[Khalid, 5(3): March 2018] ISSN DOI /zenodo Impact Factor

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1 GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES LORENZ TRANSFORMATION FOR FREE SPACE AND FIELDS USING MAXWELL S EQUATIONS AND NEWTON'S LAWS Nuha Abdelrahman Khalid*, Mubarak Dirar Abdallah, Zoalnoon Ahmed Abeid Allah 3 & Sawsan Ahmed Elhouri Ahmed 4 * Sudan University of Siene &Tehnology-College of Siene-Department of Physis- Khartoum- Sudan Sudan University of Siene &Tehnology-College of Siene-Department of Physis & International University of Afria- College of Siene-Department of Physis- Khartoum-Sudan 3 King Khalid University- Abha-(KSA) 4 University of Bahri- College of Applied & Industrial Sienes Department of Physis-Khartoum- Sudan ABSTRACT The epression of Lorentz fore and Mawell's equations were used to drive Lorentz transformation in terms of eletri and magneti fields. This epression is typial to that of speial relativity. The Lorentz transformation that takes are of the effet of fields on the physial system was also derived using Newtonian laws speially the veloity aeleration relation. The relation obtained is typial to that of generalized speial relativity. Keywords: Lorentz fore, Lorentz transformation, average veloity, speial relativity, generalized speial relativity. I. INTRODUCTION Mawell s equations desribe the relation between eletri and magneti fields. It also desribes how they are generated. They are derived from Gauss law, ampere's law beside Faraday's law. It relates also flues to field are strengths via permittivity. Mawell s equations are shown to be invariant under Lorentz transformation of oordinate. Lorentz transformation (LT) is of the orner stone s of speial relativity (SR). Usually LT is used to relate spae and time oordinates for different inertial frames. Lorentz transformation is suessful in desribing the inertial motion, but unable to desribe the motion of partiles in fields. Different attempts were made to aount for the effet of fields by using Lorentz transformation [].These transformations are mainly dependent on spae and time oordinates. Nothing is done for Mawell s equations. This work is devoted to use Mawell's equations to derive Lorentz transformation that aounts for the effet of fields on spae, time and mass[,3]. II. LORENTZ TRANSFORMATION AND ELECTROMAGNETIC FILED The fore is given by in the frame S F = e E + v B () In the frame S it is given by F = e E + v B () Assume that e is onstant and the eletromagneti fore is tans forms frame S to frame S as ee = eγ E + v B (3) If one assumes that the harge is at rest in frames, thus no magneti field is eerted therefore the fore is S is given by (assume the eletri field) E z = γ E z + vb y (4) If in ontrary, the harge is at rest in S, hene: F = ee z And 8

2 ee z = eγ E z vb y (5) Using Mawell s equations E = B t 6 i j k E = y z E E y E z = y E z z E y i E z z E j + E y y E k The j omponent is given by Let Sub (0) and () in (8) yields Sub () in (4) and (5) thus = B t i B y t j B z t k(7) E z + E z = B y t (8) i k.r ωt E z = E 0 e B y = B 0 e i k.r ωt 9 k. r = k + k y y + k z z k = k y = k z = k E z = ik. E z 0 B y t = iωb y ike z = +iωb y B y = k ω E z = πe z λ πf = E z λf = E z B y = E z B y = E z E z = γ E z v E z = γ v E z 3 E z = γ E z + v E z = γ + v E z (4) E z = γ + v E z γ + v v = Whih is ordinary SR epression v γ = γ = 5 v v 8

3 III. GENERALIZED SPECIAL RELATIVITY FARADAY ELECTROMAGNETIC LORENTZ TRANSFORMATION For a partile moving with aeleration a the veloity is given by at v = v 0 + at = v 0 = φt + v 0 6 Where V = potential = F = ma φ = potentialperunitmass 7 = V = a 8 m Sub in (5) and assuming the relation hold for all physial system 9 v For photon For no potential Whih is ordinary SR epression 83 φt + v 0 = t() φt + v 0 t φ + v 0 φ = O v 0 One an also use the average veloity to find Lorentz transformation in the presene of fields. To do these assume again the relation. v 4 (5) (0) () (3) For partile in a field moving with onstant aeleration the veloity is given by: v = v 0 at 6 But Where v m is the mean veloity This is given by = v 0 t at 7 v 0 + v v m = v 0 + v t = v m t 9 30

4 Replaing v by v m in (5) one gels Using the relation Inorporating (33) in (30) and (3) one gets v m (3) v = v 0 a = v 0 φ 3 v 0 = v + φ v 0 = v + φ 33 When no filed eists v + v + φ φ = 0 (34) From (7) Therefore equation (30) reads Using (38) in equation (3) given v 35 v = v 0 at = v 0 a t v = v 0 φ t 36 v 0 = v + φ t 37 v m = v 0 + v = v + φ t 38 Assuming this relation is general. For pulse of light v + φ t = t v + φ t t v + φ IV. DISCUSSION The SR Lorentz transformation an be found by using the eletromagneti fore relation for a harged eletron moving in an eletromagneti field, as shown by equations (), (3), (4), and (5). 84

5 Using Mawell equations, onerning generation of eletri field by variable magneti field, one gets a relation between Yomponent of the magneti field and Z omponent of eletri field in equation (). Using all above relations the Einstein SR oeffiient γ is shown to be typial to that of SR.Coeffiient γ for partiles moving in a field is found by using ordinary relation between veloity, aeleration and potential per unit mass [see equations ( to 8). fortunately this relation redues to that of SR in the absene of afield, as equation (9) indiate replaing the veloity v with the average veloity v m in SR Einstein oeffiient in (0), one gets γ in terms ofv m. Again using the relations between veloity and potential per unit mass, one gets two different epressions for γ depending on the time and time free relation of v and v 0 [see equations (9), (6)]. Fortunately the two epressions redues to that of SR, as shown by equations (0) and (7) V. CONCLUSION Lorentz transformation of SR an be found by using Lorentz fore epression and Mawell s equations. The effet of fields on spae and time is also derived. It was also found that suh transformation redues to that of SR. REFERENCES. Kuhn T.SThe struture of sientifi revolutions. In University of Chiago Press 0 Chiago, IL: University of Chiago Press. Lee A.R, Kalotas T.M-0 Lorentz transformations from the first postulate. Am. J. Phys. 43, doi:0.9/ Open Url 3. Lévy-Leblond J.-M-03 One more derivation of the Lorentz transformation. Am. J. Phys. 44, doi:0.9/ Open Url 85