Khmelnik. I. Lrentz Fre, Ampere Fre and Mmentum Cnservatin Law Quantitative. Analysis and Crllaries. Abstrat It is knwn that Lrentz Fre and Ampere fre ntradits the Third Newtn Law, but it des nt ntradit the mre general Law f Mmentum Cnservatin, as the eletrmagneti field has a mmentum. Frm this it fllws that the Lrentz and Ampere fres must be balaned by the flw f eletrmagneti field mmentum. Hwever, as far as the authr knws, there is n rrespnding quantitative mparisn and therefre it is disussed belw. In partiular, it is shwn that sme f the rllaries f the Mmentum Cnservatin Law an be fund. Cntent 1. Intrdutin. The Field s Cnfiguratin 3. The Lrentz Fre 4. The Ampere Fre 5. Disussin Referenes 1. Intrdutin It is knwn that Ampere fre ntradits the Third Newtn Law, but it des nt ntradit the mre general Law f Mmentum Cnservatin, as the eletrmagneti field has a mmentum. It is imprtant t nte that a statinary eletrmagneti field an als have a mmentum and therefre the Ampere fre des nt ntradit law f nservatin f mmentum, als in the ase when it urs in njuntin with a permanent magneti field. Frm this it fllws that the Ampere fre must be balaned by the flw f eletrmagneti field mmentum. Hwever, as far as the authr knws, a quantitative mparisn f the Ampere fre with the flw f eletrmagneti field 1
mmentum des nt exist. Therefre this mparisn will be disussed belw. Here we shall als define sme parameters, and taking them int aunt we shall shw that the Lrentz fre and Ampere fre an be regarded as rllaries f the existene f eletrmagneti field mmentum and the law f mmentum nservatin.. The Field s Cnfiguratin Fr an eletrmagneti field let us dente: W - the energy density (salar), kg m -1 s, - the energy flw density (vetr), kg s 3, p - the mmentum density (salar), kg m s 1, f - the eletrmagneti field mmentum density (vetr), kg m - 1 s, - the eletrmagneti field vlume (salar), m 3. The fig. 1 shws ndutr arrying urrent I and length f L that is lated in a magneti field with indutin B and is mving at the speed v under the atin f the Ampere fre F. etrs f the intensity E f the urrent-reating eletri field, and f the indutin B - are mutually perpendiular. Therefre there appears a flw f eletrmagneti energy with a density, shwn in the fig. 1 by irles. It may be presented in the frm f tw spheres, united in the bdy f ndutr and penetrating the ndutr in the vertial diretin. This flw is equivalent t the flw f an eletrmagneti field mmentum f. F B I L Fig. 1. Fig. 1a learly shws several lines f urrent, indutin and flw. The"frest" f brwn lines f the flw begins at the intersetin pints
f the lines f urrent and the lines f indutin, as shwn by irles. The flw lines penetrate the bdy, pass ut f the bdy and are lsed as shwn by hrizntal arrws. On Fig. 1 these lsing lines are shwn by irles. F F B I Fig. 1а. It is knwn [1, ], that f W. (1) W, () p W, p, (3) f p, f. (4) The integral f the density by vlume will be dented as A A d. (4а) The energy flw may exist als in a statinary eletrmagneti field [3]. Therefre the mmentum flw f exists als in a statinary eletrmagneti field reated by diret urrent and permanent magneti field. The law f mmentum nservatin fr a devie interating with eletrmagneti field an be written in a fllwing frm [3]: J p f, (5) t t where J mehanial mmentum f the devie, 3
- the vlume f devie; vlume in whih the eletrmagneti field mmentum interats with the devie (the summary mmentum flw in all vlume f the field is equal t zer). It is knwn that the fre ating n the devie is F J. (6) t Cnsequently, F p f. (7) t Cmbining (7) and (3, 4), we get: F. (8) t Thus, if the devie is in the flw f eletrmagneti energy, then it is influened by a fre (8), depending nly n the flw f eletrmagneti energy. This fre exists als fr a permanent flw, and then F. (9) In this ase, if the flw f eletrmagneti energy eletrmagneti energy flux is distributed in the material with relative permittivity and permeability, then in the frmulas (8, 9) the light speed in vauum shuld be replaed by the light speed in material s (10) Let us nsider the ase (shwn n the fig. 1), when vetr f permittivity E and permeability H are perpendiular. Then EH (11) Let als the field in the devie is unifrm and is nentrated in the vlume. Then frm (8, 10, 11) we get EH EH F. (1) t If, besides that, the field is permanent, then EH F. (13) 4
3. The Lrentz Fre Let us nsider the magneti Lrentz fre, ating n a bdy with harge q, mving with speed v perpendiularly t the vetr f magneti indutivity B : F L qvb. (14) We shall neglet the intrinsi magneti indutin field f a mving harge (mpared with the indutin f an external magneti field) and its wn mmentum mving harge. Then we have t aept that the fre (14) is aused by the flw mmentum f eletrmagneti field that penetrates the bdy f the harge. Thus frm (13, 14), we btain: EH F L. (15) where is the bdy vlume. Frm this we get: EH qvb (16) r, fr B H, E / qv. (17) Cnsequently, inside the bdy there shuld be eletri field intensity direted alng the velity, and equal t qv E. (18) / Let us nte that and 377 (19) qv q E 377 v. (0) Cnsequently, inside a harged bdy, mving in a magneti field and being under the influene f Lrentz fre, there exists an intensity f eletri field prprtinal t the mvement speed. The example with an Eletrn 19 15 It has a harge q 1.6 10, lassial radius r.810, a vlume rrespnding t this radius, 4 3 r 45 3 9 10. Als 5
6 6 E 7 10 v. One may als say that n the diameter f the eletrn alng the speed diretin, there exists a ptentials 1 differene a vltage U Er 4 10 v. Cnsidering the arguments f Feynman [3] n the internal fres f the eletrn, restraining the eletrn harges n the surfae f the sphere, we an see that this vltage is the fre whih "pulls" lagging harges t their plae n the sphere when they mve under the atin f the Lrentz fre 4. The Ampere Fre Let us nsider the Ampere fre ating n a ndutr with urrent I, дmving with speed v perpendiularly t the vetr f magneti indutin B : F A IBL. (1) If this fre is aused by the flw f the mmentum f eletrmagneti field permeating the ndutr, then EH F A, () where is the ndutr s vlume. Frm this we find: EH IBL (3) r, fr B H, EH IHL. (4) Therefre, the intensity f eletri field in this ase will be IL E. (4а) / If s is the setin area, L - the ndutr length, then sl. (5) If the vltage n the ndutr is permanent and equal tu, then E U / L. (6) If the speifi resistane f the ndutr is equal t, then U IL s jl (7) and
Then r E j. (8) js j (9) s /. (30) Thus, the permittivity f the ndutr with urrent depends nly n and. Fr example, fr 6 1, 10 (m*m) we find 16 that 710. Fr verifiatin let us substitute (30) int () r int (13), we shall get EB FA EH. (31) and further, taking int aunt (8), we get (1). imilarly, substituting (30, 8) int (1), we get EH EH FA EH EH t t r L FA IB IBL. (3) t Hene, the Ampere fre must depend als n the speed f urrent hange r n magneti indutin. These hanges may be aused by urrent hange f by the hange f urrent psitin relative t the field. Pratially suh dependene an be deteted nly fr very high frequeny (due t the effiient 4 10 15 ). 5. Disussin Frm the abve said it fllws that the Ampere fre may be nsidered as a rllary f the existene f eletrmagneti field mmentum and f the mmentum nservatin law. But in this ase it shuld als be assumed that the permittivity f the urrent-arrying ndutr depends n and arding t (30). In this ase the dependene f the Ampere fre n the speed f urrent hange and/r magneti indutin an be revealed. 7
Cmbining (0) and (30), we find qv E. (33) r qv E. (34) Qualitatively, this effet an be explained by the fat that the free eletrns "lag" frm the bdy and aumulate in the "tail" f aelerating bdy - a phenmenn nsidered by Feynman fr aelerating eletrn [3]. Eletrial resistane f the material slws unifrm harge distributin. Fr this nsumes energy. Cnsequently, the mtin f the harged bdy at a nstant speed urs with the expenditure f energy fr thermal lsses. This ensures nstany f the energy f the eletri field inside a harged bdy. Thus, the the Lrentz fre an be regarded as a rllary f the existene f eletrmagneti field mmentum and f the mmentum nservatin law. But in this ase it shuld als be assumed that inside the harged mving bdy exists the intensity f eletri field f the frm (34), prprtinal t the mvement speed. Thus, the harged bdy mving with a ertain speed in a magneti field, turn ut t be in an eletrmagneti field with a: eletrmagneti energy flw, eletrmagneti field mmentum eletrmagneti field mmentum flw. Frm the law f mmentum nservatin it fllws that the time derivative f the mehanial mmentum f the bdy (i.e., the fre ating n the bdy) depends n: 1) the time derivative f eletrmagneti field mmentum, and ) the mmentum flw f eletrmagneti field This fre is exatly the Lrentz fre. Referenes 1. Landau L.D., Lifshitz E.M. Field thery.. Ivanv.K. General physis urse (in Russian), http://lms.physis.spbstu.ru/pluginfile.php/134/md_resure/ ntent/1/pt_1_03.pdf 3. R.P. Feynman, R.B. Leightn, M. ands. The Feynman Letures n Physis, vlume, 1964. 8