Force between two parallel current wires and Newton s. third law

Transcription

1 Foce between two paallel cuent wies and Newton s thid law Yannan Yang (Shanghai Jinjuan Infomation Science and Technology Co., Ltd.) Abstact: In this pape, the essence of the inteaction between two paallel cuent wies is analyzed. Duing the analysis we found that if the inteaction is oiginated fom the esultant Loentz foce of the chage caies in the cuents, the Newton s thid law will be violated in this system. Howeve, if the essence of the inteaction of the two cuent wies is consideed as the attaction o epulsion of the two cicula magnetic fields ceated by each cuent, the touble of violating Newton s thid law can be avoided. Intoduction When a cuent-caying wie is placed in an extenal magnetic field B, the wie will expeience a foce exeted by the magnetic field. It is widely believed that the physical oigin of this foce is fom the esultant Loentz foce of the chage caies in the cuent. The inteaction of attaction o epulsion between two cuent-caying wies is thought coming fom this foce. Each wie geneates a magnetic field, and the othe wie expeiences a magnetic foce as a consequence. Based on this, it is deived that the foce between two paallel cuent-caying wies can be expessed as, F = µ 0I 1 I 2 L (1) Whee, I 1 and I 2 ae the cuent intensities, the distance between the two wies, L the length of the wie, and µ 0 the pemeability of vacuum [1, 2, 3, 4]. It is called Ampèe's foce law. When deiving fomula (1), it is also thought by default: a) the two wies ae identical in geomety and mateial; b) the two wies ae identical in envionment, such as they ae both in vacuum o ai, etc. So, the magnitude of the foce deived fo each wie is equal. Howeve, if the two wies ae made fom diffeent mateials o they ae in diffeent envionments, like, one of them is coveed with magnetic shielding mateial, will the deived foce fo the two wies be equal o not? In the following thee cases, we have the two paallel wies made fom diffeent mateials with diffeent pemeability o they ae in diffeent envionments. The analysis shows that if we think the essence of the inteaction between the two cuent wies is the esultant Loentz foce of chage caies in the cuent, the foces fo the two wies ae not same in magnitude. Obviously, it conflicts with the Newton s thid law. Howeve, if the essence of the inteaction is consideed as the foce between the two magnetic fields poduced by the two cuents, the touble against the Newton s thid law can be avoided. Cases and analysis Case1. Two wies ae made fom diffeent mateials with diffeent pemeability When we put a mateial into an applied unifom magnetic field, the total field inside the object would then be vaies with its pemeability. Fo the two paallel wies with the same

2 electic cuent, accoding to Ampèe's cicuital law the same concentic cicles magnetic field is poduced aound each wie. Because of the diffeent pemeability of the two wies, the total magnetic field stength in each wie induced by the opposite magnetic field is diffeent. If the essence of the foce between the two cuent wies is the esultant Loentz foce of chage caies, the two wies will feel diffeent foce in the field poduce by the othe cuent wie. This is obviously against the Newton s thid law. In figue1, thee ae two paallel wies having diffeent pemeability, µ 1, µ 2. The electic cuents of the two wies ae I 1 and I 2. Accoding to Ampèe's foce law, Fig.1. Two paallel wies, with diffeent pemeability, µ 1, µ 2, have cuent I 1 and I 2. The foce exeted on wie1 is F 1 = I 1 LB 2, whee L is the length of the wie1. B 2 = µ 0µ 1 I 2 (2) F 1 = µ 0µ 1 I 1 I 2 L (3) The foce exeted on wie2 is F 2 = I 2 LB 1, B 1 = µ 0µ 2 I 1 (4) F 2 = µ 0µ 2 I 1 I 2 L (5) Because µ 1 µ 2, we get F 1 F 2. Case2. One of the two paallel wies is wapped up with a magnetic shielding tube As shown in figue2, thee ae two paallel wies with identical geomety and made fom the same mateial. But, one of them is wapped up with a tube made of magnetic shielding mateial and the othe not. If the two cuents ae in same diection and without this tube, fom the Ampee Foce Law we get that the two wies pull togethe and the foce on each wie is same in magnitude. F = µ 0I 1 I 2 L (6)

3 Now the wie2 is wapped up in a magnetic shielding tube which keeps the magnetic field poduced by cuent1 fee fom the space inside the tube, so the wie2 does not feel the magnetic field poduced by cuent1. Howeve, the wie1 feels the magnetic field poduced by cuent2. As a esult, wie1 feels a foce fom the magnetic field poduced by cuent2, but wie2 feels no foce fom the magnetic field poduced by cuent1. This does not accod with the Newton s thid law. Fig.2. Two paallel wies with identical geomety and made fom the same mateial. The wie2 is wapped up with a tube made of magnetic shielding mateial. Electic cuents I 1 and I 2 flow though the two wies. Case3. One of the two paallel wies is wapped up with a supeconducting tube Hee, the situation is quite simila to that in case2. Because of pefect diamagnetism of the supeconducting tube, the magnetic field poduced by cuent1 cannot each to wie2. Howeve, the magnetic field poduced by cuent2 can each to wie1. So, accoding to Ampee s foce law, wie1 feels a foce fom the magnetic field poduced by cuent2, but wie2 feels no foce fom the magnetic field poduced by cuent1. The esult conflicts with the Newton s thid law, again. Discussion Up to now, a question may be asked that if any foce on the tube is neglected duing the deivation above and just this esult in the touble of conflicting with Newton s thid law? At fist, let s see the case2. Suppose thee is a foce between the cuent and the magnetic shielding tube. Fom the aspect of symmety, thee should not be net inteaction between the tube and the wie2. If any inteaction, it may only exist between the tube and cuent wie1. The foce may be attactive o epulsive. Howeve, physically, the foce being attacting o epelling will have nothing to do with the diection of the cuent1, so the poblem against the Newton s thid law is still not be esolved. Fom the same logical analysis, the touble of conflicting with Newton s thid law in case3 is not esolved, eithe. By the analytical agument fo the thee cases above, we see that the conflict with the Newton s thid law is unavoidable if the Loentz foce is consideed as the essence of the inteaction between two paallel cuent wies.

4 Howeve, if the foce between the two cuent wies is egaded as the inteaction of the two magnetic fields poduced by each cuent athe than the Loentz foce, the touble violating Newton s thid law can be avoided. A cuent wie could be egaded as a kind of ba magnet. Unlike the nomal one, this ba magnet has no poles, but a cicula field aound the wie. The inteaction of the two cuent wies is vey simila to the inteaction of two paallel line chages. The foce between two paallel line chages is expessed as, F = 2λ 1λ 2 L (7) Whee λ 1 and λ 2 ae the linea chage density, the distance between the two linea chages, L the length of the linea chage [5, 6]. If we egad I 1 I 2 as the measue of stength of the magnetic field souce of the cuent wie ba magnet, fomula (1) will almost be the same as fomula (7). In fomula (1), if we think that it is not visual and intuitionistic enough to use I to indicate a magnet s stength, we can change fomula (1) into anothe fom, in which the I is eplaced with the stength of the cicula magnetic field of the cuent wie. Fo two paallel wies with cuent I 1 and I 2, and adius R 1, R 2, accoding to Ampèe's cicuital law, the cicula magnetic field ceated by each cuent wie is, B 1 () = µ 0I 1 B 2 () = µ 0I 2 (8) (9) Whee, is the pependicula distance to the wie. Fo each wie, the cicula magnetic field on the suface of the wie poduced by the cuent itself is, B 1 (R 1 ) = µ 0I 1 2πR 1 (10) B 2 (R 2 ) = µ 0I 2 2πR 2 (11) In fomula (8) and (9), we time R 1 R 1 and R 2 R 2 then we get, B 1 () = µ 0I 1 = B 1(R 1 ) R 1 (12) I 1 = 2πR 1B 1 (R 1 ) µ 0 (13) B 2 () = µ 0I 2 = B 2(R 2 ) R 2 (14) I 2 = 2πR 2B 2 (R 2 ) µ 0 (15)

5 The foce exeted on each wie is, F 1 = F 2 = I 1 B 2 L = I 2 B 1 L = 2πR 1R 2 B 1 (R 1 )B 2 (R 2 ) L (16) µ 0 Equation (16) is identical with equation (1). B 1 (R 1 ), B 2 (R 2 ) ae the magnetic field stength on the suface of the wies. R 1 B 1 (R 1 ), R 2 B 2 (R 2 ) can be egaded as the souce stength of the cuent wie ba magnets. So, the foce between the two cuent wie ba magnets is popotional to the poduct of the two ba magnets stength and invesely popotional with the distance between them. It is so simila to the inteaction of two linea chages. Conclusions 1. Fo the inteaction between two paallel cuent wies, if it is oiginated fom the esultant Loentz foce of the chage caies in the cuent, the Newton s thid law will be violated in this system. 2. If the foce between the two cuent wies is oiginated fom the inteaction of the two cicula magnetic fields ceated by each cuent, the touble of violating Newton s thid law can be avoided. Refeences: [1] Feynman, R. et al., Feynman lectues on physics volume 2, Mainly Electomagnetism and Matte, Chapte 13, Addison-Wesley,2011. [2] Raymond A Seway and John W. Jewett, J., Physics fo Scientists and Enginees with Moden Physics, Chapte 29, Ninth Edition, Technology Update, CENGAGE Leaning, [3] Raymond A Seway and John W. Jewett, J., Physics fo Scientists and Enginees with Moden Physics, Chapte 30, Ninth Edition, Technology Update, CENGAGE Leaning, [4] Pucell, Edwad M. and Moin, David J. Electicity and Magnetism, chapte 6, (Bekeley Physics Couse, Vol. 2), Cambidge Univesity Pess; 3 edition (Januay 21, 2013) [5] Pucell, Edwad M. and Moin, David J. Electicity and Magnetism, chapte 1, (Bekeley Physics Couse, Vol. 2), Cambidge Univesity Pess; 3 edition (Januay 21, 2013) [6] Feynman, R. et al., Feynman lectues on physics volume 2, Mainly Electomagnetism and Matte, Chapte 5, Addison-Wesley,2011.