The Faraday Induction Law and Field Transformations in Special Relativity

Transcription

1 Apeiron, ol. 10, No., April The Farada Indction Law and Field Transformations in Special Relatiit Aleander L. Kholmetskii Department of Phsics, elars State Uniersit, 4, F. Skorina Aene, 0080 Minsk elars -mail: The non-inariance of the Farada indction law, reealed in [1] throgh calclation of an e.m.f. along a mathematical line, is frther analed for integration oer a condcting closed circit. The principal difference of a condctor from a mathematical line is the appearance of internal electromagnetic fields indced b rearranged condction electrons. In or analsis we distingish two general cases: 1- the internal electromagnetic fields from the condction electrons contribte an indced e.m.f.; - the internal fields do not gie sch a contribtion. Case makes a condcting circit similar to a mathematical line, where the Farada law is alwas correct, while the instein relatiit principle is iolated. Howeer, in sch a case the iolation of special relatiit occrs not for a hpothetical model problem, bt in phsical realit. Kewords: Farada indction law, transformation of electromagnetic field, sccessie Lorent transformations 003 C. Ro Kes Inc.

2 1. Introdction Apeiron, ol. 10, No., April It has been shown in ref. [1] that the mathematical epression for the Farada indction law d ε ds, (1) dt does not follow from the Mawell s eqation dl ds, () t in particlar, de to the ineqalit d ds dt S Γ S S d dt S ds for SS(t), ΓΓ(t) (here Γ is the closed line enclosing the area S, and dl is the element of the circit Γ). In this connection the epression (1) was tested in [1] for its Lorent-inariance. It has been fond that the Farada law is not inariant at least at a formal mathematical leel, when e.m.f. is calclated throgh integration oer a mathematical line in space. For integration oer a condctor we hae additionall to take into accont the effect of rearrangement of the condction electrons nder a presence of eternal electromagnetic fields. The rearranged electrons create their own electric and magnetic fields, which can be negligible otside the condctor, bt significant in its inner olme. It is clear that an inflence of these fields cannot be analed in a general form de to their dependence on man factors (geometr of condctors, configration of eternal fields, etc.). Neertheless, in frther consideration we will distingish two different general cases: (3) 003 C. Ro Kes Inc.

3 Apeiron, ol. 10, No., April the electromagnetic fields, being created b sch rearranged electrons, contribte the force in integrand of q. (4): ε ( r, t) + ( r ) ( r, t) dl, (4) ( ) Γ( t ) which represents a general definition of e.m.f. []; the electromagnetic fields from rearranged electrons gie a negligible contribtion to the total force in q. (4). These cases are consectiel analed in Sections and 3. The case is especiall interesting, becase it is similar to integration oer a mathematical line, where a iolation of the instein relatiit principle has been fond [1]. Howeer, on the contrar to a hpothetical model problem of [1], here we seek a contradiction of the Farada law and the special relatiit for phsical realit.. The Farada indction law: the internal electromagnetic fields of condctor contribte an e.m.f. in a circit This case is simpl realied nder sbstittion of the closed mathematical line in ref. [1] b a condcting rectanglar loop A--C- D to be placed inside the charged condenser FC (Fig. 1). Then we ma imagine that the side A is electricall connected with the sides U-C and U1-D b means of sliding contacts. First calclate e.m.f. in the loop A--C-D for a laborator obserer (inertial frame K). Condction electrons in resting parts of the loop C, CD and DA are rearranged b sch a wa, so that to gie a resltant anishing electric field R inside the condctor: + 0, R et int 003 C. Ro Kes Inc.

4 Apeiron, ol. 10, No., April U FC C _ K U1 A D + K 0 Fig. 1. Condctie loop U-C-D-U1 is short-circited b a moing condctie bridge A inside the charged flat condenser FC. is the oltmeter. where et is the field of eternal sorce (FC), and int stands for the field, created b re-distribted condction electrons. Since the eternal magnetic field is absent in the laborator frame K, that we obtain R, R 0 inside the segments C, CD and DA. De to a homogeneit of field transformations, the same eqalit r ', r ' 0 for the segments C, CD and DA remains alid for an other inertial obserer, and these segments do not contribte an e.m.f. in an inertial frame. In the moing bridge A the rearranged condction electrons indce the electric field ( int ) γ ( γ 1 1 c ), as well as the magnetic field along the ais ( ) γ c int, which preents frther rearrangement of the condction electrons, in order to reach R 0. Hence, a resltant force, acting along the ais per nit condction electron inside of A, is 003 C. Ro Kes Inc.

5 F R Apeiron, ol. 10, No., April ( 1 γ ) γ + γ c 1. From there the e.m.f. in the circit A--C-D is ( 1 1 ) l c ε, (5) 0 l γ where l is the length of A (hereinafter we adopt the accrac of calclations to the order c - ). The magnetic fl across the area ACD is defined b weak magnetic field, created b moing rearranged condction electrons in A otside this segment. If the distance between the segments A and CD is large, that the magnetic fl across the area ACD does not change with time nder motion of A. Hence, the Farada indction law is not correct in the frame K, becase of non-anishing e.m.f. in q. (5). The reslt clearl indicates that, in general, the internal electromagnetic fields contribte an e.m.f., and this effect is dropped in the Farada law. Howeer, the reealed deflection from the Farada indction law is impractical, becase een nder 300 m/s (speed of sond), l 10 4 (potential 9 difference between the plates of FC), ε , that is a negligible ale. Now let s compte the e.m.f. in an inertial frame K 0, wherein the frame K moes at the constant elocit along the ais (Fig. 1). In this case for the segments C, CD and DA, as we mentioned aboe, ', ' 0, and the can be eclded from frther integration. The electric and magnetic fields inside the segment A can be fond ia the field transformations from K to K 0 R [ R ] R [ c ] ( ) γ ( ) + ( ) ( ' ) ( ) ( ) R R + taking into accont that in K ', γ, ( γ 1 1 c ) (6) R 003 C. Ro Kes Inc.

6 Apeiron, ol. 10, No., April ( ) γ, ( ) γ c R Sbstitting qs. (7) into qs. (6), one gets: ( ' ) γ γ γ γ γ c R 003 C. Ro Kes Inc. R,. (7) ( ' ) γ γ c + γ c γ γ c R The segment A moes in the frame K 0 at the constant elocit + '. (8) 1+ c Hence, the e.m.f. in the loop A--C-D is eqal to ε l γ [( ' ) ' ( ' ) ] l r r ( 1 ' c ) + γ γ l( ' ) c γ γ l( 1 ' c ) Using the ale of ' from q. (8), we derie 1 1 ' c, ' γ 1 c γ ( ) +. ( c ) 1+ Sbstitting these eqalities into q. (9), we obtain ( 1 1 γ ) l l c ε γ 0 ( 1+ c ) ε. (9). (10) Ths, the e.m.f. in the frames K and K 0 is the same to the adopted accrac of calclations, that is in agreement with relatiistic transformation of e.m.f. []. Simltaneosl we notice that in the frame K 0 the magnetic fl across the area ACD, as for mathematical line, is eqal to d dt Φ c 1 c l l c ( 1+ c )

7 K 0 Apeiron, ol. 10, No., April (K ) A L C D (K 1 ) Fig.. The inertial frame K 1 is attached to the sqare condctie loop, while the inertial frame K is attached to the flat condenser. The pper lead A of the loop lies inside the condenser. The profile leads of loop pass across the tin holes C and D in the lower plate of condenser. (see, q. (18) of Ref. [1]), which being taken with the opposite sign, differs from q. (10). Ths, we reeal that the Farada indction law is incorrect in K 0. Phsical reason for sch a iolation of this law, as mentioned aboe, is a contribtion of the internal electromagnetic fields of condctor to the indced e.m.f. 3. The Farada indction law: the internal electromagnetic fields of condctor do not inflence an e.m.f. in a circit In this section we consider a phsical problem as follows. Let there is a condcting rectanglar loop with the elongated segment A inside a flat charged condenser FC (Fig. ). The thin ertical wires of the loop enter into the condenser ia the tin holes C and D in its lower plate, so that a distortion of electric field inside the condenser is negligible. An inertial frame K 1 is attached to the loop, while an inertial frame K is attached the FC. There is some eternal inertial reference frame K 0, wherein the frame K 1 moes at the constant elocit along the ais, and the frame K moes at the constant 003 C. Ro Kes Inc.

8 Apeiron, ol. 10, No., April {, ) K K 0 K 1 Fig. 3. The motion diagram of inertial reference frames K 1 and K in the third inertial frame K 0. elocit {, } in the -plane (Fig. 3). For sch a motion diagram, the frame K moes onl along the ais of K 1. One reqires to find an e.m.f. in the loop (indication of the oltmeter ). One can see that the internal electric field, being indced b redistribted condction electrons in the presence of electric field of FC, do not inflence the integral (4) along the ais (segment A). esides, the elocit of this segment in K 0 is parallel to its ais A-, and an internal (or eternal) magnetic fields do not create a force along this segment. The magnetic forces, being indced b the internal magnetic fields in the sides AC and D, compensate each other de to eqal elocit of these sides in an inertial frame. (Strongl speaking, sch a compensation is tre to the adopted order of approimation c - ). Hence, an e.m.f. in the circit is fll determined b the eternal electromagnetic fields of moing condenser, that makes the loop similar to a mathematical line. At the 003 C. Ro Kes Inc.

9 Apeiron, ol. 10, No., April same time, we hae alread proed in [1] that throgh integration oer a mathematical line, the Farada indction law is correct and the instein relatiit principle is iolated. In this connection the problem nder consideration looks non-triiall and especiall interesting, becase a magnetic fl and its time deriatie hae a propert to eist/disappear for different obserers (see below). Hence, according to the Farada law, an e.m.f. in the loop shold also eist or disappear in different inertial frames, that means a contradiction with the instein relatiit principle. Or remaining problem is to demonstrate this conclsion b concrete calclations. Let s determine the electric and magnetic fields in the frame K 0. We take into accont that in the frame of FC (K ) 0, 0,, where is the electric field in space region between the plates of FC. In intermediate calclations, we introdce into consideration the inertial frames K r, K 0r, whose aes are parallel to. Then in K r 0, sin α, cosα, 0, where α is the angle of with the ais of K 0. Sbstitting these ales into the field transformation from K r to K 0r, ' ' ; ' γ ; ' γ ( γ 1 1 c ), we get ( + ); ' γ ( ); [ ( c ) ]; ' γ + ( c ) 0 r sin α, 0 r 0; 0r γ cosα, 0 r 0; [ ], (11) 003 C. Ro Kes Inc.

10 Apeiron, ol. 10, No., April , γ ( c ) cosα 0 r 0 0 r. (1) Then the electric and magnetic fields in the frame K 0 are (to the order of approimation c - ): cosα sin α 0 0r sin α cosα γ sin α cosα c 0 sin 0r α + γ cosα sin α sin α + cos 0r ; c 0r α cosα (13a) (13b) + cos α + ; c c 0 0; (13c) 0 0r cosα 0r sin α 0; (13d) 0 0r sin α + 0r cosα 0; (13e) 0 0r γ ( c ) cosα. (13f) c Ths, in K 0 the magnetic field inside the condenser is not eqal to ero, and its non-anishing -component is defined b q. (13f). Simltaneosl one can see that nder motion of FC at the elocit {, }, and motion of loop at the elocit along the ais, the area ADC between the lower plate of FC and pper line of loop (the gra area in Fig. 4, where the magnetic field 0 eists) decreases with time. Therefore, in the frame K 0 the total time deriatie of magnetic fl across the area ACD decreases with time, too. One can eas find that this time deriatie is eqal to 003 C. Ro Kes Inc.

11 Apeiron, ol. 10, No., April (K ) _ A 0 K 0 C (K 1 ) D + Fig. 4. Obserer in the frame K 0 sees that nder motion of the frames K 1 and K, the gra area ADC decreases with time and hence, the magnetic fl across the condcting loop also decreases. dφ ds ADC 0 L, (14) dt dt c where L is the length of the side A. (In the adopted accrac of calclations a contraction of this length in K 0 is not significant). Hence, the Farada indction law reqires the appearance of e.m.f. in the loop. Under calclation of e.m.f. we assme that the electric and magnetic fields below the lower plate of FC are negligible. Then we ma write the conter-clockwise integral (4) as + dl Γ ( ) ( + 0 ) d + d + ( 0 ) D A AC. (15) d 003 C. Ro Kes Inc.

12 Apeiron, ol. 10, No., April c ϕ ϕ a) Condenser A L ϕ ϕ C ϕl D K 0 b) Fig. 5: a the ais of the frame K constittes the angle ϕ with the ais of K 0 de to the scale contraction effect in the frame K 0 ; b de to this effect, an obserer in the frame K 0 fies that the plates of condenser constitte the angle ϕ with the ais (and with the line A). Here we take into accont that for the ector to be parallel to the 003 C. Ro Kes Inc.

13 ais, the integral ( ) (13a), we get: Then Apeiron, ol. 10, No., April dl A is eqal to ero. Frther, sing q. d L. (16) c A ε L + ( + 0 )( D AC) c. (17) We notice that the segments D and AC are not eqal to each other at an fied time moment (chosen as integration moment) of the frame K 0. The reason is that the ais of K is not parallel to the ais of K 0 de to the scale contraction effect. Indeed, the projection of ais of K onto the direction to be collinear to the ector ( ) contracts b γ times, while the projection of this ais onto the direction to be orthogonal to the ector ( ), remains nchanged (see, Fig. 5,a). As a reslt, the ais of K is trned ot with respect to the ais of K 0, and the trn angle can be eas calclated: ϕ c. Ths, the moing condenser in the frame K 0 trns ot at the negatie angle ϕ, as depicted in Fig. 5,b. As a reslt, the length of segment D is longer than the length of segment AC b the ale ϕ L L c. Hence, integrating oer the loop we obtain ε L + 0 c L + + c c c ( + )( D AC) L L c 003 C. Ro Kes Inc. (18)

14 Apeiron, ol. 10, No., April (here we sbstitte the magnetic field 0 from q. (13f)). Comparing qs. (14), (18), we find that in the frame K 0 ε dφ dt. As we epected, the Farada indction law is correct for the considered circit. We pa attention to the fact that the ector is not orthogonal to the srface of (condcting) plates of FC. Indeed, one can see from Fig. 5 (red fragments) that the angle of electric field with the normal to the plates of FC for an obserer in K 0 is ϕ' ϕ + c ϕ. This reslt is natral in K 0, becase the condction electrons on the internal srfaces of FC are sbject to an action of magnetic force, and its component onto the srfaces is eqal to 0 cosϕ /c. Hence, an eqilibrim state of condction electrons is onl possible nder non-ero projection of the electric field to these srfaces, which is eqal to -/c -ϕ. It is jst the case of Fig. 5,b. Frther, let s write a transformation from K 0 to K 1 : ; ; 1 1 γ γ ( 0 0) ; 1 γ ( ); ( + ( c ) ); γ ( c ) 0 0 Sbstitting qs. (13) into qs. (19), one gets: 1 (19) ( ). 1 (0a) c γ + 1 γ 1, (0b) c c c C. Ro Kes Inc.

15 Apeiron, ol. 10, No., April , (0c) ( ) ( ) 0 γ c c +. (0d) c Ths, the magnetic field is the frame K 1 disappears, while the electric field has a non-ero projection onto the ais (q. (0a)). A trn of the ector at the angle ϕ c has a simple phsical meaning in special relatiit, if we take into accont the Thomas-Wigner rotation of the aes of K 1 and K frames for the motion diagram in Fig. 3. The angle of this rotation is [3] Ω c ϕ. It means that the ector is orthogonal to the ais of K, and an obserer in K 1 frame sees a simple space trn of FC, as depicted in Fig. 6. At the same time, as known in electrostatics, an trn of a charged condenser does not indce an e.m.f. in a closed loop passing throgh the condenser. Ths, we hae fond that in the frame K 0 an e.m.f. in the loop eists, while in the frame K 1 e.m.f. disappears. It occrs in a fll accordance with the Farada indction law: in the frame K 0 the magnetic fl across the area ACD eists and changes with time, while in the frame K 1 the magnetic fl disappears. Howeer, a presence of e.m.f. in the frame K 0, and its absence in the frame K 1 obiosl contradict to the instein relatiit principle. In another words, a conception abot eqialence of all inertial reference frames comes into a deep contradiction with casalit: a crrent in the loop A--C-D cannot eist in one inertial frame and be absent in another inertial frame. As a reslt, we hae to recognie that the Farada indction law, discoered man decades before creation of relatiit and being non-inariant in its natre, alread disproed this theor. 003 C. Ro Kes Inc.

16 Apeiron, ol. 10, No., April Condenser (K ) A L C Ω ΩL D K 1 Fig. 6. De to the Thomas-Wigner rotation between the frames K 1 and K, an obserer in K 1 sees the space trn of condenser at the angle Ω-ϕ. Net problem is to eplain the non-inariance of the Farada indction law in the ether theories, adopting an eistence of an absolte space. Consistent analsis of this problem will be done in a separate paper. 4. Conclsions The non-inariance of the Farada indction law with respect to field transformations in special relatiit, reealed earlier for e.m.f. along a mathematical line, was frther analed throgh integration oer a condcting closed circit. The principal difference of a condctor from a mathematical line is the appearance of internal electromagnetic fields indced b rearranged condction electrons. In the case where sch internal fields contribte an e.m.f. in a condcting circit, the Farada indction law is iolated, while the instein relatiit principle remains alid. In these conditions it is 003 C. Ro Kes Inc.

17 Apeiron, ol. 10, No., April especiall interesting to anale condcting circits in which the rearranged condction electrons do not contribte an e.m.f. In this case the circit becomes similar to a mathematical line, where the Farada indction law is tre, while the instein relatiit principle is iolated. A phsical problem of jst this kind has been fond, and actal iolation of relatiit has been confirmed. We stress that the latter reslt does not reeal an mathematical imperfection of the relatiit theor. It reflects a simple fact that the empiricall discoered Farada indction law is not Lorentinariant. As a reslt, we conclde that the special theor of relatiit, in its application to electromagnetism, was disproed b Farada as long as seeral decades before its creation. Acknowledgements The athor warml thanks Walter Potel, Thomas. Phipps, George Galecki, Oleg Misseitch and ladimir Onoochin and ictor dokimo for helpfl discssions. References [1] A.L. Kholmetskii, On the non-inariance of the Farada law of indction, Apeiron, 10, (003) []. G. Cllwick, lectromagnetism and Relatiit, London, Longmans, Green & Co. (1957). [3] C. Møller, Theor of Relatiit, Clarendon Press, Oford. (197). 003 C. Ro Kes Inc.