2. Equation of generalized Dynamics. Let rectangular right hand coordinate triple is fixed in three-dimensional Euclidian space.

Transcription

1 Genealized Dynamis about Foes Ating on Chage Moving in Capaito and Solenoid. J.G. Klyushin, Ph. D. Aademy of Civil Aviation, hai of applied mathematis; mail: Intenational Club of Sientists, Kazanskay st., 6, 9 St. Petesbug, RUSSIA. Kaufman s pape published in 9 on eletons deviation in eleti and magneti fields beame a one stone in expeimental substantiation of speial elativity theoy. It is shown in this pape that genealized eletodynamis poposed by the autho implies this esult.. Intodution. Fundamental step in eletiity theoy was taken in 846 when W. Webe poposed foe fomula genealizing Coulomb law fo the ase of moving hages. This fomula desibes foe appeaing between two hages moving with etain veloities and aeleations. This fomula showed beautiful oodination with expeiments and beame a basis fo an investigation pogam in the poess of whih othe fomulas desibing hages inteation wee poposed. Let us mention hee Neumann s, Ampee s, Gassman s, Whittake s fomulas whih auate investigation and ompaison with Webe s fomula may be found in the autho s pape published in Russian and now tanslating into English (also see Mainov s pape []. A little bit late Maxwell in geat Bitain poposed his field appoah to eletodynamis desiption. This appoah thanks to Hez expeiments and Heaviside effots beame pevailing and patly elipsed Webe s one. But these new onepts didn t have as plausible physial meaning as Webe s foe ones. Suh meaning was allotted to field onept by well-known fomula appaently poposed by Heaviside and late alled Loentz foe fomula. It an be eadily shown this fomula in field tems epeat Gassman s fomula attahed foe meaning to field onept. If Maxwell equations ae onsideed as postulates whih desibe hanges in suounding spae whih a etain set of hages eates then Loentz fomula is an additional axiom desibing onsequenes to whih hanges lead fo a etain hage alled pobe. It is assumed that pobe hage does not have its own field although it moves just in the same way as ative hage whih eates the field. It is quite nonsymmetially assumed that eleti field ats on stati pat and magneti field ats on dynami pat of the pobe hage. It is mentioned above that Loentz fomula gives not moe and not lese in ompaison with Gassman fomula i.e. it desibes athe a naow lass of phenomena. This fat ealization led to attempt of diet foe intepetation of Maxwell equation, i.e. to flow ules whih ae used in moden physis togethe with Loentz foe fomula. Although flow ules widened the lass of desibing phenomena but fist it is not suffiient and the seond (and this is the main point it tuned to be absolutely nonsatisfatoy in logial aspet. Maxwell equations desibe only fields (hanges in spae oiginated by a etain set of hages and they ae not apable to desibe fields inteation. Suh inteation should be desibed by a etain additional fomula. Although Loentz foe fomula is not univesal enough its ole in eletodynamis is just this one, it is additional to Maxwell equations axioms, whih desibes fields inteation. Reently one additional poblem beame lea. As we undestand now Maxwell equations self ae not univesal enough. Thei multiple expeimental poves led to thei dogmatisation. The gaps wee filled with additional postulates in the famewok of elativity theoy, etaded potentials et. These additional axioms togethe with plain mathematial mistakes led to efusal of Galileo invaiane and to some othe suh paadoxial assumptions suh that new theoy insanity is now onsideed to be neessay ondition fo its validity. Populaizes of physis and jounalists inulated suh insane theoies into publi mentality eating auas of mystey and mystiism aound physial theoies. In physis self it led to pedomination of oveompliated mathematis ove sobe physial mentality. Appaently it is a high time to etun to soues and eundestand many habitual assumptions. Suh an attempt was done by the autho in his pape []. A digest of it is epodued in setion hee. Setion is devoted to explanation of Kaufman s expeiment in the poposed tems.. Equation of genealized Dynamis. Let etangula ight hand oodinate tiple is fixed in thee-dimensional Eulidian spae. Let =(,, be a point in this spae, t be time and i, j, k be ots. Let q, q be eleti hages and. V, V, and, be thei veloities and aeleations. If othe assetion is not delaed, hages q and q ae onsideed to be evenly distibuted in a ball of adius. Let E, E, B, B be eleti and magneti fields tensities oiginated by the hages in spae. Double index fom below means field tensity

2 eated by the hage whose index goes fist in the point whee the hage whose index goes seond is situated. Fo instane means eleti field tensity eated by the seond hage in the point whee the fist hage is situated. Let be the adius-veto fom hage to hage, be its modulus, >>, and be eleti onstant. Genealized fomula fo Loentz foe. Chage podues the following foe on hage : d F = gad[ ( B E ] [ ( B B ] (. Gadient is alulated with espet to passive hage oodinates. Hee and eveywhee below = [ i j] k, whee is light veloity. This quantity is alled pseudosala light veloity. Two notion of foe ae used in moden physis: idea inheited afte Newton and Desates as an impulse deivative with espet to time and idea inheited afte Huygens and Leibnitz as an enegy gadient. (. ombines both ideas. Evey hage eates eleti and magneti fields in spae. Sala podut of passive hage magneti field and ative hage eleti field desibes inteative enegy density oiginated by the hages. Its integal is in squae bakets in the fist item. Veto podut of hages' magneti field defines inteative impulse density. Its integal is in squae bakets in the seond item. One needs to find the field oiginated by the hages. One an get it fom equation desibing the fields. Maxwell equations is suh a system in lassial theoy. But Maxwell equations should be modenized in ode to be oodinated to fomula (.. Genealized Maxwell equations. Eleti hage q distibuted in the spae with density, oiginates eleti and magneti fields whih ae solution of following system: ρ dive = (. ote = db (. ρ divb = (.4 otb = de (.5 Let us begin ou explanations with the equation (.5. de E = ( V gad E (.6 t whee V is the hage veloity. The fist item in the ight hand pat of (.6 genealizes the idea of a uent in lassial theoy and omes to it if E satisfies some additional onditions. j ( V gad E = Vdiv E ot( E V = ot( E V, whee j is uent density. So ight hand pat of (.5 ontains a oto omponent in addition to lassial one. This item is manifested fo instane in a light wave. O in eation foe lines not enveloping uents. (.4 means that equations (. - (.5 genealize the idea of magneti field. Magneti field B that is the solution of (. - (.5 possesses not only oto but divegent omponents as well. Divegent omponent of B is defined by pseudosala eleti hage (usual eleti hage divided by mixed podut of ots and light veloity. B opens to be pseudoveto just like in lassial theoy. Right hand pat of (.4 may be onsideed as "anothe inanation" fo eleti hage beause existene of eleti hage is neessay and suffiient fo its existene. One may onside it as a "magneti hage". But it is neessay to emphasize that suh a "magneti hage" does not oinide with Dia's monopole. Let us pinpoint some of the diffeenes.. Suh a "magneti hage" is pseudosala, i.e. its sign hanges when ight hand oodinate tiple is hanged fo left hand one.. It is times less than eleti hage, oespondingly its dimension diffes fom eleti hage dimension.. And last but not least (. implies that two stati "magneti hages" do not inteat, beause seond item in (. esponsible fo magneti field' inteation is zeo in this ase. I ask eade to pay attention to this fat beause "odinay physial mentality" usually identifies field and foe.

3 The ight hand pat of (. looks as de B = ( V gad B (.7 t So (. diffes fom lassial one in that it inludes gadient deivative of B oiginated by eleti hage (and oespondingly "magneti hage" movement with veloity V. Classial theoy assoiates the appeaane of magneti field just with the movement of eleti hages but does not inlude the oiginating movement into (. equation. (. oinides with lassial one. (. - (.5 define in diffeential fom the fields E and B oiginated by moving hages. Just this fields one needs in ode to use fomula (.. Mathematially system (. - (.5 dissoiates into two goups. Equations (. and (.5 defines E and B whih ae thei solutions. Equations (. and (.4 fix bounday-value onditions in a peulia Neumann poblem: not a gadient but a divegene is defined on the bounday, i.e. in the point whee ρ. When E and B ae found on the bounday they ae extended on the whole domain. It is possible beause potential is hamoni. E and B got fom these onditions define pat of fields' tensities. If veloity V does not depend with espet to spae oodinates then equations (. - (.5 imply dρ ρ = V gadρ = (.8 t This oelation defines an intensified hage onsevation law: the hage is not only onseved but it behaves like inompessible liquid. Let us investigate ase when is independent with espet to t expliitly, i.e. ρ t = Then (.8 implies beause of abitainess of V: gad ρ = (. It is supposed that V is independent with espet to spae oodinates and is a funtion with espet to only t. (.9 V = V( t (. Chages evenly distibuted in a ball of adius << evidently satisfy onditions (.9 and (.. if these onditions ae valid then one an define one patial solution of (. - (.5: ρ ( V E = (. ρ ( V B = (. whee is adius-veto fom the hage to obsevation point. Let us veify by diet substitution that (. and (. ae solutions of modified Maxwell equations (. - (.5. gadρ ( V ρ ( V ρ dive = = gadρ ( V ρ ( V ρ divb = = Let us find left and ight hand pats of (. ot ρ V ρ ρv E = gad [ ( V gad ( gad V V( div ] = d dρ V ρ V V a ρ a B = V = V. But the fist item in last squae bakets is adiated by a hanging field. So one get finally:

4 d ρv B = (.5 is veified in the same way. Let us wite down in an expliit way fo this ase all the items inluded in (.. q V q V. B = = π 4 q V. E = Let us find gadient of sala podut of these fields alulating the oesponding deivatives with espet to passive fist hage oodinates. ( V ( V qq. B E = 6 6π 4. gad[ ( B E ]= (( V ( V ( V V ( V V ( V V qq = q V 5. B = q V 6. B = qq 7. ( ( V ( V ( V V B B = Radius-veto deivatives d d = V V, = a. If the time of signal's lagging behind is not essential with espet to the poblem's onditions then the deivatives ae alulated at the same time t. Othewise the ative hage veloity and aeleation should be alulated at the pevious time. d 8. 4 ( B B τ = t qq [ ] = ( V V ( V V [ ( ] π { ( V V [ ( ( V V ] [ ( a ] ( V V [( V ( V ] [( V ( a ( V ( a ] One gets finally: the foe whih the seond hage podues on the fist one is qq qq F = ( V V V ( V V ( V (( V ( V [( ( ( ] ( V V V V V V [ ( ( V V ] [ ( ( a ] ( V V [( V ( V ] [( V ( a ( V ( a ] (.4 One gets anothe fom fo the foe when veto tiple poduts ae evealed: 4

5 F q q qq ( V V V ( V V ( V (( V ( V = [ ( ( ( ( ] ( V V V V V V V V [ ( ( V V ( V V ] [ ( ( ( ] ( V V ( ( V V ( V a ( V a a Let us wite out additional fom of (.5 using expliitly the angles between vetos. Let - be angle between and V - be angle between and V - be angle between V and V 4 - be angle between [ ] (.5 and ( V V 5 - be angle between and ( a 6 - be angle between and ( V V 7 - be angle between ( V and a 8 - be angle between ( V and a qq qq F [ V V Cos V V Cos V V ( Cos Cos ] = { Cos [ ( V V ( Cos 4 ( V V V V Cos 4 ] [ a ] ( V V V V Cos Sin [ a V Sin Cos a Sin Cos ] 6 7 V 8 (.6 All deivatives hee ae alulated with espet to laboatoy fame of efeene fo "nude hages" and with espet to ondutos fo uents in neutal ondutos. Let us etun to funtion (. and (.. The seond item in thei ight hand pats define stati omponent and is manifested only fo "nude hages". The fist one defines dynami omponent and is manifested not only fo haged but fo neutal uents as well. This quality is inheited when these omponents ae multiplied and when deivatives ae alulated in fomula (.. fo instane the fist item in (.4 - (.6 is got as a gadient of stati omponents' podut. Theefoe it is valid only fo "nude hages" (Coulomb foe. On the ontay the fist squae baket is a esult of dynami omponents' podut. So it is valid fo neutal uents as well. One an easily see that this squae baket is a symmetization of lassial Loentz foe. The fist two items oespond to this lassial ase and the seond two ones wok in symmetial ases. The seond squae baket (.4 - (.6 is podut of dynami and stati omponents. So it is equal to zeo between two neutal uents. It is valid if at least one of the uents is haged. This squae baket depends on the hages veloities' diffeene and pedits some effets of Relativity theoy. It also pedits the foe podued on the "nude hage" at est nea the neutal uent. The thid squae baket depends on the hages' aeleations and desibes field adiation. It is valid fo all kinds of uents beause adiated field should be onsideed as a "nude one". It usually pedits the same esults as lassial theoy but it an eadily be show that it pedits no adiation fo an eleton otating aound nuleus. The last two items in baes have the thid small ank with espet to light veloity. They ae appaently essential in eleo-weak inteation.. Foe Ating on a Chage Moving in a Capaito. Let us find foes ating on hage q nea an infinite plane haged with density using the fomulas of the pevious setion. We ae inteested in the ase when the plain hages ae immovable, i.e. V =. qδ qδv F = [ os ] (. wee (, v is the angula between adius-veto fom point x on the haged plane to q and veloity veto V of the hage q. Let us note that only the fist item in (. is taken into aount in 5

6 moden physis. The seond one is not taken into aount beause it is assumed that Coulomb field ats only on stati pat of q. Usually we ae inteested not to undestand foe ating fom a sepaate point of the plane but to define integal foe ating fom the whole plane. We shall find it if integate foe (. fom a to infinity, whee a is the distane q fom the plane. When having integated the fist item one gets the following foe qδ F = (. whih is dieted fom the plane and is independent with espet to q distane fom it. The seond item integal ontains a omponent dieted along the plane in geneal ase beause this foe depends on the osines between adius-veto and V. But we ae inteested in the ase when V is paallel to the plane. Fo this ase os = (. whee is the distane fom the plane points. If (. is put into the seond item of (. and it is integated one gets qδa V a qδv F = [ ]d = 4 (.4 a This foe is also pependiula to the plane. qδ V F = F F = [ ] (.5 The seond item in (.5 depends on the hage veloity and helps Coulomb foe. This item is not taken into aount in moden physis and was not used when Kaufman s expeiments wee explained. Suh explanation was poposed by elativity theoy. The foe ating an a hage in an infinite apaito is doubled qδ V F = [ ] (.6 Fo this ase 4. Foe moving in a haged tube with uent. δ V E = [ ] (4. δ V B = [ ] (4. q V B = [ ] (4. Hee is the hage density on the tube, V is the hage veloity moving on the tube sufae, V is q veloity moving inside the tube, is adius-veto fom the points on the tube sufae to hage q. One gets having epeated the alulation fom setion : foe ating fom point x on the tube sufae on q is: qδ [( V ( V ] F = { V ( V V ( V ( V V ( ( V V ( V V [ ( ( a ]} (4.4 Foe (4.4 does not depend on adius-veto modulus and depends only on its dietion. In any avity (not only in infinite tube o sphee fo any dietion fom the sufae thee exists an invese one fom the symmetial point. Null ontibution to integal foe is done by an item if its oeffiient is onstant. The fist item in baes (Coulomb foe is suh an item in (4.4. The seond the thid the fouth and the fifth items do ontibution of full value to integal foe beause V also hange sign when adius-veto hange it. 6

7 Theefoe foes ating fom symmetial sufae points ae added onstutively, i.e. ae doubled. The thid and the fifth items ae just lassial Loentz foe. It beomes espeially evident if they ae desibed as tiple podut V ( V ( V V = V ( V (4.5 Veto podut in bakets defines tube magneti field. We ae inteested in the ase of solenoid, i.e. the ase when V is dieted along dietix tangent. Radius-veto in (4.5 is dieted fom the solenoid sufae inside it. Theefoe the seond item in the left-hand pat of (4.5 pedits foe dieted fom solenoid axis. But expeiment shows existene just entipetal and not entifugal foe. One finds explanation investigating the sixth item in (4.4. One gets fo two symmeti points on solenoid sufae ( V V ( V V = 4( V V This foe is added to the adial omponent of Loentz foe. We neglet hage aeleation, i.e. assume the last item in (4.4 to be null. One finally eeives integal foe ating on q fom two symmetial points on solenoid sufae: q δ ( V ( V ( ( V V F= [ V ( V V ( V ( V V ] (4.6 What is the physial essene of diffeent items? The seond one is a pat of lassial Loentz foe dieted along ylinde dietix. It otates q. The foth item modulo oinide with the seond pat of Loentz foe but is oppositely dieted. The thid and fifth items podue additional adial foe depending on osines of angles between and V and V. The fist item pedit appeaane of a foe dieted along V. It is shown in the autho s pape [] that extenal foe aeleate hage only duing shot time afte whih it just maintains onstant hage veloity. Let q omes into solenoid stitly pependiula to its axis. What is its tajetoy? Rotating foe V ( V and adial foes uve its tajetoy. In steady mode the tajetoy beomes iumfeene (Lamo obit. If then hage omes with a etain angel to the axis, i.e. V possesses nonzeo pojetion on the axis beause of the fist item. The tajetoy beomes helix wih is shown in expeiment. This fist item appeas only in genealized dynamis poposed by the autho in []. This item symmeties Loentz foe fomula and emoves its ontadition to the thid Newton law. Fo this ase it pedits helix haate of the hage movement whih is not done by lassial Loentz foe. Let us biefly onside the ase when the tube uent is dieted along geneatix, i.e. the usual ase of uent in a onduto. The fist Coulomb item podues null ontibution into integal foe just as peviously. Now V. Theefoe the seond and the fifth items ae null identially. If V is pependiula to ylinde axis then the fouth item is also null beause V V and the sixth and the seventh items ae mutually annihilated beause V. Only the thid item dieted along V, i.e. along ylinde axis is peseved. Suh hage q is swept along uent. If V is not stitly pependiula to ylinde axis thee appeas a adial foe on aount of the fouth, the sixth and the seventh items. This foe gows with deease of the angle between V and the uent. Meanwhile the axis foe defined by the thid item deeases. When V V this thid item beomes zeo and the adial foe omes to its maximum. The seventh item beomes null. Appaently just this foe is esponsible fo uent flow along onduto sufae. In geneal the foe is q ( ( ( ( ( δ V V F = V V V V V V. And what is this foe distibution inside the tube? In the ase of solenoid foe is distibuted steadily along its setion. But it is not so hee. Integal foe is null on the ylinde axis beause of the ylinde sufae symmety with espet to it. If hage q is on a ylinde dietix adius with distane a fom the axis then symmety is peseved only with espet to ylinde geneatix but is invalid with espet to its dietix. Foe fom the geate a exeeds the foe fom the mino one and the integal foe is equal to thei diffeene. 7

8 Let ϕ be angle leaning on the mino a and is the dietix adius. The geate a A is equal to and the mino a B is equal to ϕ. Thei diffeene atio to the iumfeene length is ( π ϕ A B π ϕ =, ϕ [ π,]. π π ϕ an be expessed with the help of the distane a fom the ylinde axis to hage q a = osϕ Hene ϕ = aosa When a=, ϕ = π, when a =, ϕ =. One finally gets : when V V the integal foe qδ π os a F = π when V V qδ F = [ ( ] [ π os( a ]( V V ( V V V [ ] π Hee is adius-veto fom the ylinde sufae going though q and ylinde axis. Equi-foe sufaes hee ae ylindes oaxial to the initial tube. Suh foes lines wee expeimentally deteted by E. A. Gigoiev. Let us not that suh foe lines do not envelope uent but ae ontained in it. This fat ontadits the othodox theoy but is implied by genealized eletodynamis. Let as summaize ou naation. Genealized dynamis equations poposed by the autho pedit additional integal foes ation on hage moving as inside apaito as inside solenoid. The foes depend on the hage veloity and final distibution of the hages on seen in fat depends on thei veloities. In ode to undestand the haate of suh dependene one should desibe hage movement unde suh foes ation. This poblem is investigated in the seond atile of the autho in this olletion []. P.S.: It was assumed above that is steady (in aodane to the autho's onept is aethe density []. Fo apaito this means that spae between its plates is not filled with matte. If this spae is filled with dieleti then an additional foe appeas beause of spae hange. This additional foe is qδ V F = ( Cos gad (4.7 Hee q is the hage inside dieleti, is the angle between adius-veto fom the plate to q, is its module, V is hage veloity on the apaito's plates. The fist item hee has multipliity of Coulomb foe and is dieted against gad. Just this foe is the eason why plasti of dieleti is dawn into apaito. To-day theoy explains this effet by dieleti polaization. The majoity of expeimental fats known to the autho an be explained eithe by othodox o by the poposed theoy. Theefoe it is essential to find an expeiment whih would help to selet between these two appoahes. Let us investigate a apaito with dieleti of deeasing between its plates. Othodox theoy does not pedit any onsideable hanges. It believes that field inside apaito is deeasing as integal value. The poposed theoy pedits a foe ating on apaito's plates and dieted to deease, i.e. suh a apaito moves. The seond item in (4.7 has a multipliity of magneti foe, it is dieted along gad and appaently defines paamagneti and diamagneti haateistis of matte. It is neessay to note that these foes inease with. 8

9 Refeenes [] S. Mainov. Divine Eletomagnetism., p. 8, (East-West, Gaz, 99 [] J.G. Klyushin. A field Genealization fo the Loentz Foe Fomula, Galilean Eletodynamis, v. N5 (is to appea, [] J.G. Klyushin. "On eleton s dynamis"., this olletion. [4]......,,