A new approach in classical electrodynamics to protect principle of causality

Transcription

1 A new appoach in classical electodynamics to potect pinciple of causality Biswaanjan Dikshit * Lase and Plasma Technology Division Bhabha Atomic Reseach Cente, Mumbai INDIA * Coesponding autho bdikshit73@yahoo.co.in Pepint of aticle published in Jounal of Theoetical Physics and Cyptogaphy, vol. 5 (Mach, 014) ABSTRACT In classical electodynamics, electomagnetic effects ae calculated fom solution of wave equation fomed by combination of fou Maxwell s equations. Howeve, along with etaded solution, this wave equation admits advanced solution in which case the effect happens befoe the cause. So, to peseve causality in natual events, the etaded solution is intentionally chosen and the advance pat is just ignoed. But, an equation o method cannot be called fundamental if it admits a wong esult (that violates pinciple of causality) in addition to the coect esult. Since it is the Maxwell s fom of equations that gives bith to this acausal advanced potential, we ewite these equations in a diffeent fom using the ecent theoy of eaction at a distance (Biswaanjan Dikshit, Physics essays, 4(1), 4-9, 011) so that the pocess of calculation does not geneate any advanced effects. Thus, the long-standing causality poblem in electodynamics is solved. Key wods: Advanced solution, Causality, Maxwell s equations, Wave equation, Action at a distance, Reaction at a distance 1

2 1. Intoduction: Electomagnetism is one of the oldest natual phenomena studied by moden science. Laws of electomagnetism gadually evolved fom the vaious expeimental obsevations. In this pocess, Coulomb s law was the fist one which established the 1/ dependence of foce between two chages. Then, Faaday discoveed the induction of voltage by changing magnetic field and Ampee quantified the magnetic field geneated by electic cuent. Finally, Maxwell intoduced the concept of displacement cuent (i.e. time vaying electic field giving ise to magnetic field) and wote all the laws of electomagnetism in an elegant fom which ae commonly known Maxwell s equations. These equations in diffeential fom ae given by, ρ E = (1) ε B = 0 () B E = (3) E B = µ J + µε (4) Eq. (1) known as Gauss law which is an altenative fom of Coulombs law, Eq. () and Eq. (3) (Faaday s law) ae all based on expeimental obsevations. Fist tem in the ight hand side of E Eq. (4) is due to Ampee s law and the second tem ( µε ) was added by Maxwell to save continuity equation in expeiments (such as chaging of a capacito). Thus, all these fou equations ae just mathematical epesentations of expeimental esults. Howeve, a scientific law is useful if it can accuately pedict a paticula vaiable fom the knowledge othe vaiables. To achieve this objective, thee was a need to devise a method to calculate electic and magnetic foce (o field) in an assembly of abitaily moving chages. Geneally in textbooks [1-3], this is done by taking diffeentials and foming the wave equations fo scala potential φ and vecto potential A given by, φ ρ φ µε = ε (5)

3 A A µε = µ J Although the wave equations Eq. (5) and (6) admit both etaded and advanced solutions fo φ and A, conventionally only the etaded solution is chosen and the advanced pat is just ignoed [1,3-4]. It is done because in the advanced solution, time uns backwad (i.e. t is negative) and thus the effect happens befoe the cause. This violates pinciple of causality and we have neve obseved any acausal event in the natue till date. Although one can ague in suppot of advanced potential fo its utility in calculation of adiation eaction foce in some appoaches such as action-at-a-distance [5], this adiation eaction and enegy loss fom an acceleating chage can also be deived without use of advanced potential by consideing self foce [6-9] ceated in the chaged paticle. Coulomb gauge is in no way bette than Loenz gauge as in this case also pinciple of causality fo the potential is violated [1-,10]. Electic and magnetic fields ae calculated by following equations. A E = φ (9) B = A (10) Although Maxwell s equations ae tue epesentations of expeimental esults, but an equation o law can be called fundamental only if thee exists a way to pedict only the coect answe and no wong answes in the pocess. To explain it, let us take an example. Let thee be a vaiable called x and suppose the tuth is that x=5. This tuth can be epesented by many equations such as x-10=0 o x =5 o dx/dt=0. Among these equations, equation x-10=0 can be simplified to get only one value of x i.e. 5, which is the coect answe. But x =5 gives a value of 5 o 5 fo x and dx/dt=0 pedicts that x is any constant including the tue value 5. Thus, although all the equations ae tue, equation x-10=0 is the fundamental equation as it pedicts only the coect answe and nothing else. Aguing along these lines, as thee is only one way that uses Maxwell s equations to calculate the electomagnetic effects and it geneates along with the coect answe (i.e. etaded solution) a pactically impossible answe (i.e. advanced solution), we cannot call the Maxwell s fom of equations as fundamental. Rathe this is just one of the many possible mathematical epesentations of the expeimental obsevations. (6) 3

4 So, it will be bette if we can find an altenative way to coectly pedict electomagnetic effect on any chage without simultaneously ceating any unusual esult (like advanced solution) duing the pocess of calculation. Thus, the causality in natual events can be potected. To achieve this objective, Jefimenko [11] has ecently assumed the final foms of equations fo the electic and magnetic fields as the fundamental equations and then he has deived the Maxwell equations fom these equations. But, as his method does not give any explanation o eason fo that paticula complex fom of expession fo electic and magnetic fields, it looks like that we ae being foced to conside a complex equation as fundamental just to avoid violation of causality. But in this pape, we popose a diffeent fom fo laws of electomagnetism which we can call fundamental as they ae shown to pedict expeimentally obseved effects without simultaneously ceating any unusual esult like advanced solution duing the pocess of calculation. Thus, the long-standing causality poblem in electodynamics is solved.. Fomulation of laws of electomagnetism in a diffeent fom: In ou fomulation, we will use only the established expeimental esults given by Coulomb, Ampee, Faaday, Maxwell and add no additional hypothesis duing fomulation. Ou aim is to coectly calculate the foce on a test chage Q, which is placed in any electomagnetic envionment. We ae not concened about whethe o not the electic field o magnetic field exists as a physical entity at paticula point in space. In othe wods, ou analysis will be a fieldless theoy diectly calculating the foce on a test chage due to position and state of othe chages distibuted in the univese. This is in ageement with the theoy of action-at-a-distance [5, 1] poposed by Feynman and theoy of eaction-at-a-distance [13-14] published by us ealie, which ule out the existence of electic field. Although we will use the symbol E fo convenience, we emphasize that the electic field E just epesents the foce expeienced by a unit test chage at a paticula point of space. But if test chage is not thee, E does not exist at that point. So, the poposed basic laws of electomagnetism coesponding to each expeimental obsevation ae given below. a) As the Coulomb s law states that foce between two chages is diectly popotional to the 1/ and the electic field is gadient of potential (fo electostatic case), we will take the potential to 4

5 E vay as, φ α q/. We also know that Maxwell added the tem µε in the Ampees law to save continuity equation in expeimental situations (such as chaging of a capacito). The µε ultimately manifested as 1/c in wave equation whee c is the speed of electomagnetic influence. But, in stead of binging the Maxwell s tem µε (o 1/ c ) into laws of electomagnetism, we can simply state that, Any chage can be electomagnetically influenced by anothe chage only if = ct is satisfied, whee is the distance of the acting chage fom affected chage and t is the time elapsed. This is exactly same as the pinciple of eaction-at-a-distance poposed in ou ealie publication [13-14] fo solving the poblems of invaiance of speed of light and wavepaticle duality. In theoy of eaction-at-a-distance, the obseve is affected by all the past and pesent events of the univese on its own (no caie equied), which is a fundamental popety of matte. Even if some matte is ceated just now (say by electon positon pai poduction), it gains the knowledge about all the events of the univese as soon as it is bon. Howeve, one can eact exactly afte a time delay, t=/c, which is the inheent chaacte of all matte. Thus, nothing (such as electomagnetic infomation) comes fom the souce; obseve only eacts to the souce based on its own absolute knowledge about the events that have occued in space-time. So, using this law, potential at any point 1 will depend upon distibution of chages in space at etaded time and mathematical expession fo scala potential φ will be, 1 ρ(, t 1 / c) φ ( 1, t) = dv( ) (11) 4πε vol Whee, 1 =Distance of the volume element dv at fom field point 1 1 ρ=chage density in the elemental volume dv( )at etaded time t- 1 /c c = Constant of popotionality elating distance and eaction time delay (o Speed of electomagnetic wave in conventional theoy) Above Eq. (11) fo scala potential is same as the etaded genealized solution fo φ deived by Feynman [3] fom wave equation. Then using Eq. (11) and the method of sweeping integal as given in [3] o method of eaction line [15], we can also calculate the electic potential at any point due to a moving point chage which is commonly known as Lienad-Wiechet potential. 5

6 b) Just as the Ampee s law gives the elation between the magnetic field and cuent, we state that the elation between magnetic vecto potential and cuent distibution is given by (afte incopoating the concept of eaction time delay as ealie), 1 J(, t 1 / c) A ( 1, t) = dv( ) 4πε c vol Whee, 1 =Distance of the volume element dv at fom field point 1 J =Cuent density in the elemental volume dv( )at etaded time t- 1 /c c) Faaday s law is accounted by incopoating the tem field. So, in place of E = φ, we take E as, A E = φ 1 (1) A in the expession fo electic t d) Net foce on Q is expeimentally found out be, F = Q[ E + v ( A)] (14) Substituting a new vaiable B fo ( A ), we get the conventional equations given by, F = Q[ E + v B)] and B = 0 Thus, we can calculate the foce on any chaged paticle due to othe chages by using Eq. (11) to Eq.(14). We note hee that the E and B above ae just intemediate mathematical tems to calculate the net foce on Q fom knowledge of potentials and they do not exist as physical entities in space in accodance with the theoy of eaction-at-a-distance [13-14]. 3. Conclusion: We have epesented the Maxwell s equations expessing the expeimental esults in a diffeent fom so that the poblem of advanced fields does not aise duing the pocess of calculation of electomagnetic effects (o foce) which othewise happens in the wave equation appoach. We have constucted this altenative fom of Maxwell s equations using the theoy of eaction-at-distance, which is a field-fee electomagnetic theoy. Thus, the long-standing causality poblem in electodynamics could be solved. (13) 6

7 REFERENCES: [1] David J. Giffiths, Intoduction to Electodynamics, (PHI Leaning Pivate Limited, New Delhi, 009), p 4, 45,41, 431 [] J D Jackson, Classical Electodynamics (Wiley, New Yok, 1975) p 1, 3 [3] Richad Philips Feynman, Robet B Leighton and Matthew Sands, The Feynman Lectues on Physics (vol. II), (Nasoa Publishing House, New Delhi, 199), p1-3, 0-14, 1-5, 1-10 [4] F Rohlich, Causality, the Coulomb field, and Newton s law of gavitation, Ameican Jounal of Physics, 70 (4), (00) [5] John Achibald Wheele and Richad Philips Feynman, Inteaction with the absobe as the mechanism of adiation, Reviews of Moden Physics, 17 (-3), (1945) [6] F. Rohlich, The self-foce and adiation eaction, Ameican Jounal of Physics, 68 (1), (000) [7] David J Giffiths, Thomas C Pocto and Daell F Schoete, Abaham-Loentz vesus Landau-Lifshitz, Ameican Jounal of Physics, 78 (4), (010) [8] Philip Peale, Classical electon models in book Electomagnetism Paths to Reseach, (Edited by Dois Teplitz, Plenum pess, New Yok, 198), p [9] Biswaanjan Dikshit, A new appoach in the deivation of the Lamo fomula fo adiation loss fom an acceleating chaged paticle, Canadian Jounal of Physics, 91(7), (013) [10] C W Gadine and P D Dummond, Causality in the Coulomb gauge: A diect poof, Physical eview A, 38 (9), (1988) [11] Oleg D Jefimenko, Pesenting electomagnetic theoy in accodance with the pinciple of causality, Euopean Jounal of Physics, 5, (004) [1] John Achibald Wheele and Richad Philips Feynman, Classical Electodynamics in tems of Diect Intepaticle Action, Reviews of Moden Physics, 1 (3), (1949) [13] Biswaanjan Dikshit, A new concept of light: Solving the poblems of invaiance of speed of light and wave-paticle duality, Physics Essays, 4(1), 4-9 (011) [14] Biswaanjan Dikshit, Deivation of Loentz tansfomation equations fom the new concept of eaction-at-a-distance, Physics Essays, 4(4), (011) [15] Biswaanjan Dikshit, Space-time diagam appoach in deivation of Lienad-Wiechet potential fo a moving point chage, Canadian Jounal of Physics, 91(7), (013) 7