Deriving 0 and 0 from First Principles and Defining the Fundamental Electromagnetic Equations Set

Transcription

1 International Journal of Engineering Reearch and Developent e-issn: 78-67X, p-issn: 78-8X, Volue 7, Iue 4 (May 3), PP Deriving and fro Firt Principle and Defining the Fundaental Electroagnetic Equation Set André Michaud SRP Inc Service de Recherche Pédagogique Québec Canada Abtract:- In 94, renowned phyicit Juliu Ada Stratton wrote in hi einal work titled 'Electroagnetic Theory' : "In the theory of electroagneti, the dienion of and that link repectively D and E and B and H, in vacuu, are perfectly arbitrary." Thi aertion i till valid today in the context of claical electrodynaic ince thee two contant have never been derived fro firt principle. It will be hown in thi paper how they can be derived fro firt principle. Keyword:- Perittivity contant, pereability contant, firt principle, peed of light, Maxwell equation, Lorentz equation, fundaental electroagnetic dienion C. I. BRIEF HISTORY The electrotatic perittivity contant of vacuu wa defined and then experientally accurately eaured by ean of an experient carried out with a parallel-plate capacitor of known area "A" with fixed ditance plate eparation "d" acro which a known voltage V wa aintained, which allowed calculating the exact correponding capacitance "C" involved. Thi allowed an exact calculation of in ter of very accurately eaured quantitie, a clearly decribed in ([], Section 3.): Cd ε A However, depite having been eaured in thi anner with the highet preciion, reain to thi date an ad hoc value that never wa derived fro any underlying theory. The agnetic pereability contant of vacuu on it part, wa aigned the arbitrary value 4-7 and the current ued to define the Apere with the current balance experient that wa ued in the proce, wa adjuted by convention o that would have the exact value needed in the following equation, a clearly laid out in ([], Section 34. to 34.4): F μ i l πd In thi cae alo, contant never wa derived to thi date fro any underlying theory, on top of having had it value arbitrarily aigned. All electroagnetic phenoena atheatically decribed with the help of both contant deontrate out of any doubt that thee contant are exact and required, even if no theory can currently atheatically explain their origin. Many attept have been ade to derive the fro known theorie but with inconcluive reult. II. THE SPEED OF LIGHT CALCULATED FROM MAXWELL'S EQUATIONS Let u exaine the equation allowing calculating the peed of light by ean of thee two fundaental contant ([], p.689), an equation that wa elaborated by Maxwell a he dicovered ore than 5 year ago that the reciprocal of the product of the perittivity and pereability contant wa equal to the quare of the peed of light: c () ε μ where i the electrotatic perittivity contant, with a value of = /(4c -7 ) = E- Farad per eter. Thi contant wa experientally eaured ([], p. 746). Contant on it part i the agnetic pereability contant, which wa calculated to be exactly: 3

2 Deriving and fro Firt Principle and Defining the Fundaental Electroagnetic... = 4-7 = E-6 Henri per eter fro Apere' Law ([], p. 848): B dl μ o i () It i very intereting to note that the perittivity contant ( ) i in reality a eaure of tranvere capacitance per eter (uually ybolized by capital C), related to the "preence of electric energy" in electrodynaic. The pereability contant ( ) on the other hand i in reality a eaure of tranvere inductance per eter (uually ybolized by capital L), related to the "preence of agnetic energy" in electrodynaic. Equation () in fact, turn out to be the only ean ever dicovered for calculating the peed of light fro theory. Thi equation i actually arrived at fro identitie drawn fro Faraday equation and Maxwell forth equation: Edl d Φ B dt and Bdl μoi εodφe dt (3) Without going into full decription, let u ention that fro thee two equation, equivalent econd partial derivative of the intantaneou agnetic field of a propagating electroagnetic wave in vacuu with repect to ditance and tie can be equated in the following anner (note that the ae relation can be arrived at for the related electric field): B B μ ε and/or E E μ ε (4) x t x t The iilarity of thi for with the equation giving the velocity of a tranvere wave on an elatic cord wa obviou: y L y y y which reolve to (5) x F t x v t Where F i the tenion applied to the cord, expreed in Newton. L i the lineic a (a per eter) and y = A co(kx-ωt + f), that i, the intantaneou tranvere aplitude of the travelling wave. Since F/ L = v, which i the quare of the longitudinal velocity of the wave along the rope, the iilarity wa obviou with the wave equation derived fro Maxwell equation and o it i by iilarity that equation () cae about, ince the product will obviouly be equal to the invere of a quared velocity, which in thi cae i /c. Conequently, ince the local condition exiting in all of vacuu that deterine the value of and can be aued to be univerally contant, thi i ufficient in and of itelf to conclude that the peed of light can only be contant in vacuu, a tate of fact that wa extenively confired by experient. Note alo that iple dienional analyi of the ratio L /F of equation (5) directly lead to the ratio /c. Firt, fro equation E=c, can obviouly be redefined a =E/c, which give the dienion Joule/c. Thi give L the dienion Joule/(c ). The Newton on i part, which i the dienion of F ultiately reolve to J/ (Joule per eter). Here i how J/ can be derived fro the uual Newton kg/ definition. The kg i a eaure of a, and fro =E/c, kg = J/ /, that i J /. If we replace kg by thi equivalent in kg/, we obtain J /, which, upon iplifying, becoe J/, which i Joule per eter. So the dienion of L /F reolve to J/(c ) J/, that i (J )/(c j), and iplifying: /c. III. DIMENSIONAL ANALYSIS OF AND Obviouly, and are rather abtract value and it i oewhat difficult to iagine what caue their product to reain contant. The traditional unit of (farad per eter) and that of (Henry per eter) give no idea of the real dienion of thee contant. But if we reolve the down to their International Syte (SI) dienion, we find that i an expreion involving charge in Coulob, a in kg, tie in econd and ditance in eter (Q /kg 3 ). Siilarly, we oberve that alo involve into charge in Coulob, a in kg, and ditance in eter (kg / Q ). Upon iplifying, we obtain / a the end reult: (3) c Q kg kg Q Thi level of detail allow uch better undertanding of why the firt equation preented in thi chapter allow calculating a velocity in eter per econd, that i, the peed of light. But it ay ee urpriing that a 33

3 Deriving and fro Firt Principle and Defining the Fundaental Electroagnetic... unit of a, the kilogra, hould be part of the SI dienion of two contant eant to deterine the velocity of free electroagnetic energy in vacuu! The iue i reolved if we keep in ind that, a we aw at the end of Section II, the kilogra i itelf a copoite unit that ultiately reolve to J /, that i Joule (a unit of energy) divided by a velocity quared. The ae obcurity proble occur with the unit of ipedance of vacuu (Z ), which i the Oh. By ean of the SI unit that we jut reolved, we can already deterine the value of the Oh in SI unit: 3 4 μ kg kg kg kg Ω (Oh) (4) 4 ε Q Q Q Q Z But reolving now kg to it ore eleentary unit: kg = J /, and ubtituting reveal already that we are really dealing with Joule econd per charge quared! kg J J Z Ω (5) Q Q Q Now the ytery deepen again if we reolve Z fro the pi related value of and : μ 7 7 Z 6π c 4πc / Ω (5b) ε which reolve the ipedance of vacuu in Oh a being eter per econd (/), that i, a velocity! We will reolve thi apparent incopatibility in Section IX. IV. FORCE AT THE PAIR DECOUPLING RADIUS The anner in which a photon of energy. MeV can logically decouple into a pair of electron and poitron wa thoroughly analyzed in a previou paper ([5]). Given the peranently fixed aount of ret energy that the electron preent at the relative decoupling ditance a the eparation becoe coplete, it i not excluded that the "unit charge" of the electron could be an apect not yet clarified of kinetic energy otion at thi decoupling ditance. The expanded tri-patial geoetry ([3]) that allow explaining how thi decoupling can echanically proceed definitely give hint in thi direction. We will now ue the well known claical force equation F=a, where a=v /r to initiate thi new tage in our analyi. We know the ret a of the electron, it velocity in electrotatic pace at the decoupling radiu(r d ), that i c, a well a it decoupling radiu proper a analyzed in ([5]): pair: c K rd roα E 3 (6) π Ee In line with our hypothei, let u calculate the force that would apply at thi decoupling radiu of the e c Fe.3666 N (7) r d V. MAXIMUM TRANSVERSE DECOUPLING VELOCITY In a eparate paper ([3], Section 7.5), we verified that the axiu tranvere velocity of the part of a photon' energy that i cycling between electrotatic and agnetotatic pace wa reached when half of it energy had left either pace during each cycle. Knowing that the correponding part of the electron energy ha the ae pulating otion between agnetotatic and noral pace and having jut deterined the force that applie at the decoupling radiu, we are in a poition to calculate that axiu tranvere velocity of the ocillating energy, a axiu tranvere velocity that applie by definition to photon alo, given the identity of tructure that thi odel reveal between electron and photon. We know fro the electron tructure in the tri-patial geoetry a decribed in previou paper ([5], Section ) that only half of it energy dynaically pulate o we ut ue = e / a the a that ut be conidered. Let u note that, a i traditionally done with contant and, we will ue a a a convenient place-holder for the energy of the electron a we introduce the concept of tranvere acceleration ince the acceleration equation i traditionally deeed to apply to a. The known decoupling radiu alo allow eaily deterining that half the ditance between axiu aplitude and the tri-patial junction will be: d = r d / = /4 (8) Once again, fro F=v /r, we can derive F=v /d and 34

4 Deriving and fro Firt Principle and Defining the Fundaental Electroagnetic... Fd v 99,79, / (9) Thu interetingly confiring that axiu tranvere velocity of a photon' or electron' energy during it internal cyclic pulating otion i preciely the peed of light, and that thi velocity i reached at preciely half of it aplitude in either electrotatic and agnetotatic pace. VI. DEFINING THE REQUIRED 3-SPACES MAJOR UNIT VECTORS SET Before proceeding to the actual derivation, it i ueful to put in perpective the ajor and inor unit vector et required in the 3-pace expanded geoetry to allow proper repreentation. Thi unit vector et wa copletely decribed and jutified in previou paper ([3]), but we will uarily reproduce the unit vector bae here for convenience. Fig.: The 3-pace unit vector et The traditional ˆ i, ˆj and kˆ unit vector et wa of coure defined to repreent vectorial propertie in noral pace ince electroagnetic phenoena have up to now been perceived a occurring entirely within noral 3-D pace. But the expanded 3-pace geoetry involve two ore pace noral to noral pace and to each other, each of which alo requiring it own internal inor unit vector et. The three utually orthogonal pace (noral, electrotatic and agnetotatic) alo need to be repreented by a pecial unit vector et. It wa defined a a new uperet of ajor unit vector that identify the three orthogonal pace a Î, Ĵ andkˆ, or to ake notation eaier, I, J and K (Ibar, Jbar and Kbar), o each local ˆ i, ˆj and kˆ unit vector et becoe ubordinated to the ajor unit vector pecific to it local pace, all reulting unit vector (9 inor and 3 ajor) being of coure drawn fro the ae origin O. Each of the three orthogonal inor unit vector ubet (repreented in Fig. a being half folded), that i I-i, I-j, I-k for noral pace, J-i, J-j, J-k for electrotatic pace, and K-i, K-j, K-k for agnetotatic pace, allow defining the vectorial agnitude of energy in any one of the three orthogonal coexiting pace. VII. DERIVING AND FROM THE TRANSVERSE ACCELERATION EQUATION But let' coe back to equation F=v /d, that we can now forulate F=c /d in the cae of the electron at ret to account for the velocity being the peed of light at the decoupling radiu (equation (7)), that i F = c /d () which becoe after replacing d with it proper value deterined in equation (8): c F and finally 4πc F () 4π We have now eaured a force that applie in noral pace (I), produced by an acceleration that act orthogonally in electrotatic pace (J), that i, tranverally with repect to the direction of otion of the particle in noral pace: 4π c F e I J () The energy having reached it axiu velocity, that i c, after having reached half it aplitude in electrotatic pace, it will then tart decelerate and coe to a coplete top a it reache it axial aplitude, a decribed in paper ([3]). Thi energy will then re-accelerate in agnetotatic pace (K) until the peed of light i again reached in that pace, that i, when half of the energy will have left electrotatic pace. We will thu have: 35

5 Deriving and fro Firt Principle and Defining the Fundaental Electroagnetic... 4π c F I K (3) We have thu identified two acceleration in two pace orthogonal to each other (J and K) that reult in two force acting oppoite to each other in noral pace (I), which i orthogonal to both of the. Let' ee what happen when we execute an invere product (to take into account the utual orthogonality of both electrotatic and agnetotatic pace) of thee two equation ter for ter: FI 4π c K (4) F I 4π c J e The two calar quantitie and will of coure iplify ter for ter to unity, but the vectorial quantitie reain. In Section II, we entioned the value involving of fundaental contant and, that i: = /(4c -7 ) and = 4-7 (5) Don't we recognize here value having the for of acceleration identical to thoe that we jut derived fro the fundaental acceleration equation? Note that the -7 factor which i part of thee contant i a iple converion factor that wa introduced to allow proper converion between the rationalized abolute CGS unit yte and the MKS unit yte that we are uing here ([], p. 4.). The reaon why thi -7 factor reain outtanding in the MKS yte i that contrary to the CGS gra which i a natural fraction of the MKS kilogra, and to the CGS centieter which i a natural fraction of the MKS eter, the CGS erg i a unit different fro the MKS Joule, which render ipoible it direct integration ( erg = -7 Joule). So let' add utually reducible occurrence of thee value to our equation, ince we are uing the MKS yte here: 7 F I 4π c K (6) 7 F I 4π c J e which, when converted to calar quantitie give: 7 4πc 4π, and rearranging: 7 c 7 7 4πc 4πc (7) which allow clearly revealing the actual related definition of fundaental contant and, a explained in Section II: 7 4π c ε 7 μc 4πc (8) Which, after iolating c and extracting the quare root give the following relation, which i identical to equation () previouly derivable only fro Maxwell' equation: c ε (9) μ VIII. CYCLIC TRANSVERSE ACCELERATION CONSTANTS AND Haven't we jut copletely clarified here the origin up to now o yteriou of thee two contant? Let' recall what Juliu Ada Stratton wrote in 94: "In the theory of electroagneti, the dienion of and that link repectively D and E and B and H in vacuu, are perfectly arbitrary." ([], p. 7). Thi ay have been the cae in traditional electroagneti, but haven't we jut oberved that in thi odel, they definitely have the for of utually orthogonal acceleration, repectively in electrotatic pace, and in agnetotatic pace, in perfect harony with the obliged dynaic otion of any quantu of kinetic energy in the 3-pace odel? We know ince Maxwell that free electroagnetic energy cannot exit without interecting electric and agnetic field, and that uch interection ha to be cyclic for the energy to exit at all, which necearily involve cyclic acceleration, ince table tranvere velocity would prevent repeated interection. 36

6 Deriving and fro Firt Principle and Defining the Fundaental Electroagnetic... IX. THE FUNDAMENTAL C,, S DIMENSIONS SUBSET Let u now conider the new equation to calculate energy derived fro Paul Maret' work in a eparate paper ([6], Equation ()), and let u give it the for of a quantity being ubitted to an acceleration, that i, by converting it to a quantity being ultiplied by a velocity quared and divided by a length in eter. Thi require converting to it related value a follow: 7 7 e e 4π c e π c C E with dienion J (3) εα α α Exaining thi relation, we oberve that both for of thi definition of energy naturally involve a quared unit charge (e ), that i C (Coulob quared), which allow equating the Joule (J), which i the traditional unit of energy, with the dienion Coulob eter per econd, when cobined with the dienion of c and, the two reaining dienioned variable, which clearly reveal a pair of charge in the proce of being accelerated. Let u now recall that we found earlier two apparently contradictory dienional definition of the Oh in relation with the ipedance of vacuu (Z ), (See Equation (5) and (5b) above), that i: J Z Ω (3) Q If we now ubtitute the new definition of the Joule that we jut etablihed with equation (3), we effectively verify after iplifying that both definition of the Oh are equivalent: J Q Z Ω (3) Q Q Conequently, to ucceed in finally clarifying the relation between charge (in Coulob), aplitude of a photon' energy (in eter) and tie (in econd), for the reainder of thi paper we will ue thi new definition of the Joule that i C /, which will allow highlighting an underlying acceleration proce preent in all claical equation related to energy. One ore thing that the analyi of Z in Section III revealed wa the fact that the converion factor -7 preent in both and, i not part of the value that the quare root i to be extracted of (See equation(5b)), but that it ut be applied to the quare root reult for the correct ipedance to be obtained. By the ae token, thi highlight the fact that here alo thi factor ha to be external to the quared value of the charge. We will ee that thi infiniteial contant quantity (the quared charge), that ee ade up of unit quantitie, in confority with de Broglie' hypothei and that the equation preent to u a if they were charge, appear to be found at the heart of each photon. X. THE FUNDAMENTAL CHARGES ACCELERATION EQUATION To iplify notation, and to clearly highlight the iilarity in tructure with the traditional acceleration equation, let u identify the pair of quared charge with a capital X, and identify by a the aociated acceleration involving the quare of the peed of light on a tranvere ditance equal to the integrated aplitude of the abolute wavelength of the aociated particle, which in context correpond to half thi integrated aplitude on either ide of the Y-x axi (along which the photon i aued to be oving in vacuu). So fro now on, we will ue X and a with the following definition: 7 7 πc X e E 38 C and a (33) α We will convert here oe of the ajor electrodynaic equation to ore eaily perceive the progreive coplexification of thee equation fro that for energy to that for energy denity, and oberve that all of the involve the "Xa" tranvere acceleration relation that we jut identified. XI. DEFINING THE FUNDAMENTAL EM EQUATION SET Before going further, let u ay a few word regarding what ut be undertood by "integrated wavelength", and "integrated aplitude" of the energy of a particle. The relation between the wavelength of a localized particle and calculation of it energy by ean of pherical integration i tudied in detail in eparate paper ([6]). The analyi carried out in that paper allowed defining the integrated wavelength of a particle a correponding to the abolute wavelength of the energy of thi particle (=hc/e, or =h/c) ultiplied by the fine tructure (). The integrated aplitude of a particle will thu be fro equation (33): α Aα, whence c π c a a defined in (33) (34) π Aα α 37

7 Deriving and fro Firt Principle and Defining the Fundaental Electroagnetic... We can now define a general equation equivalent a well to E=c (for the ret energy of a aive particle) a equal to E=hf (for the energy of a photon), conidering the iilarity in internal tructure of both particle, that i 7 hc e π c E hf c Xa with dienion C J (35) α Contrary to the two traditional equation however, the new equation directly highlight the tranvere electroagnetic acceleration of two unit charge, a already entioned. The expert reader will no doubt iediately ee the relation with the Coulob equation that preciely calculate the force between two eparate charge, and the related energy when we divide thi energy by the ditance between the two charge, and that would directly apply inide a photon ince it can now be een a containing unit charge. If we conider in context two unit charge, we obtain: 7 Q e 4π c J C 4πε Xa d F r with dienion α The agnetic force relation involving an interaction between two electroagnetic particle can alo be expreed in thi anner ([8], equation ()). Fro the definition of the agnetic dipole oent etablihed in paper ([6], Equation (35a)), that i =E/B=ec /4 we can write 7 3μ μ 3α e 3π c J C Xa d N (36) F 3 4π r r 4 with dienion N (37) Fro the equation for energy (35), a can of coure be defined a energy divided by c. 7 7 X e πc e π Aα αc with dienion C kg (38) α Thi lat redefinition of a i of particular interet ince it i the for that allow unifying all claical force equation in eparate paper ([7]). The general equation for a calculation of down quark, up quark and electron repectively, that we will ee in a coing paper, and copletely analyzed in ([9], Section 7.) will thu forulate a: 7 9 e 9 X A [d,u,e] Cα n where C i the electron Copton wavelength. Planck' contant on it part reolve to π C α n where (n=,,3) (39) 7 X π c e π c h α with dienion C J (4) α c We can oberve that the expreion for ħ take u further away yet fro the for of a quantity being ubitted to an acceleration : 7 h Xc e πc h π α with dienion C J (4) α πc The alternate ditance baed contant that wa teporarily naed electroagnetic intenity contant a it wa being defined it in paper ([3], Section J) becoe: 7 Xπc e πc H hc α with dienion C J (4) α We can now undertand that H i in reality a tranvere electroagnetic acceleration contant, poibly the ot fundaental contant that can poibly be defined in electroagneti. One only need to divide it by the abolute wavelength of any particle, aive or not, to obtain it energy. By the ae token, we can clearly ee that h, Planck' contant, necearily alo act tranverally and i in reality a tranvere electroagnetic action contant! So, If we redefine the firt and econd Planck radiation contant uing tranvere electroagnetic acceleration contant (H), they becoe: c = hc = Hc and c = hc/k = H/k, 38

8 Deriving and fro Firt Principle and Defining the Fundaental Electroagnetic... and it becoe uch ore obviou that Planck' black body radiation equation entioned previouly i very directly linked to Maxwell' equation ince H i obtained directly fro electroagnetic conideration. We can alo eaily reforulate the electron electrotatic energy induction contant (K) that wa defined previouly in ter of contant H and alo aociate it to electroagnetic acceleration 7 C C e c C K c H π 4π 3 with dienion C J α π (43) The expreion of a local electrical field for an individual localized photon decribed in paper ([6]) drawn fro the Lorentz equation becoe: 7 F Xa e π c π αe Aα e J C E with dienion α α αe C (44) The expreion for the correponding local agnetic field will be 7 Xa e πc π T Aα ec B with dienion J C T α α αec C (45) And the expreion for energy denity of a localized photon within volue V, a defined in paper ([6], Equation (4i)) becoe: 7 Xa e πc π πe J C U V with dienion 3 α α α αe α (46) Let u note alo that B Fπ 3 U εe J/ 4 μ α (47) XII. CONCLUSION We thu oberve concluively how both and are directly related to the tranvere acceleration of free energy in the expanded 3-pace geoetry and can be derived fro the fundaental acceleration equation, which alo allow defining the fundaental electroagnetic equation et. REFERENCES []. Juliu Ada Stratton. Electroagnetic Theory, McGraw-Hill, 94. []. Robert Renick & David Halliday. Phyic. John Wyley & Son, New York, 967. [3]. André Michaud. The Expanded Maxwellian Space Geoetry and the Photon Fundaental LC Equation. International Journal of Engineering Reearch and Developent, e-issn: 78-67X, p- ISSN: 78-8X, Volue 6, Iue 8 (April 3), PP ( [4]. André Michaud. Fro Claical to Relativitic Mechanic via Maxwell. International Journal of Engineering Reearch and Developent e-issn: 78-67X, p-issn: 78-8X, Volue 6, Iue 4 (March 3), PP. - ( [5]. André Michaud. The Mechanic of Electron-Poitron Pair Creation in the 3-Space Model. International Journal of Engineering Reearch and Developent, e-issn: 78-67X, p-issn: 78-8X, Volue 6, Iue (April 3), PP , ( iue/f63649.pdf). [6]. André Michaud. Field Equation for Localized Individual Photon and Relativitic Field Equation for Localized Moving Maive Particle. International IFNA-ANS Journal, No. (8), Vol. 3, 7, p. 3-4, Kazan State Univerity, Kazan, Ruia. (Alo available online at [7]. André Michaud. Unifying All Claical Force Equation. International Journal of Engineering Reearch and Developent, e-issn: 78-67X, p-issn: 78-8X, Volue 6, Iue 6 (March 3), PP ( [8]. André Michaud. On the Magnetotatic Invere Cube Law and Magnetic Monopole, The General Science Journal 6: ( [9]. André Michaud. Expanded Maxwellian Geoetry of Space, 4 th edition, SRP Book, 4. Available in variou ebook forat at: 39