London-Proca-Hirsch Equations for Electrodynamics of Superconductors on Cantor Sets

Transcription

1 London-Poca-Hisch Equaions fo Elecodynamics of Supeconducos on Cano Ses Vico Chisiano * Absac In a ecen pape published a Advances in High Enegy Physics (AHEP) jounal, Yang Zhao e al. deived Maxwell equaions on Cano ses fom he local facional veco calculus. I can be shown ha Maxwell equaions on Cano ses in a facal bounded domain give efficiency and accuacy fo descibing he facal elecic and magneic fields. Howeve, so fa hee is no deivaion of equaions fo elecodynamics of supeconduco on Cano ses. Theefoe, in his pape I pesen fo he fis ime a deivaion of London-Poca-Hisch equaions on Cano ses. The name of London-Poca-Hisch is poposed because he equaions wee based on modifying Poca and London-Hisch s heoy of elecodynamics of supeconduco. Consideing ha Poca equaions may be used o explain elecomagneic effecs in supeconduco, I sugges ha he poposed London-Poca-Hisch equaions on Cano ses can descibe elecomagneic of facal supeconducos. I is hoped ha his pape may simulae fuhe invesigaions and expeimens in paicula fo facal supeconduco. I may be expeced o have some impac o facal cosmology modeling oo. Key Wods: Cano ses, Hisch heoy, London equaions, local facional veco calculus, Maxwell equaions, Poca equaions, elecodynamics, supeconduco.. Inoducion Accoding o J.E. Hisch, fom he ouse of supeconduciviy eseach i was assumed ha no elecosaic fields could exis inside supeconducos and his assumpion was incopoaed ino convenional London elecodynamics.[] Hisch suggess ha hee ae difficulies wih he wo London equaions. To summaize, London s equaions ogehe wih Maxwell s equaions lead o unphysical pedicions.[] Hisch also popose a new model fo elecodynamics fo supeconducos. [][] The pesen pape is inended o be a follow-up pape of ou fou ecen papes: one pape eviews Shpenkov s inepeaion of classical wave equaion and is ole o explain peiodic able of elemens and ohe phenomena [], and he second one pesens a deivaion of * Coespondence: Vico Chisiano, vicochisiano@gmail.com. Independen eseache and adminisao of Phone: (6) URL: hp://

2 GavioElecoMagneic Poca equaions in facional space [], he hid one pesens an ouline of cosmology based on he concep of facal vibaing sing [3], and he fouh one pesens a deivaion of Poca equaions on Cano ses [4]. In his egad, in a ecen pape Yang Zhao e al. deived Maxwell equaions on Cano ses fom he local facional veco calculus.[3] I can be shown ha Maxwell equaions on Cano ses in a facal bounded domain give efficiency and accuacy fo descibing he facal elecic and magneic fields. Howeve, so fa hee is no deivaion of equaions fo elecodynamics of supeconduco on Cano ses. Theefoe, in his pape I pesen fo he fis ime a deivaion of London-Poca-Hisch equaions on Cano ses. The name of London-Poca- Hisch is poposed because he equaions wee based on modifying London equaions, Poca equaions and Hisch s heoy of elecodynamics of supeconduco. Theefoe he aim of he pesen pape is o popose a combined vesion of London- Poca-Hisch model fo elecodynamics of supeconduco. Then I exend fuhe his poposed model fo elecodynamics of supeconduco on Cano ses. Consideing ha Poca equaions may be used o explain elecomagneic effecs in supeconduco [4]-[8], I sugges ha he poposed London-Poca-Hisch equaions on Cano ses can descibe elecomagneic of facal supeconducos. I is hoped ha his pape may simulae fuhe invesigaions and expeimens in paicula fo facal supeconduco. I may be expeced o have some impac o facal cosmology modeling oo.. Hisch s model o evise London s equaions Accoding o J.E. Hisch, fom he ouse of supeconduciviy eseach i was assumed ha no elecosaic fields could exis inside supeconducos and his assumpion was incopoaed ino convenional London elecodynamics.[] Hisch suggess ha hee ae

3 difficulies wih he wo London equaions. Theefoe he concludes ha London s equaions ogehe wih Maxwell s equaions lead o unphysical pedicions.[] Howeve he sill uses fou- vecos J and A accoding o Maxwell s equaions: π A = J, () c 4 And c J J = ( A A ). () 0 0 4π Theefoe Hisch poposes a new fundamenal equaion fo elecodynamics fo supeconducos as follows: [] ( A A ) = ( A A ), (3a) 0 0 whee London peneaion deph λ L is defined as follows:[] 4π ne, λ = (3b) s L mc e And d Alembeian opeao is defined as: [] = c. (3c) Then he poposes he following equaions: [] ( F F ) = ( F F ), (4) 0 0 3

4 And ( J J ) = ( J J ), (5) 0 0 whee F is he usual elecomagneic field enso and F 0 is he field enso wih enies E v 0 and 0 fom E v and B v especively when expessed in he efeence fame a es wih espec o he ions. In he meanime, i is known ha Poca equaions can also be used o descibed elecodynamics of supeconducos, see [4]-[8]. The diffeence beween Poca and Maxwell equaions is ha Maxwell equaions and Lagangian ae based on he hypohesis ha he phoon has zeo mass, bu he Poca s Lagangian is obained by adding mass em o Maxwell s Lagangian.[7] Theefoe, he Poca equaion can be wien as follows:[7] μν 4π ν μf + mγ Aν = J, (6a) c whee m γ ω = is he invese of he Compon wavelengh associaed wih phoon mass. [7] In c ems of he veco poenials, equaion (6a) can be wien as [7]: 4π ( + mγ ) Aμ = Jμ. (6b) c Similaly, accoding o Kuglov [5] he Poca equaion fo a fee paicle pocessing he mass m can be wien as follows: + = (6) νϕμν( x) m ϕμ( x) 0, 4

5 Now, he similaiy beween equaions () and (6b) ae emakable wih excepion ha equaion () is in quadaic fom. Theefoe I popose o conside a modified fom of Hisch s model as follows: = (8a) ( mγ )( F F0) ( F F 0), And = (8b) ( mγ )( J J0) ( J J 0). The elevance of he poposed new equaions in lieu of (4)-(5) should be veified by expeimens wih supeconducos [6]. Fo convenience, he equaions (8a)-(8b) can be given a name: London-Poca-Hisch equaions. 3. A eview of pevious esul - Maxwell equaions on Cano ses I will no e-deive Maxwell equaions hee. Fo a good efeence on Maxwell equaions, see fo example Julian Schwinge e al. s book: Classical Elecodynamics [9]. Zhao e al. wee able o wie he local facional diffeenial foms of Maxwell equaions on Cano ses as follows [3, p.4-5]: - Gauss s law fo he facal elecic field: D = ρ, (9) D - Ampee s law in he facal magneic field: H = Ja +, (0) 5

6 - Faaday s law in he facal elecic field: B E =, () - magneic Gauss s law in he facal magneic field: B = 0, () and he coninuiy equaion can be defined as: ρ J =, (3) whee and ae defined as follows:.. In Cano coodinaes [0, p. ]: u u u u = div u = x x x3, (4) = u = u u + u u + u 3 3 u cul u e e e 3 x x3 x3 x x x.. In Cano-ype cylindical coodinaes [3, p.4]:. (5) R θ R z = + + +, R R θ R z (6) θ θ R z θ R R = er eθ e R θ z z R R R R θ z. (7) 3. London-Poca-Hisch Equaions on Cano Ses I can be shown ha Poca equaions can be deived fom fis pinciples [6], and also ha Poca equaions may have link wih Klein-Godon equaion [7]. Howeve, in his pape I will no aemp o e-deive Poca equaions. Insead, I will use Poca equaions as descibed in [6]. Then I will deive he London-Poca-Hisch equaions on Cano Ses, in accodance wih Zhao e al. s appoach as oulined above [3]. Accoding o Blackledge, Poca equaions can be wien as follows [7]: 6

7 ρ E = ε 0 κφ, (8) B = 0, (9) B E =, (0) E B = μ0j+ εμ κ A, () whee: A φ = E, () B = A, (3) mc 0 κ = h. (4) Theefoe, by using he definiions in equaions (4)-(7), we can aive a Poca equaions on Cano ses fom (8) hough (3), as follows: = ε ρ E κφ 0, (5) B = 0, (6) B E =, (7) E B = μ0ja + εμ κ A, (8) whee: A φ = E, (9) B = A (30) and Del opeao φ can be defined as follows [0, p.]: 7

8 φ φ φ = + + φ e e e 3 x x x3. (3) Since accoding o Blackledge, he Poca equaions can be viewed as a unified wavefield model of elecomagneic phenomena [7], heefoe we can also egad he Poca equaions on Cano ses as a fuhe genealizaion of Blackledge s unified wavefield model. Now, having defined Poca equaions on Cano Ses, we ae eady o wie down London-Poca-Hisch on Cano ses using he same definiion, as follows: κ F F = F F (3) ( )( 0) ( 0), And κ J J = J J (33) ( )( 0) ( 0), whee = c. (34) As fa as I know, he above London-Poca-Hisch equaions on Cano Ses have neve been pesened elsewhee befoe. Povided he above equaions can be veified wih expeimens, hey can be used o descibe elecodynamics of facal supeconducos on Cano ses. As a las noe, i seems ineesing o emak hee ha Kuglov [5] has deived a squaeoo of Poca equaions as a possible model fo hadon mass specum, heefoe pehaps equaions (3)-(34) may be facoized oo o find ou a model fo hadon masses (on Cano ses). Howeve, his poblem is lef fo ohe pape. 8

9 Concluding emaks In a ecen pape Yang Zhao e al. deived Maxwell equaions on Cano ses fom he local facional veco calculus. I can be shown ha Maxwell equaions on Cano ses in a facal bounded domain give efficiency and accuacy fo descibing he facal elecic and magneic fields. Howeve, so fa hee is no deivaion of equaions fo elecodynamics of supeconduco on Cano ses. Theefoe, in his pape I pesen fo he fis ime a deivaion of London-Poca-Hisch equaions on Cano ses. The name London-Poca-Hisch is poposed because he equaions wee based on modifying London equaions, Poca equaions and Hisch s heoy of elecodynamics of supeconduco. Theefoe he aim of he pesen pape is o popose a combined vesion of London- Poca-Hisch model fo elecodynamics of supeconduco. Then I exend fuhe his poposed model fo elecodynamics of facal supeconduco on Cano ses. Consideing ha Poca equaions may be used o explain elecodynamics in supeconduco, he poposed London- Poca-Hisch equaions on Cano ses may be able o descibe elecomagneic of facal supeconducos. I is hoped ha his pape may simulae fuhe invesigaions and expeimens in paicula fo facal supeconduco. I may be expeced o have some impac o facal cosmology modeling oo. Acknowledgmen: I d like o expess my sincee gaiude o D. Geoge Shpenkov fo sending his books and papes. And special hanks o Pof D. K. Raja Rama Gandhi fo publishing my wo ealie papes a IJMSEA and BMSA. Noneheless, he ideas pesened hee ae my sole esponsibiliy. 9

10 Refeences [] Hisch, J.E Elecodynamics of supeconducos. axiv:cond-ma/0369 [condma.s-el.] [] Hisch, J.E. 0. Coecing 00 yeas of misundesanding: elecic fields in supeconducos, hole supeconduciviy, and he Meissne effec. axiv: 0.85 [condma.sup-con] [3] Zhao, Y., Baleanu, D., Caani, C., Cheng, D-F., & Yang, X-J. 03. Maxwell s equaions on Cano Ses: A Local Facional Appoach. Advances in High Enegy Physics Vol. 03 Aicle ID 68637, hp://dx.doi.og/0.55/03/68637, hp://downloads.hindawi.com/jounals/ahep/03/68637.pdf [4] Tajma, M Elecodynamics in Supeconduco explained by Poca equaions. axiv:cond-ma/ [5] de Maos, C.J., &Tajma, M Gaviomagneic London Momen and he Gavion Mass inside a Supeconduco. axiv:g-qc/06059 [6] Gondan, Michel Poca equaions deived fom fis pinciples. axiv: [quan-ph], URL: hp://axiv.og/pdf/ pdf [7] Blackledge, Jonahan M An Appoach o Unificaion using a Linea Sysems Model fo he Popagaion of Boad-band Signals.ISAST Tansacion on Eleconics and Signal Pocessing, Vol., No., 007. URL: hp://eleceng.di.ie/papes/00.pdf [8] Demi, Suleyman. 03. Space-ime algeba fo he genealizaion of gaviaional field equaions. Pamana Vol. 80 No. 5 (Indian Academy of Sciences), May 03, URL: hp:// [9] Schwinge, Julian, DeRaad, J., Lese L., Milon, Kimball A., & Tsai, Wu-yang Classical Elecodynamics. Reading, Massachuses: Peseus Books. 59 p. [0] Yang, X-J., Baleanu, D., & Teneio Machado, J.A. 03. Sysems of Navie-Sokes on Cano Ses. Mahemaical Poblems in Engineeing Vol. 03, Aicle ID 76974, 0

11 hp://dx.doi.og/0.55/03/76974, URL: hp:// o URL: hp://ecipp.ipp.p/biseam/0400./3455//art_teneiomachado_03_dee.pdf [] Chisiano, Vico. 04. A Review of Schödinge Equaion and Classical Wave Equaion. Pespaceime Jounal, May 04. URL: Also available a: hp://vixa.og/abs/ [] Chisiano, Vico. 04. A deivaion of GavioElecoMagneic Poca equaions in facional space. Pespaceime Jounal, May 04. URL: [3] Chisiano, Vico. 04. An Ouline of Cosmology based on Inepeaion of The Johannine Pologue. Bull. Soc. Mah. Sevices and Sandads (BSOMASS), Sep. 04. URL: [4] Chisiano, Vico & Rahul, Biudugani. 04. A deivaion of Poca equaions on Cano Ses: A Local Facional Appoach. Bull. Mahemaical Sciences and Applicaions (BMSA), Nov. 04. URL: [5] Kuglov, S.I Squae oo of he Poca equaion: Spin-3/ field equaion. axiv:heph/ [6] Dessel, M. 03. Elecodynamics of Meallic Supeconducos. Adv. Cond. Ma. Phys. Aicle ID hp://dx.doi.og/0.55/03/04379 [7] Candemi, Nuay; Tanish, Mua; Ozdas, Kude and Demi, Suleyman Hypebolic Oconionic Poca-Maxwell equaions. Z. Naufosch. 63a, 5-8 (008). URL: hp:// Documen hisoy: Vesion.0: Dec 5 h, 04 VC, vicochisiano@gmail.com