On refinement of certain laws of classical electrodynamic

Transcription

1 On efinement of etain laws of lassial eletodynami F.F Mende -mail : mende_fedo@mail.u Abstat The poblems onsideed efe to the mateial equations of eleti- and manetoele ti indution. Some ontaditions found in fundamental studies on lassial eletodynamis have been explained. The notion manetoeleti indution has been intodued whih pemits symmetial witin of the indution laws. It is shown that the esults of the speial theoy of elativity an be obtained fom these laws thouh the Galilean tansfomations. The pemittivity and pemeability of mateials media ae shown to be independent of fequeny. The notions manetoeletoineti and eletomanetopotential waves and ineti apaity have been intodued. It is shown that alon with the lonitudinal anmui esonane the tansvese esonane is possible in nonmanetized plasma and both the esonanes ae deeneate. A new notion sala-veto potential is intodued whih pemits solution of all pesent-day poblems of lassial eletodynamis. The use of the sala-veto potential maes the maneti field notion unneessay. Intodution Until now some poblems of lassial eletodynamis involvin the laws of eletomaneti indution have been intepeted in a dual o even ontavesal way. As an example let us onside how the homopola opeation is explained in diffeent wos. In [] this is done usin the Faaday low speified fo the disontinuous motion ase. In [] the ule of flow is ejeted and the opeation of the homopola eneato is explained on the basis of the oentz foe atin upon haes. The ontaditoy appoahes ae most evident in Feynman s wo [] (see pae 53): the ule of flow states that the ontou e.m.f. is equal to the opposite-sin ate of hane in the maneti flux thouh the ontou when the flux vaies eithe with the hanin field o due to the motion of the ontou (o to both). Two options the ontou moves o the field hanes ae indistinuishable within the ule. Nevetheless we use these two ompletely diffeent laws to explain the ule fo the two ases: [ B ] fo the movin ontou and B fo the hanin field. And fu-

2 the on: Thee is hadly anothe ase in physis when a simple and auate eneal law has to be intepeted in tems of two diffeent phenomena. Nomally suh beautiful enealization should be based on a unified fundamental piniple. Suh piniple is absent in ou ase. The intepetation of the Faaday law in [] is also ommonly aepted: Faaday s obsevation led to the disovey of a new law elatin eleti and maneti fields: the eleti field is eneated in the eion whee the maneti field vaies with time. Thee is howeve an exeption to this ule too thouh the above studies do not mention it. Howeve as soon as the uent thouh suh a solenoid is haned an eleti field is exited extenally. The exeption seem to be too numeous. The situation eally auses onen when suh noted physiists as Tamm and Feynman have no ommon appoah to this seeminly simple question. It is nowin [3] that lassial eletodynamis fails to explain the phenomenon of phase abeation. As applied to popaation of liht the phenomenon an be explained only in tems of the speial theoy of elativity (STR). Howeve the Maxwell equations ae invaiant with espet to the ovaiant STR tansfomations and thee is theefoe evey eason to hope that they an funish the equied explanation of the phenomenon. It is well nown that eleti and maneti indutivities of mateial media an depend on fequeny i.e. they an exhibit dispesion. But even Maxwell himself who was the autho of the basi equations of eletodynamis believed that and µ wee fequeny-independent fundamental onstants. How the idea of and µ-dispesion appeaed and evolved is illustated vividly in the monoaph of well-nown speialists in physis of plasma [4]: while woin at the equations of eletodynamis of mateial media G. Maxwell looed upon eleti and maneti indutivities as onstants (that is why this appoah was so lastin). Muh late at the beinnin of the XX entuy G. Heavisid and R.Wull put fowad thei explanation fo phenomena of optial dispesion (in patiula ainbow) in whih eleti and maneti indutivities ame as funtions of fequeny. Quite eently in the mid- 5ies of the last entuy physiists aived at the onlusion that these paametes wee dependent not only on the fequeny but on the wave veto as well. That was a evolutionay beaaway fom the uent onepts. The impotane of the poblem is lealy illustated by what happened at a semina held by. D. andau in 954 whee he inteupted A.. Ahieze epotin on the subjet: Nonsense the efative index annot be a funtion of the efative index. Note this was said by. D. andau an outstandin physiist of ou time. What is the atual situation? Runnin ahead I an admit that Maxwell was iht: both and µ ae fequeny independent onstants haateizin one o anothe mateial medium. Sine dispesion of eleti and maneti indutivities of mateial media is one of the basi poblems of the pesent day physis and eletodynamis the system of views on these questions has to be adially alteed aain (fo the seond time!). In this ontext the hallene of this study was to povide a ompehensive answe to the above questions and thus to aive at a unified and unambiuous standpoint. This will etainly equie a evision of the elevant intepetations in many fundamental wos.. quations of eletomaneti indution in movin oodinates The Maxwell equations do not pemit us to wite down the fields in movin oodinates poeedin fom the nown fields measued in the stationay oodinates. Geneally this an be done thouh the oentz tansfomations but they so not follow fom lassial eletodynamis. In a homopola eneato the eleti fields ae measued in the stationay oodinates but they ae atually exited in the elements whih move elative to the stationay oodinate system. Theefoe the pin-

3 iple of the homopola eneato opeation an be desibed oetly only in the famewo of the speial theoy of elativity (STR). This bins up the question: Can lassial eletodynamis funish oet esults fo the fields in a movin oodinate system o at least offe an aeptable appoximation? If so what fom will the equations of eletomaneti indution have? The oentz foe is F (.) e + e B. [ ] It beas the name of oentz it follows fom his tansfomations whih pemit witin the fields in the movin oodinates if the fields in the stationay oodinates ae nown. Henefowad the fields and foes eneated in a movin oodinate system will be indiated with pimed symbols. The lues of how to wite the fields in movin oodinates if they ae nown in the stationay system ae available even in the Faaday law. et us speify the fom of the Faaday law: d d Φ l d t The speified law o moe peisely its speified fom means that and B. (.) dl should be pimed if the ontou inteal is souht fo in movin oodinates and unpimed fo stationay oodinates. In the latte ase the iht-hand side of q. (.) should ontain a patial deivative with espet to time whih fat is eneally not mentioned in liteatue. The total deivative with espet to time in q. (.) implies that the final esult fo the ontou e.m.f. is independent of the vaiation mode of the flux. In othe wods the flux an hane eithe puely with time vaiations of B o beause the system in whih dl the spatially vayin field B. In q. (.) ΦB v whee the maneti indution B µ H d S in the movin oodinates. B d S Tain into aount q. (.3) we an find fom q. (.) Sine d + ad d t B is measued is movin in (.3) is measued in the stationay oodinates and the element d d t d l B d S we an wite [ B ] d l d S d l div B d S In this ase ontou inteal is taen ove the ontou d l ovein the spae S. Henefowad we assume the validity of the Galilean tansfomations i.e. (.5) funishes the well-nown esult: + [ B]. (.4). (.5) d d l d l and d S d S. q. (.6) 3

4 whih suests that the motion in the maneti field exites an additional eleti field desibed by the final tem in q. (.6). Note that q. (.6) is obtained fom the slihtly speified Faaday law and not fom the oentz tansfomations. Aodin to q. (.6) a hae movin in the maneti field is influened by a foe pependiula to the dietion of the motion. Howeve the physial natue of this foe has neve been onsideed. This bins onfusion into the explanation of the homopola eneato opeation and does not pemit us to explain the eleti fields outside an infinitely lon solenoid on the basis of the Maxwell equations. To lea up the physial oiin of the final tem in q. (.6) let us wite B and in tems of the A B : A B B ot AB. (.7) maneti veto potential Then q. (.6) an be e-witten as and futhe: A A B B + [ ot A ] ( ) A + ad ( A ) B B B (.8). (.9) The fist two tems in the iht-hand side of q. (.9) an be onsideed as the total deivative of the veto potential with espet to time: d A ( A ) B + ad B. (.) d t As seen in q. (.9) the field stenth and hene the foe atin upon a hae onsists of thee omponents. The fist omponent desibes the pue time vaiations of the maneti veto potential. The seond tem in the iht-hand side of q. (.9) is evidently onneted with the hanes in the veto potential aused by the motion of a hae in the spatially vayin field of this potential. The oiin of the last tem in the iht-hand side of q. (.9) is quite diffeent. It is onneted with the potential foes beause the potential eney of a hae movin in the potential field equal to e ( ). The manitude e ad ( ) A B A B A B at the veloity is desibes the foe just as the sala potential adient does. Usin q. (.9) we an explain physially all the stenth omponents of the eletoni field exited in the movin and stationay oopeates. If ou onen is with the eleti fields outside a lon solenoid whee the no maneti field the fist tem in the iht-hand side of q. (.9) ome into play. In the ase of a homopola eneato the foe atin upon a hae is detemined by the last two tems in the iht-hand side of q.(.9) both of them ontibutin equally. It is theefoe inoet to loo upon the homopola eneato as the exeption to the flow ule beause as we saw above this ule allows fo all the thee omponents. Usin the oto in both sides of q. (.) and tain into aount ot ad we obtain 4

5 d B ot d t. (.) If motion is absent q. (.) tuns into Maxwell equation (.). quation (.) is etainly less infomative than q. (.): beause of ot ad it does not inlude the foes defined in tems of e ad ( ) A B. It is theefoe moe easonable to use q. (.) if we want to allow fo all omponents of the eleti fields atin upon a hae both in the stationay and in the movin oodinates. As a peliminay onlusion we may state that the Faaday aw q. (.) when examined losely explains lealy all featues of the homopola eneato opeation and this opeation piniple is a onsequene athe than an exeption of the flow ule q. (.). Feynman s statement that B fo the movin ontou and fo the vayin field ae absolutely diffeent [ B ] laws is ontay to fat. The Faaday law is just the sole unified fundamental piniple whih Feynman delaed to be missin. et us lea up anothe Feynman s intepetation. Faaday s obsevation in fat led him to disovey of a new law elatin eleti and maneti fields in the eion whee the maneti field vaies with time and thus eneates the eleti field. This oelation is essentially tue but not omplete. As shown above the eleti field an also be exited whee thee is no maneti field namely outside an infinitely lon solenoid. A moe omplete fomulation follows fom q. (.9) and the elationship d A d t B is moe eneal than ot B. This suests that a movin o stationay hae inteats with the field of the maneti veto potential athe than with the maneti field. The nowlede of this potential and its evolution an only pemit us to alulate all the foe omponents atin upon haes. The maneti field is meely a spatial deivative of the veto field. As follows fom the above onsideation it is moe appopiate to wite the otentz foe in tems of the maneti veto potential F e + e [ ot AB ] e е( ) AB + еad( AB ) (.) whih visualizes the omplete stutue of the foe. The Faaday law q. (.) is efeed to as the law of eletomaneti indution beause it shows how vayin maneti fields an eneate eleti fields. Howeve lassial eletodynamis ontains no law of manetoeleti indution showin how maneti fields an be exited by vayin eleti fields. This aspet of lassial eletodynamis evolved alon a diffeent pathway. Fist the law H d l (.3) I was nown in whih I was the uent ossin the aea of the inteation ontou. In the diffeential fom q. (.3) beomes j whee σ is the ondution uent density. Maxwell supplemented q. (.4) with displaement uent ot H (.4) j σ 5

6 ot H j D + σ. (.5) Howeve if Faaday had pefomed measuement in vayin eleti indution fluxes he would have infeed the followin law whee Φ DdS D H d d Φ l d t is the eleti indution flux. Then D D (.6) H d l d S + [ D ] d l + div Dd S v. (.7) Unlie div B in maneti fields eleti fields ae haateized by div ρ and the last tem in the iht-hand side of q. (.7) desibes the ondution uent I i.e. the Ampee law follows fom q. (.6). q. (.7) ives H [ D ] (.8) whih was ealie obtainable only fom the oentz tansfomation. Moeove as was shown onvininly in [] q. (.8) also leads out of the Biot-Savat law if maneti fields ae alulated fom the eleti fields exited by movin haes. In this ase the last tem in the iht-hand side of q. (.7) an be omitted and the indution laws beome ompletely symmetial. d l d S Hd l d S+ D B [ B ] d l D [ D ] d l. + [ B] H H [ D]. alie qs. (.) wee only obtainable fom the ovaiant oentz tansfomations i.e. in the famewo of speial theoy of elativity (STR). Thus the STR esults ay ate to the ~ tems (.9) (.) an be deived fom the indution laws thouh the Galilean tansfomations. The STR esults auate to the tems an be obtained thouh tansfomation of q (.9). At fist howeve we shall intodue anothe veto potential whih is not used in lassial eletodynamis. et us assume fo votex fields [5] that D ot (.) A D 6

7 whee o o A D is the eleti veto potential. It then follows fom q. (.9) that A H D A H H + [ ] AD ad [ AD ] D d A [ ot A ] D ad [ A ] D (.) (.3) D. (.4) d t These equations pesent the law of manetoeleti indution witten in tems of the eleti veto potential. To illustate the impotane of the intodution of the eleti veto potential we ome ba to an infinitely lon solenoid. The situation is muh the same and the only hane is that the vetos B ae eplaed with the vetos D. Suh situation is quite ealisti: it ous when the spae between the flat apaito plates is filled with hih eleti indutivities. In this ase the displaement flux is almost entiely inside the dieleti. The attempt to alulate the maneti field outside the spae D oupied by the dieleti (whee ) uns into the same poblem that existed fo the alulation beyond the fields of an infinitely lon solenoid. The intodution of the eleti veto potential pemits a oet solution of this poblem. This howeve bins up the question of pioity: what is pimay and what is seonday? The eleti veto potential is no doubt pimay beause eleti votex fields ae exited only whee the oto of suh potential is non-zeo. As follows fom qs. (.) if the efeene systems move elative to eah othe the fields and H ae mutually onneted i.e. the movement in the fields H indues the fields and vie vesa. But new onsequenes appea whih wee not onsideed in lassial eletodynamis. Fo illustation let us analyze two paallel ondutin plates with the eleti field in between. In this ase the sufae hae ρ S pe unit aea of eah plate is Е. If the othe efeene system is made to move paallel to the plates in the field Е at the veloity this motion will eneate an additional field Н Е. If a thid efeene system stats to move at the veloity within the above movin system this motion in the field Н will eneate Е µ Е whih is anothe ontibution to the field Е. The field thus beomes stone in the movin system than it is in the stationay one. It is easonable to suppose that the sufae hae at the plates of the initial system has ineased by µ as well. This tehnique of field alulation was desibed in [6]. If we put and H fo the field om- and H fo the pependiula omponents the final ponents paallel to the veloity dietion and fields at the veloity an be witten as 7

8 8 ] [ ] [ h s Z h H H H H h s H Z h + (.5) whee µ Z is the spae impedane µ is the veloity of liht in the medium unde onsideation. The esults of these tansfomations oinide with the STR data with the auay to the ~ tems. The hihe-ode oetions do not oinide. It should be noted that until now expeimental tests of the speial theoy of elativity have not one beyond the ~ auay. As an example let us analyze how qs. (.5) an aount fo the phenomenon of phase abeation whih was inexpliable in lassial eletodynamis. Assume that thee ae plane wave omponents H Z and X and the pimed system is movin alon the x-axis at the veloity X. The field omponents with in the pimed oodinates an be witten as. h H H x sh H X Z Z Z Y X X (.7) The total field Е in the movin system is h X X Y X +. (.8) Hene the Poyntin veto no lone follows the dietion of the y-axis. It is in the xy-plane and tilted about the y-axis at an anle detemined by qs. (.7). The atio between the absolute values of the vetos Е and Н is the same in both the systems. This is just what is nown as phase abeation in lassial eletodynamis.

9 . Is thee any dispesion of eleti and maneti indutivities in mateial media? It is noted in the intodution that dispesion of eleti and maneti indutivities of mateial media is a ommonly aepted idea. The idea is howeve not oet. To explain this statement and to ain a bette undestandin of the physial essene of the poblem we stat with a simple example showin how eleti lumped-paamete iuits an be desibed. As we an see below this example is dietly onened with the poblem of ou inteest and will ive us a bette insiht into the physial pitue of the eletodynami poesses in mateial media. In a paallel esonane iuit inludin a apaito С and an indutane oil the applied voltae U and the total uent I Σ thouh the iuit ae elated as whee I Σ I C + I d U C + d t U d t d U I C C is the uent thouh the apaito I d t U d t indutane oil. Fo the hamoni voltae U U sin t I C U os t (.) is the uent thouh the Σ. (.) The tem in baets is the total suseptane σ х of the iuit whih onsists of the apaitive σ с and indutive σ omponents q. (.) an be e-witten as I σ σ σ C x +. (.3) C U os t Σ (.4) whee is the esonane fequeny of a paallel iuit. C Fom the mathematial (i.e. othe than physial) standpoint we may assume a iuit that has only a apaito and no indutane oil. Its fequeny dependent apaitane is Anothe appoah is possible whih is oet too. q. (.) an be e-witten as *( C C. (.5) 9

10 I U os t Σ. (.6) In this ase the iuit is assumed to inlude only an indutane oil and no apaito. Its fequeny dependent indutane is Usin the notion of qs. (.5) and (.7) we an wite o I I *( C* ( U os t. (.7) Σ (.8) U *( os t Σ. (.9) qs (.8) and (.9) ae equivalent and eah of them povides a omplete mathematial desiption of the iuit. Fom the physial point of view C *( and *( do not epesent apaitane and indutane thouh they have the oespondin dimensions. Thei physial sense is as follows: σ X C ( i.e. C *( is the total suseptane of this iuit divided by fequeny: * (.) * ( (.) σ and *( is the invese value of the podut of the total suseptane and the fequeny. Amount C *( is onstited mathematially so that it inludes C and simultaneously. The same is tue fo *(. We shall not onside hee any othe ases e.. seies o moe omplex iuits. It is howeve impotant to note that applyin the above method any iuit onsistin of the eative omponents C and an be desibed eithe thouh fequeny dependent indutane o fequeny dependent apaitane. But this is only a mathematial desiption of eal iuits with onstant value eative elements. It is well nown that the eney stoed in the apaito and indutane oil an be found as C U I C *( and *( X WC (.) W. (.3) But what an be done if we have )? Thee is no way of substitutin them into qs. (.) and (.3) beause they an be both positive and neative. It an be shown eadily that the eney stoed in the iuit analyzed is

11 o o W W W d σ X U d Σ (.4) d [ C*( ] d U Σ (.5) d *( U d Σ. (.6) Havin witten qs. (.4) (.5) o (.6) in eate detail we aive at the same esult: W C U + I Σ (.7) whee U is the voltae at the apaito and I is the uent thouh the indutane oil. Below we onside the physial meanin jo the manitudes ( and µ( fo mateial media.. Plasma media A supeonduto is a pefet plasma medium in whih hae aies (eletons) an move without fition. In this ase the equation of motion is d m d t e (.8) whee m and e ae the eleton mass and hae espetively; is the eleti field stenth is the veloity. Tain into aount the uent density we an obtain fom q. (.8) j j n e (.9) n e d t m In qs. (.9) and (.) n is the speifi hae density. Intoduin the notion we an wite m n e. (.) (.) j d t. (.) Hee is the ineti indutivity of the medium. Its existene is based on the fat that a hae aie has a mass and hene it possesses inetia popeties.

12 Fo hamoni fields we have sin t and q. (.) beomes j os t qs. (.) and (.3) show that j is the uent thouh the indutane oil. In this ase the Maxwell equations tae the followin fom ot µ ot H j C H + j +. (.3) d t (.4) whee and µ ae the eleti and maneti indutivities in vauum j C and j ae the displaement and ondution uents espetively. As was shown above j is the indutive uent. q. (.4) ives H µ ot ot H + µ + H. (.5) Fo time-independent fields q. (.5) tansfoms into the ondon equation whee λ is the ondon depth of penetation. µ µ ot ot H + H (.6) As q. (.4) shows the indutivities of plasma (both eleti and maneti) ae fequeny independent and equal to the oespondin paametes fo vauum. Besides suh plasma has anothe fundamental mateial haateisti ineti indutivity. qs. (.4) hold fo both onstant and vaiable fields. Fo hamoni fields (.4) ives ot H os t sin t q.. (.7) Tain the baeted value as the speifi suseptane σ x of plasma we an wite whee and ϖ ρ ( σ * whee X ot H σ X os t (.8) ρ *( ρ is the plasma fequeny. (.9)

13 Now q. (.8) an be e-witten as o ρ ot H os t (.3) ot H ( os t. (.3) * The *( paamete is onventionally alled the fequeny-dependent eleti indutivity of plasma. In eality howeve this manitude inludes simultaneously the eleti indutivity of vauum aid the ineti indutivity of plasma. It an be found as It is evident that thee is anothe way of witin σ Х whee σ X σ X ( *. (.3) ρ (.33) *( σ X ρ *. (.34) *( witten this way inludes both and. qs. (.9) and (.33) ae equivalent and it is safe to say that plasma is haateized by the fequeny-dependent ineti indutane *( athe than by the fequeny-dependent eleti indutivity *(. q. (.7) an be e-witten usin the paametes *( and *( o ot H ( os t (.35) * ot H os t. (.36) *( qs. (.35) and (.36) ae equivalent.thus the paamete *( is not an eleti indutivity thouh it has its dimensions. The same an be said about *(. We an see eadily that σ X *( (.37) *(. (.38) σ These elations desibe the physial meanin of *( and *(. Of ouse the paametes *( and *( ae hadly usable fo alulatin eney by the followin equations X 3

14 and W W (.39) Fo this pupose the q. (.5)-type fotmula was devised in [7]: Usin q. (.4) we an obtain W Σ The same esult is obtainable fom W j j. (.4) d [ * ( ]. (.4) d + + j. (.4) W d *( d. (.43) As in the ase of a paallel iuit eithe of the paametes *( and *( similaly to C*( and *( haateize ompletely the eletodynami popeties of plasma. The ase *( *( (.44) oesponds to the esonane of uent.it is shown below that unde etain onditions this esonane an be tansvese with espet to the dietion of eletomaneti waves. It is nown that the anmui esonane is lonitudinal. No othe esonanes have eve been deteted in nonmanetized plasma. Nevetheless tansvese esonane is also possible in suh plasma and its fequeny oinides with that of the anmui esonane. To undestand the oiin of the tansvese esonane let us onside a lon line onsistin of two pefetly ondutin planes (see Fi..). Fist we examine this line in vauum. If a d.. voltae (U) soue is onneted to an open line the eney stoed in its eleti field is whee W Σ a b z C ΣU (.45) U is the eleti field stenth in the line and a b z C Σ (.46) a is the total line apaitane. C medium (plasma) in SI units (F/m). b a is the linea apaitane and is eleti indutivities of the 4

15 Fi... Two-onduto line onsistin of two pefetly ondutin planes. The speifi potential eney of the eleti field is W. (.47) If the line is shot-iuited at the distane z fom its stat and onneted to a d.. uent (I) soue the eney stoed in the maneti field of the line is Sine I H we an wite b W whee HΣ is the total indutane of the line of the medium (vauum) in SI (H/m). The speifi eney of the maneti field is HΣ µ H a b z H Σ I. (.48) a z H µ b a H µ b Σ (.49) is linea indutane and µ is the indutivity W H µ H. (.5) To mae the esults obtained moe illustative henefowad the method of equivalent iuits will be used alon with mathematial desiption. It is seen that C ЕΣ and HΣ inease with owin z. The line sement dz an theefoe be eaded as an equivalent iuit (Fi..а). 5

16 If plasma in whih hae aies an move fee of fition is plaed within the open line and then the uent I is passed thouh it the hae aies movin at a etain veloity stat stoin ineti eney. Sine the uent density is the total ineti eney of all movin haes is On the othe hand m n e I j n e b z whee Σ is the total ineti indutane of the line. Hene Thus the manitude oespondin ineti indutivity of the medium. (.5) m n e a b z W Σ a b z j I. (.5) W I Σ Σ (.53) m n e a b z Σ. (.54) m n e (.55) alie we intodued this manitude by anothe way (see q. (.)).q. (.55) oesponds to ase of unifomly distibuted d.. uent. As we an see fom q. (.54) HΣ unlie C ЕΣ and Σ deeases when z ows. This is lea physially beause the numbe of paallel-onneted indutive elements ineases with owin z. The equivalent iuit of the line with nondissipative plasma is shown in Fi..б. The line itself is equivalent to a paallel lumped iuit: C b z and a a. (.56) It is howeve obvious fom alulation that the esonane fequeny is absolutely independent of whateve dimension. Indeed b z n e ρ m. (.57) C 6

17 Fi... а. quivalent iuit of the two-onduto line sement; б. quivalent iuit of the two-onduto line ontainin в. quivalent iuit of the two-onduto line sement ontainin dissipa tive plasma. This bins us to a vey inteestin esult: the esonane fequeny of the maosopi esonato is independent of its size. It may seem that we ae dealin hee with the anmui esonane beause the obtained fequeny oesponds exatly to that of the anmui esonane. We howeve now that the anmui esonane haateizes lonitudinal waves. The wave popaatin in the phase veloity in the z-dietion is equal to infinity and the wave veto is z whih oesponds to the solution of qs. (.4) fo a line of pe-assined onfiuation (Fi..). qs. (.5) ive a wellnown esult. The wave numbe is 7

18 The oup and phase veloities ae whee µ / ρ is the veloity of liht in vauum. z. (.58) ρ (.59) ρ F (.6) Fo the plasma unde onsideation the phase veloity of the eletomaneti wave is equal to infinity. Hene the distibution of the fields and uents ove the line is unifom at eah instant of time and independent of the z-oodinate. This implies that on the one hand the indutane HΣ has no effet on the eletodynami poesses in the line and on the othe hand any two planes an be used instead of ondutin planes to onfine plasma above and below. qs. (.58) (.59) and (.6) indiate that we have tansvese esonane with an infinite Q- fato. The fat of tansvese esonane i.e. diffeent fom the anmui esonane is most obvious when the Q-fato is not equal to infinity. Then z and the tansvese wave is popaatin in the line alon the dietion pependiula to the movement of hae aies. Tue we stated ou analysis with plasma onfined within two planes of a lon line but we have thus found that the pesene of suh esonane is entiely independent of the line size i.e. this esonane an exist in an infinite medium. Moeove in infinite plasma tansvese esonane an oexist with the anmui esonane haateizin lonitudinal waves. Sine the fequenies of these esonanes oinide both of them ae deeneate. alie the possibility of tansvese esonane was not onsideed. To appoah the poblem moe ompehensively let us analyze the eney poesses in loss-fee plasma. The haateisti esistane of plasma deteminin the elation between the tansvese omponents of eleti and maneti fields an be found fom whee µ µ ρ Z z y Z Hx Z is the haateisti esistane in vauum. / (.6) The obtained value of Z is typial fo tansvese eletomaneti waves in waveuides. When ρ Z and H x. At > ρ both the eleti and maneti field omponents ae pesent in plasma. The speifi eney of the fields is W µ H y + Hx. (.6) 8

19 Thus the eney aumulated in the maneti field is ρ times lowe than that in the eleti field. This taditional eletodynami analysis is howeve not omplete beause it diseads one moe eney omponent the ineti eney of hae aies. It tuns out that in addition to the eleti and maneti waves ayin eleti and maneti eney thee is one moe wave in plasma the ineti wave ayin the ineti eney of hae aies. The speifi eney of this wave is W The total speifi eney thus amounts to ρ j. (.63) W H j y + µ Hx + j. (.64) Hene to find the total speifi eney aumulated in unit volume of plasma it is not suffiient to allow only fo the fields Е and Н. At the point ρ W H (.65) W W i.e. thee is no maneti field in the plasma and the plasma is a maosopi eletomehanial avity esonato of fequeny ρ.. At > ρ the wave popaatin in plasma aies thee types of eney maneti eleti and ineti. Suh wave an theefoe be-alled manetoeletoineti. The ineti wave is a uentdensity wave j d t. It is shifted by π/ with espet to the eleti wave. Up to now we have onsideed a physially unfeasible ase with no losses in plasma whih oesponds to infinite Q-fato of the plasma esonato. If losses ou no matte what physial poesses aused them the Q-fato of the plasma esonato is a final quantity. Fo this ase the Maxwell equations beome ot µ ot H σ p. ef H + + d t. (.66) The tem σ р.ef allows fo the loss and the index ef nea the ative ondutivity emphasizes that we ae inteested in the fat of loss and do not ae of its mehanism. Nevetheless even thouh we do not ty to analyze the physial mehanism of loss we should be able at least to measue σ р.ef. Fo this pupose we hoose a line sement of the lenth z whih is muh shote than the wavelenth in dissipative plasma. This sement is equivalent to a iuit with the followin lumped paametes b z C (.67) a 9

20 whee G is the ondutane paallel to C and. The ondutane G and the Q-fato of this iuit ae elated as d (.68) b z b z G σ ρ. ef (.69) a G Q Tain into aount qs. (.67) (.69) we obtain fom q. (.7) σ ρ C. (.7) ρ. ef. (.7) Qρ Thus σ р.ef. an be found by measuin the basi Q-fato of the plasma esonato.usin qs. (.7) and (.66) we obtain ot µ ot H Q ρ H + + d t The equivalent iuit of this line ontainin dissipative plasma is shown in Fi..b. et us onside the solution of qs. (.7) at the point p. Sine We obtain. (.7) + d t. (.73) H ot µ ot H Q P. (.74) The solution of these equations is well nown. If thee is intefae between vauum and the medium desibed by qs. (.74) the sufae impedane of the medium is whee σ p. ef. Q p Z H t t µ p σ p. ef. ( + i) (.75)

21 Thee is of ouse some unetainty in this appoah beause the sufae impedane is dependent on the type of the field-uent elation (loal o non-loal). Althouh the appoah is simplified the qualitative esults ae quite adequate. Tue a moe ioous solution is possible. z z i δef δef The wave popaatin deep inside the medium deeases by the law e e. In this ase the phase veloity is σ (.76) whee F p. ef δ p. ef µ pσ is the effetive depth of field penetation in the plasma. The above ela- p. ef σ ef tions haateize the wave poess in plasma. Fo ood ondutos we usually have >>. In suh a medium the wavelenth is λ πδ. (.77) I.e. muh shote than the fee-spae wavelenth. Futhe on we onentate on the ase λ >> λ at the point р i.e. F р >>.. Disussion of esults. We have found that ( is not dieleti indutivity pemittivity. Instead it inludes two fequeny-independent paametes and. What is the eason fo the physial misundestandin of the paamete (? This ous fist of all beause fo the ase of plasma the d t - type tem is not expliitly pesent in the seond Maxwell equation. Thee is howeve anothe eason fo this seious mistae in the pesent-day physis [7] as an example. This study states that thee is no diffeene between dieletis and ondutos at vey hih fequenies. On this basis the authos suest the existene of a polaization veto in ondutin media and this veto is intodued fom the elation P Σ e n e (.78) m m whee n is the hae aie density m is the uent hae displaement. This appoah is physially eoneous beause only bound haes an polaize and fom eleti dipoles when the extenal field oveomin the attation foe of the bound haes aumulates exta eletostati eney in the dipoles. In ondutos the haes ae not bound and thei displaement would not podue any exta eletostati eney. This is espeially obvious if we employ the indution tehnique to indue uent (i.e. to displae haes) in a in onduto. In this ase thee is no estoin foe to at upon the haes hene no eleti polaization is possible. In [7] the polaization veto found fom q. (.78) is intodued into the eleti indution of ondutin media D + P (.79) whee the veto P of a metal is obtained fom q. (.78) whih is won. Sine

22 fo fee aies then fo plasma and Thus the total aumulated eney is D*( ) v m P*( e m (.8) n e m (.8) p + P*(. (.8) W + Σ. (.83) Howeve the seond tem in the iht-hand side of q. (.83) is the ineti eney (in ontast to dieletis fo whih this tem is the potential eney). Hene the eleti indution veto D*( does not oespond to the physial definition of the eleti indution veto. The physial meanin of the intodued veto P *( is lea fom σ P*( ). (.84) The intepetation of ( as fequeny-dependent indutivity has been hamful fo oet undestandin of the eal physial pitue (espeially in the eduational poesses). Besides it has dawn away the eseahes attention fom some physial phenomena in plasma whih fist of all inlude the tansvese plasma esonane and thee eney omponents of the manetoeletoineti wave popaatin in plasma. Below the patial aspets of the esults obtained ae analyzed whih pomise new data and efinement of the uent views..3 Patial aspets. Plasma an be used fist of all to onstut a maosopi sinle-fequeny avity fo development of a new lass of eletoineti plasma lases. Suh avity an also opeate as a band-pass filte. At hih enouh Q the maneti field eney nea the tansvese esonane is onsideably lowe p than the ineti eney of the uent aies and the eletostati field eney. Besides unde etain onditions the phase veloity an muh exeed the veloity of liht. Theefoe if we want to exite the tansvese plasma esonane we an put whee jct ot Q is the extinsi uent density. p + + d t j (.85) CT

23 Inteatin q. (.84) ove time and dividin it by obtain p jct p + +. (.86) Qp Inteatin q. (.86) ove the sufae nomal to the veto and tain Φ d S we have p Φ + Q Φ + p Φ p I CT (.87) whee I CT is the extinsi uent. q. (.87) is the hamoni osillato equation whose iht-hand side is typial of two-level lases [8]. If thee is no exitation soue we have a old. ase avity in whih the osillation dampin follows the exponential law P i t t P QP Φ ( t) Φ () e e (.88) i.e. the maosopi eleti flow Ф (t) osillates at the fequeny р. The elaxation time an be ound as τ Q P. (.89) If this avity is exited by extinsi uents the avity will opeate as a band-pass filte with the p pass band. Q p Tansvese plasma esonane offes anothe impotant appliation it an be used to heat plasma. Hih-level eleti fields and hene hih hane-aie eneies an be obtained in the plasma esonato if its Q-fato is hih whih is ahievable at low onentations of plasma. Suh avity has the advantae that the haes attain the hihest veloities fa fom old planes. Usin suh haes fo nulea fusion we an eep the poess fa fom the old elements of the esonato. Suh plasma esonato an be mathed easily to the ommuniation line. Indeed the equivalent esistane of the esonato at the point р is R экв. P a QP G (.9) b z The ommuniation lines of sizes а and b should be onneted to the avity eithe thouh a smooth juntion o in a stepwise manne. If b b the mathin equiement is a b µ p a Q a z a Q b z (.9) p µ. (.9) It should be emembeed that the hoie of the esonato lenth z must omply with the equiement z << λ p. 3

24 Development of devies based on plasma esonato an equie oodination of the esonato and fee spae. In this ase the followin ondition is impotant: o µ p a Q b z a Q (.93) b z p µ. (.94) Suh plasma esonatos an be exited with d.. uent as is the ase with a monoton miowave osillato [9]. It is nown that a miowave diode (the plasma esonato in ou ase) with the tansit anle of ~5/π develops neative esistane and tends to self-exitation. The equiement of the tansit anle equal to 5/π oelates with the followin d.. voltae applied to the esonato: U 3a 4π p e m 3a n e 4π µ (.95) whee а is the distane between the plates in the line. It is quite pobable that this effet is esponsible fo the eletomaneti osillations in semiondutive lases..4 Dieleti media. Applied fields ause polaization of bound haes in dieletis. The polaization taes some eney fom the field soue and the dieleti aumulates exta eletostati eney. The extent of displaement of the polaized haes fom the equilibium is dependent on the eleti field and the oeffiient of elastiity β haateizin the elastiity of the hae bonds. These paametes ae elated as β + m e m m m (.96) whee m is the hae displaement fom the equilibium. Puttin fo the esonane fequeny of the bound haes and tain into aount that β m we obtain fom q. (.96) The polaization veto beomes Sine we obtain m e m ( ) n e o. ( ) P * m m. (.97) (.98) P ( ) (.99) n e *(. m (.) 4

25 ' The quantity *( is ommonly alled the elative fequeny dependably leti indutivity. Its absolute value an be found as * n e ( ( ). m (.) One aain we aive at the fequeny-dependent dieleti pemitlivity. et us tae a lose loo at m the quantity *(. As befoe we intodue and p. and see n e immediately that the vibatin haes of the dieleti have masses and thus possess inetia popeties. As a esult thei ineti indutivity would mae itself evident too. q. (.) an be e-witten as p ( ( ). * It is appopiate to examine two limitin ases: >> and <<. If >> (.) p ( ( ) (.3) * and the dieleti behaves just lie plasma. This ase has pompted the idea that at hih fequenies thee is no diffeene between dieletis and plasma. The idea seved as a basis fo intoduin the polaization veto in ondutos [7]. The diffeene howeve exists and it is of fundamental impotane. In dieletis beause of inetia the amplitude of hae vibations is vey small at hih fequenies and so is the polaization veto. The polaization veto is always zeo in ondutos. Fo << p ( ( + ) (.4) * p ( and the pemittivity of the dieleti is independent of fequeny. It is + ) times hihe than in vauum. This esult is quite lea. At >> the inetia popeties aeinative and pemittivity appoahes its value in the stati field. The equivalent iuits oespondin to these two ases ae shown in Fis..3а and b. It is seen that in the whole ane of fequenies the equivalent iuit of the dieleti ats as a seies osillatoy iuit paallel-onneted to the apaito opeatin due to the eleti indutivity of vauum (see Fi..3b). The esonane fequeny of this seies iuit is obviously obtainable fom д p. (.5) 5

26 Fi..3. quivalent iuit of two-onduto line sement with a dieleti: а - >> ; б << ; в the whole fequeny ane. 6

27 ae in the ase of plasma is independent of the line size i.e. we have a maosopi esonato whose fequeny is only tue when thee ae no bonds between individual pais of bound haes. ie fo plasma *( is speifi suseptane of the dieleti divided by fequeny. Howeve unlie plasma this paamete ontains thee fequeny-independent omponents: and the stati pemittivity of the dieleti *(. p. In the dieleti esonane ous when Thee waves-maneti eleti and ineti-popaate in it too. ah of them aies its own type of eney. It not is not poblemati to alulate them but we omit this hee to save oom..5 Maneti media. The esonane phenomena in plasma and dieletis ae haateized by epeated eletostatiineti and ineti-eletostati tansfomations of the hae motion eney duin osillations. This an be desibed as an eletoineti poess and devies based on it (lases mases filtes et.) an be lassified as eletoineti units. Howeve anothe type of esonane is also possible namely maneti esonane. Within the uent onepts of fequeny-dependent pemeability it is easy to show that suh dependene is elated to maneti esonane. Fo example let us onside feomaneti esonane. A feite manetized by applyin a stationay field Н paallel to the z-axis will at as an anisotopi manet in elation to the vaiable extenal field. The omplex pemeability of this medium has the fom of a tenso []: whee µ T *( µ i α i α µ *( T µ (.6) Ω γ M γ M µ T *( α µ (.7) µ ( Ω ) µ ( Ω ) Ω γ Н. (.8) Bein the natual pofessional fequeny and М µ (µ)н (.9) is the medium manetization. Tain into aount qs. (.8) and (.9) fo µ T *( we an wite Ω ( µ ) *( Ω µ T. (.) Assumin that the eletomaneti wave popaates alon the x-axis and thee ae Н y and Н z omponents the fist Maxwell equation beomes 7

28 ot Tain into aount q. (.) we obtain Fo >>Ω ot ot x H Z y µ µ T. (.) µ Ω ( µ ) H Ω µ y. (.) Ω ( µ ) H y. (.3) Assumen H y H y sin t and tain into aount that q. (.3) ives o Fo << Ω The quantity H y H d t H y ot + Ω µ ( µ ) H y d t y. (.4) µ (.5) H y ot + H y d t C µ. (.6) ot H µ µ y. (.7) µ Ω ( µ C (.8) ) an be desibed as ineti apaitane. What is its physial meanin? If the dietion of the manetimoment does not oinide with that of the extenal maneti field the veto of the moment stats peessional motion at the fequeny Ω about the maneti field veto. The maneti moment m m B has the potential eney U m. ie in a haed ondense U m is the potential eney beause the peessional motion is inetialess (even thouh it is mehanial) and it stops immediately when the maneti field is lifted. In the maneti field the poessional motion lasts until the aumulated potential eney is exhausted and the veto of the maneti moment beomes paallel to the veto H. The equivalent iuit fo this ase is shown in Maneti esonane ous at the point Ω and µт*(. It is seen that the esonane fequeny of the maosopi maneti esonato is independent of the line size and equals Ω. 8

29 Рис..4. quivalent iuit of two-onduto line inludin a manet. Thus the paamete Ω ( µ ) µ H *( µ Ω (.9) is not a fequeny-dependent pemeability. Aodin to the equivalent iuit in Fi..4 it inludes µ µ and С. It is easy to show that thee waves popaate in this ase-eleti maneti and a wave ayin potential eney of the peessional motion of the maneti moments about the veto H. The systems in whih these types of waves ae used an also be desibed as eletomanetopotential devies. Conlusions Thus it has been found that alon with the fundamental paametes and µµ haateizin the eleti and maneti eney aumulated and tansfeed in the medium thee ae two moe basi mateial paametes and C. They haateize ineti and potential eney that an be aumulated and tansfeed in mateial media. was sometimes used to desibe etain physial phenomena fo example in supeondutos [] C has neve been nown to exist. These fou fundamental paametes µµ and C laify the physial pitue of the wave and esonane poesses in mateial media in applied eletomaneti fields. Peviously only eletomaneti waves wee thouht to popaate and tansfe eney in mateial media. It is lea now that the onept was not 9

30 omplete. In fat manetoeletoineti o eletomanetopotential waves tavel in mateial media. The esonanes in these media also have speifi featues. Unlie losed planes with eletomaneti esonane and eney exhane between eleti and maneti fields mateial media have two types of esonane eletoineti and manetopotential. Unde the eletoineti esonsne the eney of the eleti field hanes to ineti eney. In the ase of manetopotential esonane the potential eney aumulated duin the peessional motion an esape outside at the peession fequeny. The notions of pemittivity and pemeability dispesion thus beome physially oundless thouh ( and µ ( ae handy fo a mathematial desiption of the poesses in mateial media. We should howeve emembe thei tue meanin espeially whee eduational poesses ae involved. 3 Maneti field poblem As follows fom the tansfomations in q. (.5) if two haes move at the elative veloity thei inteation is detemined not only by the absolute values of the haes but by the elative motion veloity as well. The new value of the inteation foe is found as [] h F 3 (3.) 4π whee is the veto onnetin the haes is the omponent of the veloity nomal to the. veto If opposite-sin haes ae enaed in the elative motion thei attation ineases. If the haes have the same sins thei epulsion enhanes. Fo q. (3.) beomes the Coulomb law. Usin q. (3.) a mew value of the potential ϕ() an be intodued at the point whee the hae is loated assumin that is immobile and only exeutes the elative motion h ϕ( ). (3.) 4π We an denote this potential as sala-veto beause its value is dependent not only on the hae involved but on the value and the dietion of its veloity as well. The potential eney of the hae inteation is W h 4π. (3.3) qs. (3.) (3.) and (3.3) appaently aount fo the hane in the value of the movin haes. Usin these equations it is possible to alulate the foe of the onduto-uent inteations and allow thouh supeposition fo the inteation foes of all movin and immobile haes in the ondutos. We thus obtain all uently existin laws of eletomanetim. 3

31 Fi. 3.. Shemati view of foe inteation between uent-aein ondutos of a two-onduto line in tems of the pesent-day model. et us examine the foe inteation of two z-spaed ondutos (Fi. 3.) assumin that the eleton veloities in the ondutos ae and. The movin hae values pe unit lenth of the ondutos ae and. In tems of the pesent-day theoy of eletomanetism the foes of the inteation of the ondutos an be found by two methods.. One of the ondutos (e.. the lowe one) eneates the maneti field H() in the loation of the fist onduto. This field is H( ). (3.4) π The field is exited in the oodinate system movin toethe with the haes of the uppe onduto: [ B] H( ) µ. (3.5) I.e. the haes movin in the uppe onduto expeiene the oentz foe. This foe pe unit lenth of the onduto is µ I I π π. (3.6) F q. (3.6) an be obtained in a diffeent way. Assume that the lowe onduto exites a veto potential in the eion of the uppe onduto. The z omponent of the veto potential is ln π I ln A Z. (3.7) π The potential eney pe unit lenth of the uppe onduto ayin the uent I in the field of the veto potential A Z is 3

32 W I I ln IAZ. (3.8) π Sine the foe is the deivative of the potential eney with espet to the opposite-sin oodinate it is witten as W I I π F. (3.9) Both the appoahes show that the inteation foe of two ondutos is the esult of the inteation of movin haes: some of them exite fields the othes inteat with them. The immobile haes epesentin the lattie do not patiipate in the inteation in this sheme. But the foes of the maneti inteation between the ondutos at just on the lattie. Classial eletodynamis does mot explain how the movin haes expeienin this foe an tansfe it to the lattie. The above models of iteation ae in unsolvable onflit and expets in lassial eletodynamis pefe to pass it ove in silene. The onflit is onneted with estimation of the inteation foe of two paallel-movin haes. Within the above models suh two haes should be attated. Indeed the indution В aused by the movin hae at the distane is B. (3.) π If anothe hae moves at the same veloity in the same dietion at the distane fom the fist hae the indution В at the loation of podues the foe attatin and. F (3.) 4π An immovable obseve would expet these haes to expeiene attation alon with the Coulomb epulsion. Fo an obseve movin toethe with the haes thee is only the Coulomb epulsion and no attation. Neithe lassial eletodynamis not the speial theoy of elativity an solve the poblem. Physially the intodution of maneti fields eflets etain expeimental fats but so fa we an hadly undestand whee these fields ome fom. In 976 it was epoted in a seious expeimental study that a hae appeaed on a shot-iuited supeondutin solenoid when the uent in it was attenuatin. The esults of [3] suest that the value of the hae is dependent on its veloity whih is fist of all in ontadition with the hae onsevation law. The autho of this study has also investiated this poblem [6 4] (see below). It is useful to analyze hee the inteation of uent-ayin systems in tems of qs. (3.) (3.) and (3.3) [ 4]. We ome ba aain to the inteation of two thin ondutos with haes movin at the veloities and (Fi. 3.). + + and ae the immobile and movin haes espetively pe unit lenth of the ondutos. + and + efe to the positively haed lattie in the lowe and uppe ondutos espetively. Befoe the haes stat movin both the ondutos ae assumed to be neutal eletially i.e. they ontain the same numbe of positive and neative haes. ah onduto has two systems of unlie haes with the speifi densities + and +. The haes neutalize eah othe eletially. To mae the analysis of the inteation foes moe onvenient in Fi. 3. the systems ae sepaated alon the z-axis. The neative-sin subsystems (eletons) have veloities and. The foe of the inteation between the lowe and uppe on- 3

33 33 Fi. 3.. Shemati view of foe inteation between uent-ayin wies of a two-onduto line. The lattie is haed positively. dutos an be onsideed as a sum of fou foes speified in Fi. 3. (the dietion is shown by aows). The attation foes F 3 and F 4 ae positive and the epulsion foes F and F ae neative. Aodin to q. (3.) the foes between the individual hae subsystems (Fi. 3.) ae. 4 3 h F h F h F F π π π π (3.) By addin up the fou foes and emembein that the podut of unlie haes and the podut of lie haes oespond to the attation and epulsion foes espetively we obtain the total speifi foe pe unit lenth of the onduto + Σ h h h F π. (3.3) whee and ae the absolute values of haes. The sins of the foes appea in the baeted expession. Assumin << с we use only the two fist tems in the expession of h i.e. h +. q. (3.3) ives I I F π π Σ (3.4)

34 whee and ae the absolute values of speifi haes and ae taen with thei sins. It is seen that qs. (3.6) (3.9) and (3.3) oinide thouh they wee obtained by diffeent methods. Aodin to Feynman (see the intodution) the e.m.f. of the iuit an be intepeted usin two absolutely diffeent laws. The paadox has howeve been laified. The foe of the enteation between the uent-ayin systems an be obtained even by thee absolutely diffeent methods. But in the thid method the motion maneti field is no lone neessay and the lattie an dietly patiipate in the fomation of the inteation foes. This was impossible with the pevious two tehniques. In patie the thid method howeve uns into a seious obstale. Assumin + and i.e. the inteation fo example between the lowe uent-ayin system and the immobile hae the inteation foe is π F Σ. (3.5) This means that the uent in the onduto is not eletially neutal and the eleti field 4π (3.6) is exited aound the onduto whih is equivalent to an exta speifi stati hae on the onduto. (3.7) Befoe [3] thee was no evidene fo eneation of eleti fields by d.. uents. When Faaday and Maxwell fomulated the basi laws of eletodynamis it was impossible to onfim q. (3.7) expeimentally beause the uent densities in odinay ondutos ae too small to detet the effet. The assumption that the hae is independent of its veloity and the subsequent intodution of a maneti field wee meely voluntaisti ats. ~ In supeondutos the uent densities pemit us to find the oetion fo the hae expeimentally. Initially [3] was taen as evidene fo the dependene of the value of the hae on its veloity. The autho of this study has also investiated this poblem [6 4] but unlie [3] in his expeiments uent was intodued into a supeondutin oil by an indutive non-ontat method. ven in this ase a hae appeaed on the oil [6 4]. The expeimental objets wee supeondutin omposite Nb Ti wies oated with oppe and it is not leat what mehanism is esponsible fo the hae on the oil. It may be bouht by mehanial defomation whih auses a displaement of the Femi level in the oppe. xpeiments on non-oated supeondutin wies may be moe infomative. Anyhow the subjet has not been exhausted and futhe expeimental findins ae of paamount impotane to fundamental physis. Usin this model we should emembe that thee is no eliable expeimental data on stati eleti fields aound the onduto. Aodin to q. (3.6) suh fields ae exited beause the value of the hae is dependent on its veloity. Is thee any physial mehanism whih ould maintain the inteatin uent-ayin systems eletially neutal within this model? Suh mehanism does exist. To explain it let us onside the uent-ayin iuit in Fi This is a supeondutin thin film whose thiness is smalle than the field penetation depth in the supeonduto. The uent is theefoe distibuted unifomly ove the film thiness. Assume that the bide onnetin the wide pats of the film is muh naowe than the est of the uent-ayin film. If pesistent uent is exited in suh a iuit the uent 34