A New Formulation of Electrodynamics

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1 . Eleromagnei Analysis & Appliaions doi:1.436/jemaa.1.86 Published Online Augus 1 hp:// A New Formulaion of Elerodynamis Arbab I. Arbab 1 Faisal A. Yassein 1 Deparmen of Physis Fauly of Siene Universiy of Kharoum Kharoum Sudan; Deparmen of Physis Fauly of Siene Alneelain Universiy Kharoum Sudan. arbab.ibrahim f.a.yassein@gmail.om aiarbab@uofk.edu Reeived Deember 7 h 9; revised uly nd 1; aeped uly 3 h 1 ASTRAT A new formulaion of eleromagneism based on linear differenial ommuaor brakes is developed. Mawell equaions are derived using hese ommuaor brakes from he veor poenial A he salar poenial φ he Lorenz gauge onneing hem. Wih he same formalism he oninuiy equaion is wrien in erms of hese new differenial ommuaor brakes. Keywords: Mahemaial Formulaion Mawell s Equaions 1. Inroduion Mawell equaions are firs order differenial equaions in spae ime. They are ompaible wih Lorenz ransformaion whih guaranees is appliabiliy o any inerial frame. A symmeri spae-ime formulaion of any heory will generally guaranee he universaliy of he heory. Wih his moivaion we adop a differenial ommuaor brake involving firs order spae ime derivaive operaors o formulae he Mawell equaions quanum mehanis. This is in addiion o our reen quaernioni formulaion of physial laws where we have shown ha many physial equaions are found o emerge from a unified view of physial variables [1]. In suh a formulaion we have found ha Mawell equaions emerge from a single equaion. Mawell equaions were originally wrien in erms of quaernions. They were iniially wrien in weny equaions [a]. However laer on Mawell equaions are hen wrien in erms of veor in he way ha we are familiar oday. In our presen formulaion Mawell equaions are desribed by a se of wo wave equaions represening he evoluion of he eleri magnei fields. This is insead of having four equaions. We aim in his paper o wrie down derive he physial equaions by vanishing differenial ommuaor brakes. We know ha seond order parial derivaives ommue for spae-spae variables. We don assume here his propery is a priori for spae ime. To guaranee his we eliminae he ime derivaive of a quaniy ha is aed by a spae derivaive followed by a ime derivaive vie-versa. In eping he differenial ommuaor brake we don ommue ime spae derivaive bu raher eliminae he ime derivaive by he spae derivaive vie versa. This differenial ommuaor brake may enlighen us o quanize hese physial quaniies. y employing he differenial ommuaor brakes of he veor A salar poenial φ we have derived Mawell equaions wihou invoking any a priori physial law. Mawell arrives a his heory of eleromagneism by ombing he auss Faraday Ampere laws. For mahemaial onsiseny he modified Ampere s law. He hen ame wih he known Mawell equaions.. Relaivisi Prelude From Lorenz ransformaions one obain v v y y z z. 1 We see ha he ommuaor brake. where we have aken ino aoun in he order of mulipliaion of he spae ime differenes. This shows ha he ommuaor is Lorenz invarian. This is a new invarian quaniy in relaiviy. We however already knew ha he square inerval is Lorenz invarian i.e. S S []. I follows from Equaion 1 ha he differenial ommuaor brake is Lorenz invarian oo i.e. '. We know ha he spaial seond order derivaives of a funion f f y is ommuaive i.e. opyrigh 1 SiRes. EMAA

2 A New Formulaion of Elerodynamis opyrigh 1 SiRes. EMAA 458 y f y f. We wonder if he ommuaions of spae ime derivaives are equally valid for all physial quaniies. Moivaed by his hypohesis we propose he following differenial ommuaor brakes o formulae he physial laws. In pariular we apply hese differenial ommuaor brakes in his work o derive he oninuiy equaion Mawell equaions. 3. Differenial ommuaors Algebra Define he hree linear differenial ommuaor brakes as follows:. 3 Equaion 3 is orre sine parial derivaives ommue i.e.. For a salar ψ a veor one defines he hree brakes as follows: I follows ha 7 8 F F F 9 for any veor F. The differenial ommuaor brakes above saisfy he disribuion rule A A A 1 where A are. I is eviden ha he differenial ommuaor brakes ideniies follow he same ordinary veor ideniies. We all he hree differenial ommuaor brakes in Equaion 3 he grad-ommuaor brake he do-ommuaor brake he ross-ommuaor brake respeively. The prime idea here is o replae he ime derivaive of a quaniy by he spae derivaive of anoher quaniy vie-versa so ha he ime derivaive of a quaniy is followed by a ime derivaive wih whih i ommues. We assume here ha spae ime derivaives don ommue. Wih his minimal assumpion we have shown here ha all physial laws are deermined by vanishing differenial ommuaor brake. 4. The oninuiy Equaion Using quaernioni algebra [3] we have reenly found ha generalized oninuiy equaions an be wrien as Now onsider he do-ommuaor of. 14 Using Equaions one obains For arbirary Equaion 15 yields he wo wave equaions Equaions show ha he harge urren densiy saisfy a wave equaion raveling a speed of ligh in vauum. I is remarkable o know ha hese wo equaions are already obained in [3]. Hene he urren- harge densiy wave equaions are equivalen o. 18

3 A New Formulaion of Elerodynamis Mawell s Equaions Mawell s equaions are firs order differenial equaions in spae ime of he eleromagnei field viz. E 19 1 E E 1. These equaions show ha harge urren densiies are he soures of he eleromagnei field. Differeniaing Equaion using Equaion 1 one obains 1 E. E 3 Similarly differeniaing Equaion 1 using Equaion one obains 1. 4 These wo equaions sae ha he eleromagnei field propagaes wih speed of ligh in wo ases: 1 harge urren free medium vauum i.e. or if he wo equaions 5 6 besides he familiar oninuiy equaion in Equaion 11 7 are saisfied. Equaion 3 4 resemble Einsein's general relaiviy equaion where spae-imes geomery is indued by he disribuion of maer presen. We see here ha he eleromagnei field is produed by any harge urren densiies disribuion in spae ime. Now define he eleromagnei veor F as i F E 8 Adding Equaion 5 Equaion 6 aording o Equaion 8 one obains 1 i i E E i. 9 Applying Equaions5 6 see [3] in Equaion 9 yields 1 F F 3 This is a wave equaion propagaing wih speed of ligh in vauum. Hene Mawell wave equaions an be wrien as a pure single wave equaion of an eleromagnei soureless omple veor field F. We all Equaions 5-7 he generalized oninuiy equaions. We have reenly obained hese generalized oninuiy equaions by adoping quaernioni formalism for fluid mehanis [3]. I is hallenging o hek wheher any real fluid saisfies hese equaions or no. We have reenly shown ha Shrodinger Dira Klein-ordon diffusion equaions are ompaible wih hese generalized oninuiy equaions [3]. Using Equaions 19 he eleri field do-ommuaor brake yields E E E. 31 This is he familiar oninuiy equaion. Hene he oninuiy equaion in he ommuaor brake form an be wrien as E. 3 Similar using Equaions 1 he magnei field do-ommuaor brake yields. 33 The eleri field ross-ommuaor brake gives E E E. 34 Using Equaions 1 his yields 1 E. 35 This equaion is nohing bu Equaion 4 above. Similarly he magnei field ross-ommuaor brake gives. 36 opyrigh 1 SiRes. EMAA

4 46 A New Formulaion of Elerodynamis Using Equaions 1 his yields 1 E. E 37 This equaion is nohing bu Equaion 3 above. Hene Equaions i.e. E. 38 represen he ombined Mawell equaions. In erms of he veor F defined in Equaion 33 he wave equaion in Equaion 35 an be wrien as F 39 whih is also eviden from Equaion Derivaion of Mawell Equaions from he Veor Salar Poenials A The eleri magnei fields are defined by he veor A he salar poenial as follows A E A. 4 These are relaed by he Lorenz gauge as 1 A. 41 omparing his equaion wih Equaion 11 reveals ha he oninuiy equaion is nohing bu a gauge ondiion. This means ha a new urren densiy ' an be found so ha he equaion is sill ina. We have reenly eplored suh a possibiliy whih showed ha i is rue [3]. Wih his moivaion he physialiy of he gauge A ehibied by Aharonov ohm effe is anamoun o ha of he urren densiy [5]. The grad-ommuaor brake of he salar poenial. 4 Using Equaions 4 41 one obains 1 A A. 43 This yields he wave equaion of he veor field A as 1 A A. 44 Similarly he do-ommuaor brake of he veor A A A A. 45 Using Equaions 4 41 one obains 1 A. 46 This yields he wave equaion of The ross-ommuaor brake of he salar poenial A A A. 48 Using Equaion 4 one finds A E. 49 This yields he Faraday s equaion E. 5 I is ineresing o arrive a his resul by using he definiion in Equaion 4 only. Now onsider he doommuaor brake of A A A A. 51 Using Equaions 4 41 he veor ideniies 5 Equaion 51 yields 1 E E A. 53 For arbirary A Equaion 53 yields he wo equaions E 54 1 E. 55 Equaions are he auss Ampere equaions. Similarly he ross-ommuaor brake of A A A A. 56 opyrigh 1 SiRes. EMAA

5 A New Formulaion of Elerodynamis 461 Using Equaions 4 41 he veor ideniy 57 Equaion 56 yields 1 E E A. 58 For arbirary A Equaion 58 yields he wo equaions E 59 1 E. 6 One again Equaions 59 6 are he Faraday Ampere equaions respeively. Hene he four Mawell equaions are ompleed. To sum up Mawell equaions are he ommuaor brakes A A. 7. Energy onservaion Equaion 61 In eleromagneism he energy onservaion equaion for eleromagnei field is wrien as u S E 6 where 1 1 E u E S. 63 The energy onservaion equaion of he eleromagnei field an be easily obain using he following veor ideniy F F F. 64 Le now E F so ha Equaion 64 beomes E E E. 65 Employing Equaions 1 63 Eq.65 yi elds u S E 66 whih is he familiar energy onservaion equaion of he eleromagnei field [5]. 8. onluding Remarks y inroduing hree vanishing linear differenial ommuaor brakes for salar veor fields A he Lorenz gauge onneing hem we have derived he Mawell s equaions he oninuiy equaion wihou resor o any oher physial equaion. Using differen veor ideniies we have found ha no any independen equaion an be generaed from he hree differenial ommuaors brakes. 9. Aknowledgemens Equaions are in he form of oupled wave equaions known as inhomogeneous Helmholz equaions. We see ha he urren densiy eners ino hese equaions in a relaively ompliaed way for his reason hese equaions are no readily soluble in general. This work is suppored by he universiy of Kharoum researh fund. We are graeful for his suppor. REFERENES [1] A. I. Arbab Z. Sai On he eneralized Mawell Equaions Their Prediion of ElerosalarWave Progress in Physis Vol. 9 pp [] W. Rindler Inroduion o Speial Relaiviy Oford Universiy Press USA [3] A. I. Arbab H. M. Widaallah The eneralized oninuiy Equaions. hp://ariv.org/abs/13.71 [4]. D. akson lassial Elerodynamis nd Ediion Wiley New York [5] Y. Aharonov D. ohm Signifiane of Eleromagnei Poenials in Quanum Theory Physial Review Vol. 115 No pp Appendi F F F F A1 F F F F A opyrigh 1 SiRes. EMAA