX century as the temporary fogging of the physicists minds.

The point particles were invented at the beginning of XX century in theoreticians circles as for convenient solution of the tasks with particles embedded into a unified field of interest. In that way there is no particle "field" any more. Only the volumetrically distributed mixture (homogenized) of point-particles and of vacuum or air, for example, field.

This theoretical point of view on particles influence quickly gained a stature via this mixture ideology. The famous mathematical tool by Dirac - the -function was created for this purpose to connect the physical presence and location of particles and for that do not work with the volumetric "particle" phases while communicating to quantum mechanics linear statistical operators, these are much easier to work with as the just mathematical points in the space.

In its turn the quantum mechanics whole existence is based on the point-like particles to treat those as merely statistical objects, events, which was really simplifying and workable idea in theoretical constructions for the times of 1920-1940s.

1) We consider as a medium number one the "active" vacuum that is being understood as a medium filling all voids in between CBR photons, other photons, other particles as electrons, within atomic "volumes", and in between the "atomic" volumes and "free" space. Atomic "volumes" are the volumes of actual variable size, as soon as outer electron shells are filled at situation.

We put some different content in the word "active" with regard of a "vacuum." It would be correct and just to use the old physical term Aether (ether) which in some areas of physics is just unavoidable - in Quantum Field Theories; orthodox Particle Physics itself; Optics homogeneous. Conventional homogeneous physicists call it shamefully as a "quantum sea," then they just prefer the "inherent property" for what they can not explain and can not accept the Aether's medium because of the dogmas of SR and GR.

Generally, the people should know - what they are not told in schools and universities, that the conventional homogeneous physics has been legitimizing the fudge in physics - any adjustment you want you can do and GO. You can get the Nobel for that. And it is not a joke. There are many examples of this award given for illusions. Just well organized.

2) Now we can not avoid or neglect the Cosmic Microwave Background Radiation (CMBR), even if we do not abridge ourselves saying that CMBR is the only medium that must be taken into account as an "active" vacuum. In this consideration we avoid the approach taken in XIX century by Maxwell and others while considering electromagnetic fields as having no carriers. Well, Maxwell actually thought as of kind of an aether, plus some "elements," then the "followers" have been misinterpreting his ideas.

As of an aether that was supported by Lorentz and all major physicists at that time while gaining a serious backing in our days, we will extend our treatments in the future by including speculative so far models of aether into the scaled electromagnetic phenomena theory.

3) Consider particles, including photons and electrons as the volumetric substances as they are. Assume that electrons have the unknown changeable volume that is bound by its surface, and which might have a spherical volume with mass and radius. That was the point of view of its discoverer J.J.Thomson, which must weight a lot. Then, his opinion was ridiculed later on over 30-40 years as of an old outdated man. And that were the "Point-Mass particles" physicists - who needed only their constructions being taken in the field as of a true value.

Nevertheless, since that time have been performed a lot of experimental work and analytical observations. At present, the strong amount of opinions has been advanced regarding the other (ring-like, torus volumetric) main sub-atomic first raw particles existence. We are specifying this with more detail in the "Particle Physics" section.

In that way while having a few media (species, phases) can be derived the Upper scale electromagnetic fields with the origin in the Lower scales fields, matters, and physical phenomena.

We are not to be free in this analysis allowing being engaged into discussion or lengthy analysis of issues - What are particles as electron, photon, nuclei? We would state for the start that the major parts of these issues are not known yet, in spite that more than 100 years passed since the electron discovery, thousands pages were written on the photon nature, etc.

So far for our purposes of deriving and comparing the scale dependable Bottom-Up strict mathematical formulation of equations for electromagnetic phenomena at atomic and continuum mechanics scales the properties spelled above for particles and "vacuum" are almost enough for this task.

As we already shown above and in our extended studies of hundred years attempts to deliver this averaging - that was not done correctly. That is why we would still need to exploit the correct mathematical tools while having advantage of current state of physics and scaled mathematics.

As soon as we consider the fields consisted of matter, that means we consider the material substance fields - the fields of free electrons, but that field will be constructed out of material particles with electron features, the fields of photons, atoms, and vacuum.

Scales of interest will be:

[m][m], or of ten orders of decimal magnitude difference. Where the Upper scale can be considered as of a Contimuum matter.

We will add here some text and basics of including that or those models for electron, photon, and nucleus while not to be engaged in this section into the specifics of scaled modeling in particle physics or nuclear one. We just want to have ability to simulate our most unusual terms on the Upper scale where the continuum electrodynamics formulates.

Averaging and Bottom-Up Scaleportation from an Atomic Scale to the "Continuum" like Meso-scale of Substance for the Maxwell-Heaviside-Lorentz Electrodynamics Equations

In Homogeneous physics - like in the tested in this section monograph by Schwinger et al. (1998) and others, there are no mathematical methods, specifics, theorems to do averaging over these kinds of real free and specific form REVs, and even there are no definitions or wrong definitions (as in Continuum Homogeneous Mechanics) of a REV.

Through the years we've put in a number of subsections of this and other websites many conceptual advancements and written physical and mathematical models regarding Heterogeneous Electrodynamics averaging, statements and two-scale solutions -

The averaging theory (incorrect) Lorentz developed and applied toward the dielectric materials fundamentals we can observe in -

The HSP-VAT concepts and methods were applied toward atomic scale phenomena, plasma theory and modeling -

solid state plasma modeling equations - the Lower scale equations that supposed to be already "averaged" - unfortunately they are not averaged -

also to Heterogeneous Continuum Mechanics Electrodynamics; that can be seen in -

as well as in our hard copy publications, some of them are in the references below.

In our further consideration we approach with scrutiny to the problem of Bottom-Up averaging and scaleportation (strict physical and mathematical communication) from the atomic scale Maxwell-Heaviside-Lorentz electrodynamics modeling equations up to the continuum electrodynamics scales.

The relevant research materials have rather lengthy content (tens of pages) that we submitting here in a concise format.

******************************************************************************

We start with the microscale simplest form of the Gauss law magnetic field equation

note that we can not assign to a system of vacuum plus charges BEFORE averaging has been performed. Thus, and we can write but only in CGSE system and before averaging.

While averaging is in progress we need to sum the three functions then we might have the sought after averaged value of where subscripts , , and stand for atomic, "vacuum", and free electrons averaged fields and volume fractions.

As for the atomic medium (after averaging all charges over the atoms occupied space) - over where subscripts , , and stand for bound electrons, nuclei, and "vacuum" within the atoms, molecules; where is the part of the "vacuum" within the atoms.

We might write (while still in the CGSE assuming that but only After averaging and at what scale? ) the Gauss law for magnetic field averaged in a matter equation as

where the surficial quantities of magnetic fields and are reflecting the interaction of medium ("phase") related averaged magnetic fields in a "vacuum", free electrons and atoms as the separate "phases."

Notice, that there is no complete field of induction as in homogeneous pseudo-averaging obtained for this Gauss law for magnetic field equation, but only the full averaged magnetic field This result has been obtained by heterogeneous averaging for the Upper scale form of this equation.

****************************************************************

Averaging over the polyphase material's media of the Gauss law for electric field equation

then we can have for dielectrics while using the conventional model for the following equation

as soon as we can write the atomic phase displacement field and averaged displacement field as

as long as in the "vacuum" the polarization vector

In this averaged Upper scale equation the surficial quantities of electric fields are reflecting the interaction of medium ("phase") related averaged electric fields in a "vacuum" and atoms, molecules as the separate "phases."

Nevertheless, when we have to use more correct and exact modeling for the we get for the averaged continuum medium Gauss law electric field equation

different again from the conventional continuum Upper scale equation, where is the polarization interface averaged term.

*******************************************************************************

The heterogeneous microscale (atomic scale) equation of Ampere-Maxwell for induced magnetic field is the second magnetic field equation in the media for Averaging

that after averaging over the sub-volumes of "vacuum," atoms, molecules, and free electrons becomes rather complicated forms with multiple variants actually for almost any kind of media.

The one is for the forms for the dielectrics with the weak influence of external fields

where and are the two terms thatare reflecting the interaction of medium ("phase") related averaged magnetic fields in a "vacuum" and atoms as the separate "phases"; is the average of magnetic orbital momentum of outer "shell" of atoms;

while and are the terms with average electric fields at atom-"vacuum" interface

and polarization average over the interface with the speed of the interface between the outer "shells" of atoms and "vacuum" displacement in a material.

For this averaged equation we can write the following summary:

5) Up to this development the communication between the atomic scale microstructure and microfields and continuum medium morphology and effective properties were based on the approximate tools followed the incomplete (incorrect to some extent) Maxwell-Heaviside-Lorentz macroscale (continuum) equations.

****************************************************************

The Faraday law of induction equation

while averaged over our heterogeneous polyphase polyatom material REV; then we can finilize with the notations as ( and get averaged equation for the Faraday law of induction for dielectrics

with the four heterogeneous interface surface averaged terms. These terms and are the two terms thatare reflecting the interaction of medium ("phase") related averaged electric fields in a "vacuum" and atoms as the separate "phases"; while are the terms with average magnetic fields at atom-"vacuum" interface average over the interface with the speed of the interface between the outer "shells" of atoms and "vacuum" displacement in a material.

For this averaged equation we can write the following summary:

4) Notice, that there is no magnetic induction field in the right hand side of this equation, but only averaged as soon as in homogeneous physics the field instantly is replaced by just before an averaging even being declared - which is incorrect.

*************************************************************************************

As we can see, the equations that are resulting after averaging are hard to compare with the conventional "bulk" media Maxwell-Heaviside-Lorentz (MHL) electrodynamics equations.

And the medium (material, substance) for which governing equations derived is not merely the one being explained with the chemical or materials science definitions, it is given with more strict definitions.

According to HSP-VAT the Loss and Gain Law (LGL) we can generally accept the qualitative transformation of fields and equations. Thus the fields at one scale physical and mathematical model and at that of even neighboring scale are different by nature of transformation by scaleportation - that might be and of different physical nature.

We can bring with force (making some more simplifications and equalizations) these Upper Scale averaged (L3D) ↑(U3D) equations to the MHL form of equations for some technical and just scientific reasons. Then, the issue will be the specifics and theory for the "effective" coefficients for MHL kind of equations.

CONCLUSIONS:

6) Lorentz himself in his "Clerk Maxwell's Electromagnetic Theory. The Rede Lecture for 1923, Cambridge" (1923) used to say that:

"Will it be possible to maintain these equations? I am not thinking here of the comparatively slight modifications that have been necessary in the theory of relativity;.....

A greater and really serious danger is threatening from the side of the quantum theory, for the existence of amounts of energy that remain concentrated in small spaces during their propagation, to which several phenomena seem to point,

is in absolute contradiction to Maxwell's equations.

However this may be, even if further development should require

profound alterations, Maxwell's theory will always remain a step of the highest importance in the progress of physics."

where he was citing Maxwell himself:

"It appears to me that while we derive great advantage from the recognition of the many analogies between the electrical current and a current of a material fluid, we must carefully avoid making any assumption not warranted by experimental evidence,......

A knowledge of these things would amount to at least the beginning of a complete dynamical theory of electricity, in which we should regard electrical action, not, as in this treatise, as a phenomenon due to an unknown cause, subject only to the general laws of dynamics,

but as the result of known motions of known portions of matter,

in which not only the total effects and final results, but the

whole intermediate mechanism and details of the motion, are taken as the object of study."

7) We obtained the additional terms based on averaging heterogeneous media techniques for every atomic scale electrodynamics averaged equation; for example the 6 additional atomic scale morphology related and explained terms obtained for the averaged Upper continuum scale Ampere-Maxwell equation for induced magnetic field. These and other equations terms should extend our abilities in understanding, modeling and design of materials, media.

8) Notice, that these effects as of and for Ampere-Maxwell equation are just absent in the homogeneous physics Maxwell-Heaviside-Lorentz continuum electrodynamics equations.

9) Notice, that there is no complete field of magnetic induction as in homogeneous pseudo-averaging obtained for the Gauss law for magnetic field equation, but only the full averaged magnetic field This result has been obtained by heterogeneous averaging for the Upper scale form of this equation.

10) Notice, that the magnetic induction field in the Ampere-Maxwell equation for induced magnetic field is written as or the sign before the magnetization term is proven as it is reversed to the notation used in homogeneous physics Maxwell-Heaviside-Lorentz equations (started from averaging by Lorentz) because in homogeneous physics the micromagnetic field (of atomic scale) instantly is replaced by the atomic scale magnetic induction field just before an averaging even being declared - which is incorrect.

Meanwhile, there is no atomic scale magnetic induction field unless we either making a "SPECIAL" theory related to a separate particle (electron, nucleus) and only, or have been averaged the field over the some specified volume that would include the "vacuum" and the particles, but not a homogenized unknown one "phase" which is called vacuum (and which is the mix of "vacuum" and of smeared particles presented as volumeless "dot-points") - as of an atom or of larger dimension as the REV and have already received that field as a result of averaging with all components of this field, but obtaned as by averaging and shown as averaged either over the atom or the REV. Otherwise, the instant unjustified change of with is just the "fixing" to the Lorentz form of continuum scale equations.

11) Notice, that there is no magnetic induction field in the right hand side of the Faraday law of induction equation, but only the averaged again because in homogeneous physics the field instantly is replaced by (see in each atomic physics, electrodynamics textbook) just before an averaging even being declared - which is incorrect.

12) Notice, that there is NO ONE additional coefficient appeared in the continuum equations for the Upper scale - that means we should seek the effective coefficients based on the only fundamental properties at atomic scale of constituent's atoms, "vacuum", and of morphology of materials at atomic scale. Plus the external fields influence.

This was never achieved before.

13) Up to this development the communication between the atomic scale microstructure and microfields and continuum medium morphology and effective properties were based on the approximate tools followed the incomplete (incorrect to some extent) Maxwell-Heaviside-Lorentz macroscale (continuum) equations.

14) There would be many formulations in electrodynamics for different matters and for different scales. Not as of the case that used right now which is the application of the same kind of Lorentz' Continuum Mechanics electrodynamics equations for any scale and any medium. Just using changed coefficients that are the adjusting parameters right now.

15) As one can observe, the Upper scale continuum averaged electrodynamics equations that have the Lower scale MHL equations, can be classified as the set of integro-differential equations. One can derive on their base the numerous implementations reflecting the specific physical task. That is the way to evaluate the standard test specific property "bulk" coefficients for conventional homogeneous medium electrodynamics. This problem is of incredible value itself.

16) The Upper scale averaged atomic scale fundamental electrodynamics equations should help to design experiment(s) aiming at study of electrons outer atomic shell spatial distribution and morphology.

17) The same kind of pseudo-averaging (continuum mechanics derivation within itself) was brought into the contemporary models, mathematical formulation in hydrodynamics that are the partial differential equations for hydrodynamics phenomena - as the Navier-Stokes equations for laminar flow.

Having these incorrect averaging procedures from atomic to continuum mechanics formulation as those used for Maxwell-Heaviside-Lorentz electrodynamics equations (besides of that, there was no knowledge about the structure of atoms and molecules at the time of Stokes, other physical and mathematical schemes of XX century are not qualified for correct scaleportation of momentum and impulse as well); it is not surprising that the oldest unsolved problem in physics called the turbulence explanation and its dynamics formulation has not been solved and can not be solved based on the accepted now laminar form equations of fluid dynamics. More than hundred years effort has given the engineering approximations that are not suitable for phenomena inherently stated with the polyscale interactions, as plasma modeling, for example, for projects like the ITER.

REFERENCES:

1) Hecht, L., "SHOULD THE LAW OF GRAVITY BE REPEALED? The Suppressed Electrodynamics of Ampere-Gauss-Weber," Editorial, 21st Century Science & Technology, Spring, (2001)

2) Hecht, L., "Science: To Be, or Not to Be Or, How I Discovered the Swindle of Special Relativity," Editorial, 21st Century Science & Technology, Winter 1999-2000, (2000)

3) Hecht, L., "The Atomic Science Textbooks Don't Teach. The Significance of the 1845 Gauss-Weber Correspondence," Editorial, 21st Century Science & Technology, pp. 21-43, (1996)

4) Jackson, J., Classical Electrodynamics, 3rd ed., Wiley and Sons, New York, NY, (1999). pp. 1-808.

5) Landau, L.D. and Lifshits, E.M., Electrodynamics of Continuous Media,Pergamon Press, Oxford, (1960).

6) Landau, L.D. and Lifshits, E.M., Electrodynamics of Continuous Media, Gos. Izd. Tekh.-Teor. Lit., Moscow, (1957). 532 pgs. (in Russian).

7) Levich, V.G., Course of Theoretical Physics, Vol. 1, Nauka Publishing, Moscow, (1969). 912 pgs.(in Russian).

8) Lorentz, H.A., The Theory of Electrons and Its Applications to the Phenomena of Light and Radiant Heat : a Course of Lectures Delivered in Columbia University in 1906, Dover, New York, N.Y., (1952). pp. 1-343.

9) Lorentz, Hendrik A., Clerk Maxwell's Electromagnetic Theory. The Rede Lecture for 1923, Cambridge University Press, Cambridge, 1923. pp. 1-35.

10) Maxwell, J.C., A Treatise on Electricity and Magnetism, Vol. 1, 3rd Ed., Oxford University Press, New York, (2002a). pp. 1-506, plus 14 pages with the figures attached.

11) Maxwell, J.C., A Treatise on Electricity and Magnetism, Vol. 2, 3rd Ed., Oxford University Press, New York, (2002b). pp. 1-500, plus 6 pages with the figures attached.

12) Dayton Miller, "The Ether-Drift Experiment and the Determination of the Absolute Motion of the Earth", Reviews of Modern Physics, Vol.5(2), p.203-242, July (1933).

13) Dayton Miller, "The Ether-Drift Experiment and the Determination of the Absolute Motion of the Earth", Nature, Vol.133, p.16-27, 3 Feb. (1934).

14) Morley, E.W. and Miller, D., "Report of an Experiment to Detect the FitzGerald-Lorentz Effect", Proceedings, Am. Acad. Arts & Sciences, 41:321-328, (1905).

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