Applie Mathematics,,, 5-9 oi:.436/am..4 Pblishe Online Febrary (http://www.scirp.org/jornal/am) Complementing the Lagrangian Density of the E. M. Fiel an the Srface Integral of the p- Vector Proct Abstract Mirwais Rashi Delft Uniersity of Technology, Delft, The Netherlans E-mail: mirwaisrashi@hotmail.com Receie October 3, ; reise December 5, ; accepte December 9, Consiering the Lagrangian ensity of the electromagnetic fiel, a 4 4 transformation matri is fon which can be se to incle two of the symmetrie Mawell s eqations as one of the Eler-Lagrange eqations of the complete Lagrangian ensity. The 4 4 transformation matri introces newly efine ector procts. In a Theorem the srface integral of one of the newly efine ector procts is shown to be rece to a line integral. Keywors: Electromagnetic Fiel, Parallel-Vertical Proct, Srface Integral. Introction In physics there are for eqations which are known as the Mawell s eqations. The two of these eqations are the Gass s law for the electric fiel an the Gass s law for the magnetic fiel an the other two are the Ampère s law an the Faraay s law []. In the theory for the massless electromagnetic ector fiel [] the Gass s law for the electric fiel an the Ampère s law are written in a sccinct form as Eqation () below an the Lagrangian ensity is fon for this Eqation (). Howeer, the Gass s law for the magnetic fiel an Faraay s law are omitte apparently becase of the fact that there are no magnetic charges etecte yet. In this article the Faraay s law an the Gass s law for the magnetic fiel are formlate as one tensor Eqation (8) an the Lagrangian ensity is written for Eqation (8). Een if there are no magnetic charges the Lagrangian ensity of the two omitte Mawell s eqations which contains the ifference between the potential energy ensity an the kinetic energy ensity of the electromagnetic fiel [] shol not be omitte since the Ampère s law alone oes not imply electromagnetic waes an it is the Ampère s law together with Faraay s law which among others implie the electromagnetic waes an there is therefore a nee to complement the eisting Lagrangian ensity []. Reference [] gies the Mawell s eqations in the integral form, howeer, sing the eisting mathematical theorems the Mawell s eqations can be written in ifferential form [3]. In this article the relation between the tensor of Eqation () an the tensor of Eqation (8) is fon as a 4 4 matri the components of which contain newly efine ector procts which are calle here as the parallel-ertical (abbreiate as the p-) ector proct an the parallel-horiontal ector proct. In fining the Lagrangian from the Lagrangian ensity one shol integrate oer the olme concerne. In this article the srface integral [4] of the parallelertical ector proct is shown to be recible to a line integral.. The Mawell s Eqations in Tensor Forms The for Mawell s eqations are written with tensor notations in more compact forms as the following two Eqations [] with an F F j () F F F () E E E3 E B3 B E B3 B E3 B B 3 (3) j, J, J, J (4) Copyright SciRes.
6 M. RASHID,,, (5) t y The Lagrangian ensity which leas to Eqation () as its Eler-Lagrange Eqation of motion [5], is the following F F j A (6) 4 with A being the ector potential with for components, the first component of which is the electrical po- tential, an the three other components of which are the components of the three imensional ector potential A, the crl of which is the magnetic fiel B. Howeer, Eqation (6) which is well known in the pblishe literatre oes not lea to Eqation () irectly as its Eler-Lagrange eqation of motion. To write the complete Lagrangian ensity of the Mawell s eqations withterms for eental magnetic monopoles, Eqation () is written in a moifie form as follows B B B3 B E3 E,,,, Y, Y, Y t y B E E 3 B E E 3 3 (7) Eqations () an () apparently seem to be less compact than the Mawell s eqations in the ifferential-form as presente in the article of the Jornal PIER, mentione here as reference [6]. Howeer, Eqation () can be mae to hae a compacter form than its presente form of Eqation () with the tensor notation, as its symmetrie form is shown as Eqation (5). Eqation () can be obtaine from Eqation (7) when the ector y, Y, Y, Y3 on the right han sie of Eqation (7) is set to be eqal to ero. Denoting the 4 4 ma- tri in Eqation (7), as K, one may write Eqation (7) in analogy with Eqation () as follows K y (8) One may presme, firstly, a transformation to eist which changes F into K as follows K T F (9) then a transformation matri of the following form is fon T BB EB EB EB 3 EB B EE EB EB 3 B EB EB EB EE EB 3 EB3 EB EB B3 EE 3 3 3 with two newly efine ector procts in the components of the matri of Eqation (), one of which is efine as follows i j k ef ef PQ P P P P P Q Q i Q Qy Q j y y y PP QQ kpp y QQ y () One may call this proct of two arbitrary ectors P an Q (haing three components) in Eqation (4), the parallel-horiontal proct. The secon newly efine ector proct is as follows i j k ef ef PQ P P P ipq PQ Q Qy Q j y y y PQ PQ kpq PQ y y () () this mltiplication of two ectors in Eqation () may be calle the parallel-ertical mltiplication. The singlarity (iision by ero) in the prefactor of the EB transformation matri () is holing in the free space [7] an the singlarity of the mentione prefactor is aoie in a space of charge ensities where the electric an magnetic fiels are not necessarily perpeniclar to each Copyright SciRes.
M. RASHID 7 other. To write the Lagrangian ensity for the Eqations (7) or (8) in a form analogos to Eqation (6) one nees to efine a new ector fiel Z sch that K Z Z (3) In this way one can see that the following eqations hol for Z Z Z, Z, Z, Z3 (4) 3 Z Z, Z, Z (5) Z E (6) Z B Z (7) t The form of Eqation (8) sggests easily the following Lagrangian ensity K K yz (8) 4 which wol gie the Eqation (8) as its eqation of motion throgh the following Eler-Lagrange eqation with the canonical coorinate being Z an its eriatie being Z (9) Z Z The total Lagrangian ensity of the electromagnetic fiel wol be the sm, the epressions of the ae terms of which can be taken, respectiely, from Eqation (6) an Eqation (8), the two eqations which hae analogos forms, bt haing their ifferently efine respectie ectors, namely, A an Z in aition to the ifference in the pper an lower tensor inices.in taking the eriatie of the secon term on the left han sie of Eqation (9), an obtaining Eqation (8), the factor 4 in Eqation (8) is cancelle otring the tensor algebra maniplations e to the Einstein smmation conention that a repeate ine in a mltiplication of tensors implies a smmation oer the repeate ine throgh all the conentional ales of the concerne repeate ine [8]. The electric fiel for a conctor or a semiconctor is proportional to the rift elocity an ths to the rift momentm as follows [,5] cp E J nq nq p mc () p nqcp m c mc When the following conition is ali p mc () Then sing the approimation an the qantm me- chanical operator p [9] i p p p nq E nq nq m mc m im () The P ector in Eqation () may be replace by the operator. The qantity of the srface integral of the parallelertical proct can be rece to a line integral aron a cre in the conterclockwise irection analogos to Stokes Theorem [4]. Let the parametric representation of a smooth ifferentiable srface be escribe by the following eqation,,, r X iy jz k (3) Let the parallel-ertical proct be written as follows where Q is a continosly ifferentiable ector fiel i j k ef i Qy Q y y Q Qy Q j Q Q k Q Qy y (4) The following ector proct is a ector fiel perpeniclar to the srface i j k r r X Y Z X Y Z (5) An the nit ector perpeniclar to the srface can be written as follows r r n (6) r r Then one can write the following mathematically state theorem Theorem: Copyright SciRes.
8 M. RASHID { r r Q n Qy Q Qy Qy Qy Q (7) Or: i j k i j k X Y Z Q y Q Q y Q y Q y Q y (8) Q X Y Z Qy Q Proof: i j k i j k X Y Z Qy Q Y Z Z Y Q Q Z X X Z y y (9) Q X Y Z Qy Q Q Q y X Y Y X y Qy Y Z Qy Z Y Q Y Z Q Z Y Q Z X Q X Z y y Q Z X Q X Z Q X Y Q Y X Q X Y Qy Y X y y y Qy Y Z X Qy Y Z X Q Z Y X y y Q Z X Y Q X Z Y Q X Y Z Q y Q y Q Q Q Q Z X X Z Y X X Y Z Y Y Z Qy Z X Qy Z X Q X Y Q X Y Q V Y Q Y Z Using Green s Theorem [4] one can write Qy Z X Qy Z X Q X Y Q X Y Q V Y Q Y Z Qy Z X Qy Z X Q X Y Q X Y Q Y Z Q Y Z Q y Q Qy Q y Qy Q Q Q Q Q y Q y Q y y (3) (3) (3) (33) (34) Copyright SciRes.
M. RASHID 9 Which proes the Theorem. 3. Conclsions The Lagrangian ensity of the electromagnetic fiel is complemente here by incling a Lagrangian ensity for two of the symmetrie Mawell s eqations.in this procere a transformation matri is fon which is incling in its components two new efinitions of ector procts which are calle here the parallel-horiontal an parallel-ertical ector procts. The Theorem of the srface integral of the parallel-ertical ector proct is shown to be rece to a line integral. 4. References [] D. H. Yong an R. A. Freeman, Uniersity Physics, 9th Eition, Aison-Wesley Pblishing Company, Inc., USA, 996. [] W. E. Brcham an M. Jobes, Nclear an Particle Psy- sics, Aison Wesley Longman Limite, Singapore, 997. [3] E. M. Prcell, Electricity an Magnetism, Berkeley Physics Corse, n Eition, McGraw-Hill, Inc., USA, Vol., 985. [4] T. M. Apostol, Calcls, n Eition, John Wiley & Sons, Singapore, Vol., 969. [5] J. B. Marion an S. T. Thornton, Classical Dynamics of Particles an Systems, 4th Eition, Harcort Brace & Co., USA, 995. [6] I. V. Linell, Electromagnetic Wae Eqation in Differential-Form Representation, Progress in Electromagnetics Research, Vol. 54, 5, pp. 3-333. oi:.58/pier5 [7] G. R. Fowles, Introction to Moern Optics, n Eition, Doer Pblications, Inc., New York, 989. [8] R. D Inerno, Introcing Einstein s Relatiity, Clarenon Press, Ofor, 995. [9] S. Gasiorowic, Qantm Physics, n Eition, John Wiley & Sons, Inc., USA, 996. Copyright SciRes.