On refinement of ertain laws of lassial eletrodynamis http://fmnauka.narod.ru/works.html F. F. Mende Abstrat mende_fedor@mail.ru In the ontemporary lassial eletrodynamis exists many unresolved problems. The law of the indution of Faraday does not desribe all known manifestations of indution. Unipolar generator is exeption. Inomprehensible is the physial ause for indution. Long time was onsidered that suh material parameters, as dieletri and magneti onstant they an depend on frequeny. It turned out that this not thus. To the examination this problems is dediated the artile. Introdution Until now, some problems of lassial eletrodynamis involving the laws of eletromagneti indution have been interpreted in a dual or even ontraversal way. As an example, let us onsider how the homopolar operation is explained in different works. In [] this is done using the Faraday low speified for the disontinuous motion ase. In [] the rule of flow is rejeted and the operation of the homopolar generator is explained on the basis of the Lorentz fore ating upon harges. The ontraditory approahes are most evident in Feynman s work [] (see page 53): the rule of flow states that the ontour e.m.f. is equal to the opposite-sign rate of hange in the magneti flux through the ontour when the flux varies either with the hanging field or due to the motion of the ontour (or to both). Two options the ontour moves or the field hanges are indistinguishable within the rule. Nevertheless, we use these two ompletely different laws to explain the rule for the two ases: [ V B B ] for the moving ontour and E for the t hanging field. And further on: There is hardly another ase in physis when a simple and aurate general law has to be interpreted in terms of two different phenomena. Normally, suh beautiful generalization should be based on a unified
fundamental priniple. Suh priniple is absent in our ase. The interpretation of the Faraday law in [] is also ommonly aepted: Faraday s observation led to the disovery of a new law relating eletri and magneti fields: the eletri field is generated in the region where the magneti field varies with time. There is however an exeption to this rule too, though the above studies do not mention it. However, as soon as the urrent through suh a solenoid is hanged, an eletri field is exited externally. The exeption seem to be too numerous. The situation really auses onern when suh noted physiists as Tamm and Feynman have no ommon approah to this seemingly simple question. It is knowing [3] that lassial eletrodynamis fails to explain the phenomenon of phase aberration. As applied to propagation of light, the phenomenon an be explained only in terms of the speial theory of relativity (STR). However, the Maxwell equations are invariant with respet to the ovariant STR transformations, and there is therefore every reason to hope that they an furnish the required explanation of the phenomenon. It is well known that eletri and magneti indutivities of material media an depend on frequeny, i.e. they an exhibit dispersion. But even Maxwell himself, who was the author of the basi equations of eletrodynamis, believed that ε and µ were frequeny-independent fundamental onstants. How the idea of ε and µ-dispersion appeared and evolved is illustrated vividly in the monograph of well-known speialists in physis of plasma [4]: while working at the equations of eletrodynamis of material, media, G. Maxwell looked upon eletri and magneti indutivities as onstants (that is why this approah was so lasting). Muh later, at the beginning of the XX entury, G. Heavisidr and R.Wull put forward their explanation for phenomena of optial dispersion (in partiular rainbow) in whih eletri and magneti indutivities ame as funtions of frequeny. Quite reently, in the mid-5ies of the last entury, physiists arrived at the onlusion that these parameters were dependent not only on the frequeny but on the wave vetor as well. That was a revolutionary
3 breakaway from the urrent onepts. The importane of the problem is learly illustrated by what happened at a seminar held by L. D. Landau in 954, where he interrupted A. L. Akhiezer reporting on the subjet: Nonsense, the refrative index annot be a funtion of the refrative index. Note, this was said by L. D. Landau, an outstanding physiist of our time. What is the atual situation? Running ahead, I an admit that Maxwell was right: both ε and µ are frequeny independent onstants haraterizing one or another material medium. Sine dispersion of eletri and magneti indutivities of material media is one of the basi problems of the present day physis and eletrodynamis, the system of views on these questions has to be radially altered again (for the seond time!). In this ontext the hallenge of this study was to provide a omprehensive answer to the above questions and thus to arrive at a unified and unambiguous standpoint. This will ertainly require a revision of the relevant interpretations in many fundamental works.. Equations of eletromagneti indution in moving oordinates The Maxwell equations do not permit us to write down the fields in moving oordinates proeeding from the known fields measured in the stationary oordinates. Generally, this an be done through the Lorentz transformations but they so not follow from lassial eletrodynamis. In a homopolar generator, the eletri fields are measured in the stationary oordinates but they are atually exited in the elements whih move relative to the stationary oordinate system. Therefore, the priniple of the homopolar generator operation an be desribed orretly only in the framework of the speial theory of relativity (STR). This brings up the question: Can lassial eletrodynamis furnish orret results for the 3
4 fields in a moving oordinate system, or at least offer an aeptable approximation? If so, what form will the equations of eletromagneti indution have? The Lorentz fore is F e E + e V B (.) It bears the name of Lorentz it follows from his transformations whih permit writing the fields in the moving oordinates if the fields in the stationary oordinates are known. Heneforward, the fields and fores generated in a moving oordinate system will be indiated with primed symbols. The lues of how to write the fields in moving oordinates if they are known in the stationary system are available even in the Faraday law. Let us speify the form of the Faraday law: d Φ d t B E d l (.) The speified law, or, more preisely, its speified form, means that E and dl should be primed if the ontour integral is sought for in moving oordinates and unprimed for stationary oordinates. In the latter ase the right-hand side of Eq. (.) should ontain a partial derivative with respet to time whih fat is generally not mentioned in literature. The total derivative with respet to time in Eq. (.) implies that the final result for the ontour e.m.f. is independent of the variation mode of the flux. In other words, the flux an hange either purely with time variations of B or beause the system, in whih E d l is measured, is moving in the spatially varying field B. In Eq. (.) Φ B B d S, (.3) 4
where the magneti indution B 5 µ H is measured in the stationary oordinates and the element d S in the moving oordinates. Taking into aount Eq. (.3), we an find from Eq. (.) d d t E d l B d S. (.4) Sine d d t + V grad, we an write t B E d l d S [ B V] d l V div B d S t. (.5) In this ase ontour integral is taken over the ontour d l, overing the spae d S. Heneforward, we assume the validity of the Galilean transformations, i.e. d l d l and d S d S. Eq. (.5) furnishes the well-known result: E E + V B, (.6) [ ] whih suggests that the motion in the magneti field exites an additional eletri field desribed by the final term in Eq. (.6). Note that Eq. (.6) is obtained from the slightly speified Faraday law and not from the Lorentz transformations. Aording to Eq. (.6), a harge moving in the magneti field is influened by a fore perpendiular to the diretion of the motion. However, the physial nature of this fore has never been onsidered. This brings onfusion into the explanation of the homopolar generator operation and does not permit us to explain the eletri fields outside an infinitely long solenoid on the basis of the Maxwell equations. To lear up the physial origin of the final term in Eq. (.6), let us write B and E in terms of the magneti vetor potential A B: 5
6 A B B rot AB, E. (.7) t Then, Eq. (.6) an be re-written as A E t B + [ V rot A ] B, (.8) and further: A E t B ( V ) A + grad ( V A ) B B. (.9) The first two terms in the right-hand side of Eq. (.9) an be onsidered as the total derivative of the vetor potential with respet to time: d A B E + grad ( V AB ). (.) d t As seen in Eq. (.9), the field strength, and hene the fore ating upon a harge onsists of three omponents. The first omponent desribes the pure time variations of the magneti vetor potential. The seond term in the right-hand side of Eq. (.9) is evidently onneted with the hanges in the vetor potential aused by the motion of a harge in the spatially varying field of this potential. The origin of the last term in the right-hand side of Eq. (.9) is quite different. It is onneted with the potential fores beause the potential energy of a harge moving in the potential field A B at e V e grad V desribes the A B the veloity V is equal to ( ). The magnitude ( ) fore just as the salar potential gradient does. Using Eq. (.9), we an explain physially all the strength omponents of the eletroni field exited in the moving and stationary ooperates. If our onern is with the eletri fields outside a long solenoid, where the no magneti field, the first term in the right-hand side of Eq. (.9) ome into play. In the ase of a 6 A B
7 homopolar generator, the fore ating upon a harge is determined by the last two terms in the right-hand side of Eq.(.9), both of them ontributing equally. It is therefore inorret to look upon the homopolar generator as the exeption to the flow rule beause, as we saw above, this rule allows for all the three omponents. Using the rotor in both sides Eq. (.) and taking into aount rot grad, we obtain d B rot E. (.) d t If motion is absent, Eq. (.) turns into Maxwell equation (.). Equation (.) is ertainly less informative than Eq. (.): beause rot grad, it does not e grad V. It is therefore more reasonable inlude the fores defined in terms ( ) to use Eq. (.) if we want to allow for all omponents of the eletri fields ating upon a harge both in the stationary and in the moving oordinates. As a preliminary onlusion, we may state that the Faraday Law, Eq. (.), when examined losely, explains learly all features of the homopolar generator operation, and this operation priniple is a onsequene, rather than an exeption, of the flow rule, Eq. (.). Feynman s statement that [ V B ] for the moving B ontour and E for the varying field are absolutely different laws is t ontrary to fat. The Faraday law is just the sole unified fundamental priniple whih Feynman delared to be missing. Let us lear up another Feynman s interpretation. Faraday s observation in fat led him to disovery of a new law relating eletri and magneti fields in the region where the magneti field varies with time and thus generates the eletri field. This orrelation is essentially true but not omplete. As shown above, the eletri field an also be exited where there is no magneti field, namely, outside an infinitely long solenoid. A more A B 7
8 omplete formulation follows from Eq. (.9) and the relationship more general than rot B E. t d AB E is d t This suggests that a moving or stationary harge interats with the field of the magneti vetor potential rather than with the magneti field. The knowledge of this potential and its evolution an only permit us to alulate all the fore omponents ating upon harges. The magneti field is merely a spatial derivative of the vetor field. As follows from the above onsideration, it is more appropriate to write the Lotentz fore in terms of the magneti vetor potential F e E + e V rot A ] e E е( V ) A + еgrad ( V A ), (.) [ B B B whih visualizes the omplete struture of the fore. The Faraday law, Eq. (.) is referred to as the law of eletromagneti indution beause it shows how varying magneti fields an generate eletri fields. However, lassial eletrodynamis ontains no law of magnetoeletri indution showing how magneti fields an be exited by varying eletri fields. This aspet of lassial eletrodynamis evolved along a different pathway. First, the law H d l I, (.3) was known, in whih I was the urrent rossing the area of the integration ontour. In the differential from Eq. (.3) beomes rot H j σ, (.4) where jσ is the ondution urrent density. Maxwell supplemented Eq. (.4) with displaement urrent 8
9 9 t D j H rot σ +. (.5) However, if Faraday had performed measurement in varying eletri indution fluxes, he would have inferred the following law Φ t d d l H d D, (.6) where Φ S Dd D is the eletri indution flux. Then + + S D d V div l d V D S d t D l Hd ] [. (.7) Unlike div B in magneti fields, eletri fields are haraterized by div ρ D and the last term in the right-hand side Eq. (.7) desribes the ondution urrent I, i.e. the Ampere law follows from Eq. (.6). Eq. (.7) gives [ V] D H, (.8) whih was earlier obtainable only from the Lorentz transformation. Moreover, as was shown onviningly in [], Eq. (.8) also leads out of the Biot-Savart law if magneti fields are alulated from the eletri fields exited by moving harges. In this ase the last term in the right-hand side Eq. (.7) an be omitted and the indution laws beome ompletely symmetrial.. ] [, ] [ + l d V D S d t D l Hd l d V B S d t B l E d (.9). ] [, ] [ D V H H B V E E + (.)
Earlier, Eqs. (.) were only obtainable from the ovariant Lorentz transformations, i.e. in the framework of speial theory of relativity (STR). Thus, V the STR results aurate to the ~ terms an be derived from the indution laws V through the Galilean transformations. The STR results aurate to the terms an be obtained through transformation of Eq (.9). At first, however, we shall introdue another vetor potential whih is not used in lassial eletrodynamis. Let us assume for vortex fields [5] that D rot, (.) A D where A D is the eletri vetor potential. It then follows from Eq. (.9) that A H t D + [ V ] AD grad [ V AD ], (.) or A H t D [ V rot AD], (.3) or d A D H grad [ V AD]. (.4) d t These equations present the law of magnetoeletri indution written in terms of the eletri vetor potential. To illustrate the importane of the introdution of the eletri vetor potential, we ome bak to an infinitely long solenoid. The situation is muh the same, and the only hange is that the vetors B are replaed with the vetors D. Suh situation is quite realisti: it ours when the spae between the flat apaitor
plates is filled with high eletri indutivities. In this ase the displaement flux is almost entirely inside the dieletri. The attempt to alulate the magneti field outside the spae oupied by the dieletri (where D ) runs into the same problem that existed for the alulation beyond the fields E of an infinitely long solenoid. The introdution of the eletri vetor potential permits a orret solution of this problem. This however brings up the question of priority: what is primary and what is seondary? The eletri vetor potential is no doubt primary beause eletri vortex fields are exited only where the rotor of suh potential is non-zero. As follows from Eqs. (.), if the referene systems move relative to eah other, the fields E and H are mutually onneted, i.e. the movement in the fields H indues the fields E and vie versa. But new onsequenes appear, whih were not onsidered in lassial eletrodynamis. For illustration, let us analyze two parallel onduting plates with the eletri field E in between. In this ase the surfae harge ρ S per unit area of eah plate is εе. If the other referene system is made to move parallel to the plates in the field Е at the veloity V, this motion will generate an additional field Н VεЕ. If a third referene system starts to move at the veloity V, within the above moving system, this motion in the field Н will generate Е µε V Е, whih is another ontribution to the field Е. The field E thus beomes stronger in the moving system than it is in the stationary one. It is reasonable to suppose that the surfae harge at the plates of the initial system has inreased by µε V E as well. This tehnique of field alulation was desribed in [6]. If we put E and H for the field omponents parallel to the veloity diretion and E and H for the perpendiular omponents, the final fields at the veloity V an be written as
, ] [,, ] [, V h s E V Z V V h H H H H V h s H V V Z V h E E E E + (.5) where ε µ Z is the spae impedane, µ ε is the veloity of light in the medium under onsideration. The results of these transformations oinide with the STR data with the auray to the ~ V terms. The higher-order orretions do not oinide. It should be noted that until now experimental tests of the speial theory of relativity have not gone beyond the ~ V auray. As an example, let us analyze how Eqs. (.5) an aount for the phenomenon of phase aberration whih was inexpliable in lassial eletrodynamis. Assume that there are plane wave omponents H Z and E X, and the primed system is moving along the x-axis at the veloity V X. The field omponents with in the primed oordinates an be written as.,, V h H H Vx sh H E E E X Z Z Z Y X X (.7) The total field Е in the moving system is
3 E EX + EY E X h V X. (.8) Hene, the Poynting vetor no longer follows the diretion of the y-axis. It is in the xy-plane and tilted about the y-axis at an angle determined by Eqs. (.7). The ratio between the absolute values of the vetors Е and Н is the same in both the systems. This is just what is known as phase aberration in lassial eletrodynamis.. Magneti field problem As follows from the transformations in Eq. (.5) if two harges move at the relative veloity V, their interation is determined not only by the absolute values of the harges but by the relative motion veloity as well. The new value of the interation fore is found as [7-9] g F V g h r 4π ε r 3, (.) where r is the vetor onneting the harges, V is the omponent of the veloity V, normal to the vetor r. If opposite-sign harges are engaged in the relative motion, their attration inreases. If the harges have the same signs, their repulsion enhanes. For V, Eq. (.) beomes the Coulomb law. Using Eq. (.), a mew value of the potential ϕ(r) an be introdued at the point, where the harge g is loated, assuming that g is immobile and only g exeutes the relative motion 3
4 V g h ϕ( r). (.) 4π ε r We an denote this potential as salar-vetor, beause its value is dependent not only on the harge involved but on the value and the diretion of its veloity as well. The potential energy of the harge interation is V g g h W. (.3) 4π ε r Eqs. (.), (.) and (.3) apparently aount for the hange in the value of the moving harges. Using these equations, it is possible to alulate the fore of the ondutorurrent interations and allow, through superposition, for the interation fores of all moving and immobile harges in the ondutors. We thus obtain all urrently existing laws of eletromagnetim. Let us examine the fore, interation of two r-spaed ondutors (Fig. ) assuming that the eletron veloities in the ondutors are V and V. The moving harge values per unit length of the ondutors are g and g. In terms of the present-day theory of eletromagnetism, the fores of the interation of the ondutors an be found by two methods. One of the ondutors (e.g., the lower one) generates the magneti field H(r) in the loation of the first ondutor. This field is g V H( r). (.4) π r 4
5 Fig.. Shemati view of fore interation between urrent-arreging ondutors of a two-ondutor line in terms of the present-day model. The field E is exited in the oordinate system moving together with the harges of the upper ondutor: [ V B] V H( ) E µ r. (.5) I.e. the harges moving in the upper ondutor experiene the Lorentz fore. This fore per unit length of the ondutor is µ g V g V I I π r. (.6) π ε r F Eq. (.6) an be obtained in a different way. Assume that the lower ondutor exites a vetor potential in the region of the upper ondutor. The z omponent of the vetor potential is g V lnr lnr A Z. (.7) π ε I π ε 5
6 The potential energy per unit length of the upper ondutor arrying the urrent I in the field of the vetor potential A Z is W I I lnr IAZ. (.8) π ε Sine the fore is the derivative of the potential energy with respet to the opposite-sign oordinate, it is written as W I I. (.9) r π ε r F Both the approahes show that the interation fore of two ondutors is the result of the interation of moving harges: some of them exite fields, the others interat with them. The immobile harges representing the lattie do not partiipate in the interation in this sheme. But the fores of the magneti interation between the ondutors at just on the lattie. Classial eletrodynamis does mot explain how the moving harges experiening this fore an transfer it to the lattie. The above models of iteration are in unsolvable onflit, and experts in lassial eletrodynamis prefer to pass it over in silene. The onflit is onneted with estimation of the interation fore of two parallel-moving harges. Within the above models suh two harges should be attrated. Indeed, the indution В aused by the moving harge g at the distane r is B g V. (.) π ε r If another harge g moves at the same veloity V in the same diretion at the distane r from the first harge, the indution В at the loation g produes the fore attrating g and g. F g g V (.) 4π ε r 6
7 An immovable observer would expet these harges to experiene attration along with the Coulomb repulsion. For an observer moving together with the harges there is only the Coulomb repulsion and no attration. Neither lassial eletrodynamis not the speial theory of relativity an solve the problem. Physially, the introdution of magneti fields reflets ertain experimental fats, but so far we an hardly understand where these fields ome from. In 976 it was reported in a serious experimental study that a harge appeared on a short-iruited superonduting solenoid when the urrent in it was attenuating. The results of [] suggest that the value of the harge is dependent on its veloity, whih is first of all in ontradition with the harge onservation law. The author of this study has also investigated this problem [] (see below). It is useful to analyze here the interation of urrent-arrying systems in terms of Eqs. (.), (.) and (.3). We ome bak again to the interation of two thin ondutors with harges moving at the veloities V and V (Fig. ). Fig.. Shemati view of fore interation between urrent-arrying wires of a two-ondutor line. The lattie is harged positively. 7
8 g +, g + and g, g are the immobile and moving harges, respetively, pre unit length of the ondutors. g + and g + refer to the positively harged lattie in the lower and upper ondutors, respetively. Before the harges start moving, both the ondutors are assumed to be neutral eletrially, i.e. they ontain the same number of positive and negative harges. Eah ondutor has two systems of unlike harges with the speifi densities g +, g and g +, g. The harges neutralize eah other eletrially. To make the analysis of the interation fores more onvenient, in Fig. the systems are separated along the z-axis. The negative-sign subsystems (eletrons) have veloities V and V. The fore of the interation between the lower and upper ondutors an be onsidered as a sum of four fores speified in Fig. (the diretion is shown by arrows). The attration fores F 3 and F 4 are positive, and the repulsion fores F and F are negative. Aording to Eq. (.), the fores between the individual harge subsystems (Fig. ) are F F F F 3 4 + + g g, π ε r g g V h π ε r + g g V + h π ε r + g g V + h π ε r V,., (.) By adding up the four fores and remembering that the produt of unlike harges and the produt of like harges orrespond to the attration and repulsion fores, respetively, we obtain the total speifi fore per unit length of the ondutor g g V V V V FΣ h + h h. (.3) π ε r 8
9 where g and g are the absolute values of harges. The signs of the fores appear in the braketed expression. Assuming V<< с, we use only the two first terms in the expression V h, i.e. h V + V. Eq. (.3) gives g V gv I I FΣ, (.4) π ε r π ε r where g and g are the absolute values of speifi harges, and V, V are taken with their signs. It is seen that Eqs. (.6), (.9) and (.3) oinide though they were obtained by different methods. Aording to Feynman (see the introdution), the e.m.f. of the iruit an be interpreted using two absolutely different laws. The paradox has however been larified. The fore of the enteration between the urrent-arrying systems an be obtained even by three absolutely different methods. But in the third method, the motion magneti field is no longer neessary and the lattie an diretly partiipate in the formation of the interation fores. This was impossible with the previous two tehniques. In pratie the third method however runs into a serious obstale. Assuming g + and V, i.e. the interation, for example, between the lower urrentarrying system and the immobile harge g the interation fore is g gv Σ. (.4) π ε r F This means that the urrent in the ondutor is not eletrially neutral, and the eletri field g V E, (.5) 4π ε r is exited around the ondutor, whih is equivalent to an extra speifi stati harge on the ondutor 9
V g g. (.6) Before [], there was no evidene for generation of eletri fields by d.. urrents. When Faraday and Maxwell formulated the basi laws of eletrodynamis, it was impossible to onfirm Eq. (.6) experimentally beause the urrent densities in ordinary ondutors are too small to detet the effet. The assumption that the harge is independent of its veloity and the subsequent introdution of a magneti field were merely voluntaristi ats. In superondutors the urrent densities permit us to find the orretion for V the harge ~ g experimentally. Initially, [] was taken as evidene for the dependene of the value of the harge on its veloity. The author of this study has also investigated this problem [], but, unlike [], in his experiments urrent was introdued into a superonduting oil by an indutive non-ontat method. Even in this ase a harge appeared on the oil []. The experimental objets were superonduting omposite Nb Ti wires oated with opper, and it is not leat what mehanism is responsible for the harge on the oil. It may be brought by mehanial deformation whih auses a displaement of the Fermi level in the opper. Experiments on non-oated superonduting wires may be more informative. Anyhow, the subjet has not been exhausted and further experimental findings are of paramount importane to fundamental physis. Using this model, we should remember that there is no reliable experimental data on stati eletri fields around the ondutor. Aording to Eq. (.5), suh fields are exited beause the value of the harge is dependent on its veloity. Is there any physial mehanism whih ould maintain the interating urrent-arrying systems eletrially neutral within this model? Suh mehanism does exist. To explain it, let us onsider the urrent-arrying iruit in Fig. 3. This is a superonduting thin film whose thikness is smaller than the field penetration depth in the
superondutor. The urrent is therefore distributed uniformly over the film thikness. Assume that the bridge onneting the wide parts of the film is muh narrower than the rest of the urrent-arrying film. If persistent urrent is exited in suh a iruit, the urrent density and hene the urrent arrier veloity V in the bridge will muh exeed the veloity V in the wide parts of the film. Suh situation is possible if the urrent arriers are aelerated in the part d and slowed down in the part d. But aeleration and slowing-down of harges is possible only in eletri fields. If V > V, the potential differene between the parts d and d whih auses aeleration or slowing-down is determined as m V U. (.7) e This potential differene an appear only due to the harge density gradient in the parts d and d, i.e. the density of harge arriers dereases with aeleration and inreases with slowing down. The relation n > n should be fulfilled, where n and n are the urrent-arrier densities in the wide and narrow bridge parts of the film, Fig. 3. film. Shemati view of a urrent-arrying iruit based on a superonduting
respetively. It is lear that some energy is needed to aelerate harges whih have masses. Let us find out where this energy omes from. On aeleration the eletrostati energy available in the eletrostati field of the urrent arriers onverts into kineti energy. The differene in eletrostati energy between two idential volumes having different eletron densities an be written as e W n, (.8) 8π ε r where n n n, e is the eletron harge, r is the eletron radius. Sine e 8π ε r m, (.9) where m is the eletron mass, Eq. (.46) an be rewritten as This energy is used to aelerate the urrent arriers. Hene, W n m. (.) and n m V W, (.) V n n. (.) The eletron density in a moving flow is V n n. (.3)
3 3 We see that the hange in the urrent-arrier density is quite small, but this hange is just responsible for the existene of the longitudinal eletri field aelerating or slowing down the harges in the parts d and d. Let us all suh fields onfiguration fields as they are onneted with a ertain onfiguration of the ondutor. These fields are available in normal ondutors too, but they are muh smaller than the fields related to the Ohmi resistane. We an expet that a voltmeter onneted to the iruit, like is shown in Fig. 3, would be apable of registering the onfiguration potential differene in aordane with Eq. (.7). If we used an ordinary liquid and a manometer instead of a voltameter, aording to the Bernoulli equation, the manometer ould register the pressure differene. For lead films, the onfiguration potential differene is ~ -7 В, though it is not observablt experimentally. We an explain this before hand. As the veloities of the urrent arriers inrease and their densities derease, the eletri fields njrmal to their motion enhane. These two preesses ounterbalane eah other. As a result, the normal omponent of the eletri field has a zero balue in all parts of the film. In terms of the onsidered, this looks like r g g F ε π + +, V V h V V r g g F ε π,, 3 V h V r g g F + ε π (.4) V h V r g g F 4 + ε π. The braketed expressions in Eqs. (.4) allow for the motion-related hange in the density of the harges g and g.
After expanding h, multiplying out and allowing only for the ~ terms, Eqs. (.4) give 4 V / F F F 3 + + g g, π ε r g g VV π ε r + g g π ε r,, (.5) F 4 + g g π ε r. By adding up F, F, F 3 and F 4, we obtain the total fore of the interation g V gv I I Σ. (.6) π ε r π ε r F Again, we have a relation oiniding with Eqs. (.6) and (.9). However, in this ase the urrent-arrying ondutors are neutral eletrially. Indeed, if we analyze the fore interation. For example, between the lower ondutor and the upper immobile harge g (putting g + and V ), the total interation fore will be zero, i.e. the ondutor with flowing urrent is eletrially neutral. If we onsider the interation of two parallel moving eletron flows (taking g + g + and V V ), aording to Eq. (.), the total fore is F Σ g g. (.7) π ε r It is seen that two eletron flows moving at the same veloity in the absene of a lattie experiene only the Coulomb repulsion and no attration inluded into the magneti field onept. 4
5 Physially, in this model the fore interation of the urrent-arrying systems is not onneted with any now field. The interation is due to the enhanement of the eletri fields normal to the diretion of the harge motion. The phenomenologial onept of the magneti field of orret only when the harges of the urrent arriers are ompensated with the harges of the immobile lattie, the urrent arriers exite a magneti field. The magneti field onept is not orret for freely moving harges when there are no ompensating harges of the lattie. In this ase a moving harged partile or a flow of harged partiles does not exite a magneti field. Thus, the onept of the phenomenologial magneti field is true but for the above ase. It is easy to show that using the salar-vetor potential, we an obtain all the presently existing laws of magnetism. Besides, the approah proposed permits a solution of the problem of the interation between two parallel-moving harges whih ould not be solved in terms of the magneti field onept.. Problem of eletromagneti radiation Whatever ours in eletrodynami, it is onneted with the interation of moving and immobile harges. The introdution of the salar-vetor potential answers this question. The potential is based on the laws of eletromagneti and magnetoeletri indution. The Maxwell equations desribing the wave proesses in material media also follow from these laws. The Maxwell equations suggest that the veloity of field propagation is finite and equal to the veloity of light. The problem of eletromagneti radiation an be solved of the elementary level using the salar-vetor potential and the finiteness of propagation of eletri proesses. For this purpose, the retarded salar-vetor potential 5
6 V g h ϕ ( r, t), 4π ε r (3.) is introdued, where V is the veloity of the harge g at the moment r t t, normal to the vetor r, r is the distane between the harge g and point (Fig. 4), where the field is sought for at the moment t. The field at point an be found from the relation E grad ϕ. Assume that at the moment r t the harge g is at the origin of the oordinates and its veloity is V (t). The field E y at point is E y (, t) ϕ e V (t) h y 4π ε r y. (3.) Differentiation is performed assuming r to be a onstant magnitude. From Eq. (3.) we obtain E y e V (t) V (t) V (t) e s h s h 4π ε r y 4π ε r V (t) t V (t). (3.3) Using only the first term of the expansion of V s h (t) we an obtain from Eq. (3.3) E y x x vy t eay t e ( x, t) 4πε x t 4πε x. (3.4) 6
7 Fig. 4. Formation of the retarded salar-vetor potential. In this equation of a y x t is the being late aeleration of harge. This equation is wave equation and defines both the amplitude and phase responses of the wave of the eletri field, radiated by the moving harge. If we as the diretion of emission take the vetor, whih omposes with the axis of y the angle of α, then Eg. (3.4) will be written down: E y x eay t sin α ( x, t, α). (3.5) 4πε x Eg. (3.5) determines the radiation pattern. Sine there is a axial symmetry relative to the axis y, it is possible to alulate the omplete radiation pattern of the emitter examined. This diagram orresponds to the radiation pattern of dipole emission. Consequently 7
8 A H x evz t x t 4π x there is the being late vetor potential, the Eg. (3.5) an be rewritten E y x x x eay t sin AH t AH t α ( x, t, α ) µ 4πε x ε t t is again obtained omplete agreement with the equations of the being late vetor potential, but vetor potential is introdued here not by phenomenologial method, but with the use of a onept of the being late salar- vetor potential. Let us note one important irumstane. In the Maxwell equation eletri fields it appears vortex. In this ase the eletri fields bear gradient nature. Let us demonstrate the still one possibility, whih gives Eg. (3.5). It is known that in the eletrodynamis there is this onept, as the eletri dipole and dipole emission. Two harges with the opposite signs have the dipole moment: p ed, (3.6) Therefore urrent an be expressed through the derivative of dipole moment on the time of Consequently and d p ev e t t p v e t,. 8
9 v p a. t e t Substituting this equation into Eg. (.5), we obtain the law of the dipole emission r p( t ) E. (3.7) 4π rε t This is also known equation []. In the proess of flutuating the eletri dipole are reated the eletri fields of two forms. In addition to this, around the being varied dipole are formed the eletri fields of stati dipole, whih hange in the time in onnetion with the fat that the distane between the harges it depends on time. Energy of these pour on and it is expended on the emission. However, the summary value of field around this dipole at any moment of time defines as superposition pour on stati dipole pour on emissions. Laws (3.4), (3.5), (3.7) are the laws of the diret ation, in whih already there is neither magneti pour on nor vetor potentials. I.e. those strutures, by whih there were the magneti field and magneti vetor potential, are already taken and they no longer were neessary to us. Using Eg. (3.5) it is possible to obtain the laws of refletion and sattering both for the single harges and, for any quantity of them. In this ase eah moving harge emits the eletri fields, determined by Eg. (3.5). The superposition of eletrial pour on all harges in the distant zone and it is eletrial wave. If on the harge ats the eletri field E sin y E y ωt, then its aeleration takes the form of e a E sin y ωt. m Consequently 9
3 e sinα x K x y α y ω y ω 4πε x mx E ( x, t, ) E sin ( t ) E sin ( t ), (3.8) the oeffiient K e sinα an be named the oeffiient of the re-emission of 4πε m single harge in the assigned diretion. The urrent wave of the displaement aompanies the wave of eletri field: j y x vy t Ey esinα ( x, t) ε t 4π x t. If harge aomplishes its motion under the ation of the eletri field E E sinωt, then bias urrent in the distant zone will be written down as e ω x jy ( x, t) E y osω t 4π mx. (3.9) The sum wave, whih presents the propagation of eletrial pour on Eg. (3.8) and bias urrents Eg. (3.9) an be named eletrourrent wave. It is possible to introdue also magneti waves, assuming that E j ε roth, (3.) t divh introdued thus magneti field is vortex. Comparing Eg. (3.9) and Eg. (3.) we obtain: 3
3 H x t e ω α x 4π mx z (, ) sin E y osω t x. Integrating this relationship on the oordinate, we find the value of the magneti field e sinα x H z ( x, t) E y sinω t 4π mx. (3.) Thus, Egs. (3.8), (3.9) and (3.) an be named the laws of eletrial indution, sine. they give the diret oupling between the eletri fields, applied to the harge, and by fields and by urrents indued by this harge in its environment. Here harge plays the role of the transformer, whih ensures this reemission. The magneti field, whih an be alulated with the aid of Eg. (3.), is direted normally both toward the eletri field and toward the diretion of propagation, and their relation at eah point of the spae is equal of Ey ( x, t) H ( x, t) z µ Z ε ε. In this equation of Z is wave drag of free spae. Wave drag determines the ative power of losses on the single area, loated normal to the diretion of propagation of the wave: P ZE. y 3
3 Therefore eletrourrent wave, rossing this area, transfers through it the power, determined by the data by relationship. This is loated in aordane with by the Poynting theorem about the power flux of eletromagneti wave. Therefore, for finding all parameters, whih haraterize wave proess, it is suffiient examination only of eletrourrent wave and knowledge of the wave drag of spae. In this ase it is in no way ompulsory to introdue this onept as magneti field and its vetor potential, although there is nothing illegal in this. The fields, obtained thus, satisfy Helmholtz's theorem. This theorem says, that any single-valued and ontinuous vetor field F, whih turns into zero at infinity, an be represented uniquely as the sum of the gradient of a ertain salar funtion ϕ and rotor of a ertain vetor funtion C, whose divergene is equal to zero: F gradϕ + rotc divc., Consequently, must exist lear separation pour on to the gradient and the vortex. It is evident that in the expressions, obtained for those indued pour on, this separation is loated. Eletri fields bear gradient nature, and magneti - vortex. Thus, the onstrution of eletrodynamis should have been begun from the aknowledgement of the dependene of salar potential on the speed. But nature very deeply hides its serets, and in order to ome to this simple onlusion, it was neessary to pass way by length almost into two enturies. The grit, whih so harmoniously were ereted around the magnet poles, in a straight manner indiated the presene of some power pour on potential nature, but to this they did not turn attention. Therefore it turned out that all examined only tip of the ieberg, whose substantial part remained invisible of almost two hundred years. 3
33 Taking into aount entire aforesaid one should assume that at the basis of the overwhelming majority of stati and dynami phenomena at the eletrodynamis only Eg. (3.), whih assumes the dependene of the salar potential of harge on the speed, lies. From this formula it follows and stati interation of harges, and laws of power interation in the ase of their mutual motion, and emission laws and sattering. This approah made it possible to explain from the positions of lassial eletrodynamis suh phenomena as phase aberration and the transverse the Doppler effet, whih within the framework the lassial eletrodynamis of explanation did not find. Let us point out that one of the fundamental equations of indution Eg. (3.4) ould be obtained diretly from the Ampere law, still long before appeared the Maksvell equation. The Ampere law, expressed in the vetor form, determines magneti field Idl r H 3 4π r In this equation I - urrent, whih flows through the element dl, r - vetor, direted from dl to the point of x, y, z. It is possible to show that [ dlr ] dl grad dl rot rot dl 3 r r r r But the rotor of dl is equal to zero therefore dl H rot I rot A π r Consequently A H 4 H dl I 4π r... (3.) 33
34 The remarkable property of this expression is that that the vetor potential depends from the distane to the observation point as r possible to obtain emission laws. Sine of I. Speifially, this property makes it gv, where g the quantity of harges, whih falls per unit of the length of ondutor, from (3.) we obtain: A H gv dl 4π r For the single harge of e this equation takes the form: ev AH. 4π r In onnetion with the fat that eletri field is determined from the equation A E µ, t for this ase obtain v g dl t ga dl E µ µ 4π r. (3.3) 4π r In this equation a is aeleration of harge.. This equation appears as follows for the single harge: µ ea E 4π r. (3.4) In Eg. (3.3) and Eg. (3.4) it is neessary to onsider that the potentials are extended with the final speed they be late to the period µ ε r. Taking into aount the fat that for the vauum, these equations take the form the form: r r ga( t ) dl ga( t ) dl E µ 4π r, (3.5) 4πε r 34
35 E r ea( t ). (3.6) 4πε r Of Eg. (3.5) and Eg. (3.6) represent wave equations and are the solutions. Of the Maksvell equation, but in this ase they are obtained diretly from the Ampere law. To there remains only present the question, why eletrodynamis in its time is not banal by this method? 4. Is there any dispersion of eletri and magneti indutivities in material media? It is noted in the introdution that dispersion of eletri and magneti indutivities of material media is a ommonly aepted idea. The idea is however not orret. To explain this statement and to gain a better understanding of the physial essene of the problem, we start with a simple example showing how eletri lumped-parameter iruits an be desribed. As we an see below, this example is diretly onerned with the problem of our interest and will give us a better insight into the physial piture of the eletrodynami proesses in material media. In a parallel resonane iruit inluding a apaitor С and an indutane oil L, the applied voltage U and the total urrent I Σ through the iruit are related as I Σ I C + I L d U C + d t L U d t, (4.) where d U I C C is the urrent through the apaitor, I d t L U d t L is the urrent through the indutane oil. For the harmoni voltage U U sin ωt 35
36 I ω C Σ U os ω t ω L. (4.) The term in brakets is the total suseptane σ х of the iruit, whih onsists of the apaitive σ с and indutive σ L omponents Eq. (4.) an be re-written as σ x σ + σ L ω C. (4.3) ω L ω I ω C Σ U os ω t ω, (4.4) where ω is the resonane frequeny of a parallel iruit. LC From the mathematial (i.e. other than physial) standpoint, we may assume a iruit that has only a apaitor and no indutane oil. Its frequeny dependent apaitane is Another approah is possible, whih is orret too. Eq. (4.) an be re-written as ω C *( ω) C. (4.5) ω ω ω I Σ U os ω t. (4.6) ω L In this ase the iruit is assumed to inlude only an indutane oil and no apaitor. Its frequeny dependent indutane is L L*( ω) ω ω Using the notion Eqs. (4.5) and (4.7), we an write 36. (4.7)
37 IΣ ω C* ( ω) U os ω t, (4.8) or IΣ U os ω t. (4.9) ω L*( ω) Eqs (4.8) and (4.9) are equivalent and eah of them provides a omplete mathematial desription of the iruit. From the physial point of view, C *( ω) and L ( ) do not represent apaitane and indutane though they have the * ω orresponding dimensions. Their physial sense is as follows: σ X C ( ω) ω *, (4.) i.e. C *( ω) is the total suseptane of this iruit divided by frequeny: L* ( ω), (4.) ω σ and L *( ω) is the inverse value of the produt of the total suseptane and the frequeny. Amount C *( ω) is onstrited mathematially so that it inludes C and L simultaneously. The same is true for L *( ω). We shall not onsider here any other ases, e.g., series or more omplex iruits. It is however important to note that applying the above method, any iruit onsisting of the reative omponents C and L an be desribed either through frequeny dependent indutane or frequeny dependent apaitane. But this is only a mathematial desription of real iruits with onstant value reative elements. It is well known that the energy stored in the apaitor and indutane oil an be found as WC C U, (4.) 37 X
38 WL L I. (4.3) But what an be done if we have C *( ω) and L *( ω)? There is no way of substituting them into Eqs. (4.) and (4.3) beause they an be both positive and negative. It an be shown readily that the energy stored in the iruit analyzed is d σ X WΣ U, (4.4) d ω or [ ω C*( ω) ] d WΣ U, (4.5) d ω or d ω L*( ω) W Σ U. (4.6) d ω Having written Eqs. (4.4), (4.5) or (4.6) in greater detail, we arrive at the same result: W Σ C U + L I, (4.7) where U is the voltage at the apaitor and I is the urrent through the indutane oil. Below we onsider the physial meaning jog the magnitudes ε(ω) and µ(ω) for material media.. Plasma media A superondutor is a perfet plasma medium in whih harge arriers (eletrons) an move without frition. In this ase the equation of motion is 38
39 d V m e E, (4.8) d t where m and e are the eletron mass and harge, respetively; E is the eletri field strength, V is the veloity. Taking into aount the urrent density j n e V, (4.9) we an obtain from Eq. (4.8) j L n e E d t. (4.) m In Eqs. (4.9) and (4.) n is the speifi harge density. Introduing the notion we an write m n e L k, (4.) jl E d t. (4.) L Here L k is the kineti indutivity of the medium. Its existene is based on the fat that a harge arrier has a mass and hene it possesses inertia properties. For harmoni fields we have E E sin ω t and Eq. (4.) beomes j L k ω L k E osω t. (4.3) Eqs. (4.) and (4.3) show that jl is the urrent through the indutane oil. In this ase the Maxwell equations take the following form H rot E µ, t E rot H jc + jl ε + E d t, t L k (4.4) 39
4 where ε and µ are the eletri and magneti indutivities in vauum, j C and jl are the displaement and ondution urrents, respetively. As was shown above, is the indutive urrent. j L Eq. (4.4) gives H µ rot rot H + µ ε + H. (4.5) t L k For time-independent fields, Eq. (4.5) transforms into the London equation where λl Lk µ is the London depth of penetration. µ rot rot H + H, (4.6) L As Eq. (4.4) shows, the indutivities of plasma (both eletri and magneti) are frequeny independent and equal to the orresponding parameters for vauum. Besides, suh plasma has another fundamental material harateristi kineti indutivity. Eqs. (4.4) hold for both onstant and variable fields. For harmoni fields E E sin ω t, Eq. (4.4) gives k rot H εω E os ω t Lkω. (4.7) Taking the braketed value as the speifi suseptane σ x of plasma, we an write rot H σ X E os ω t, (4.8) where σ X ω ρ εω εω ω ε *( ω) ω ω, (4.9) L k ϖ and ε *( ω ) ρ ε, where ω ρ ω ε L is the plasma frequeny. k 4
4 Now Eq. (4.8) an be re-written as or ω ρ rot H ω ε E os ω t ω, (4.3) rot H ω ε ( ω) E os ω t. (4.3) * The ε*(ω) parameter is onventionally alled the frequeny-dependent eletri indutivity of plasma. In reality however this magnitude inludes simultaneously the eletri indutivity of vauum aid the kineti indutivity of plasma. It an be found as σ X ε ( ω) ω It is evident that there is another way of writing σ Х *. (4.3) where σ X ω εω, ω Lk ω L k ω ρ ω Lk * (4.33) Lk Lk *( ω) ω σ Xω. ω ρ (4.34) L k *(ω) written this way inludes both ε and L k. Eqs. (4.9) and (4.33) are equivalent, and it is safe to say that plasma is haraterized by the frequeny-dependent kineti indutane L k *(ω) rather than by the frequeny-dependent eletri indutivity ε*(ω). Eq. (4.7) an be re-written using the parameters ε*(ω) and L k *(ω) rot H ω ε ( ω) E os ω t, (4.35) * 4
or rot 4 H E os ω t. (4.36) ω L *( ω) k Eqs. (4.35) and (4.36) are equivalent. Thus, the parameter ε*(ω) is not an eletri indutivity though it has its dimensions. The same an be said about L k *(ω). We an see readily that σ X ε *( ω), (4.37) ω L k *( ω). (4.38) σ ω These relations desribe the physial meaning of ε*(ω) and L k *(ω). Of ourse, the parameters ε*(ω) and L k *(ω) are hardly usable for alulating energy by the following equations X E WE ε (4.39) and W j Lk j. (4.4) For this purpose the Eq. (4.5)-type fotmula was devised in [7]: W Using Eq. (4.4), we an obtain W The same result is obtainable from Σ E d [ ω ε * ( ω) ] E. (4.4) + d ω E E + L ε ε k. (4.4) ω Lk j 4
43 W d ω L *( ω) d ω k E. (4.43) As in the ase of a parallel iruit, either of the parameters ε*(ω) and L k *(ω), similarly to C*(ω) and L*(ω), haraterize ompletely the eletrodynami properties of plasma. The ase ε*(ω) L k *(ω) (4.44) orresponds to the resonane of urrent. It is shown below that under ertain onditions this resonane an be transverse with respet to the diretion of eletromagneti waves. It is known that the Langmuir resonane is longitudinal. No other resonanes have ever been deteted in nonmagnetized plasma. Nevertheless, transverse resonane is also possible in suh plasma, and its frequeny oinides with that of the Langmuir resonane. To understand the origin of the transverse resonane, let us onsider a long line onsisting of two perfetly onduting planes (see Fig. 5). First, we examine this line in vauum. If a d.. voltage (U) soure is onneted to an open line the energy stored in its eletri field is where W ε E a b z CE ΣU, (4.45) EΣ U E is the eletri field strength in the line, and a b z Σ ε (4.46) a C E 43