Journal of Theoretics Vol.5-3

Transcription

1 Jounal of Theoetics Vol.5-3 The aw of Electomagnetic Induction Poved to be False Using Classical Electostatics Jan Olof Jonson, Msc Östmaksgatan 5 n.b., SE-3 4 Fasta, Sweden bstact: new explanation to the effect of electomagnetic induction is poposed, while simultaneously ejecting the cuently accepted Induction aw, oiginally poposed by Neumann, 845. ccoding to the new theoy, a cuent will be induced by any changing electic field, due to the Continuity Equation of Electicity and the aw of Electic Displacement, two of Maxwell s Equations. It has been shown elsewhee that a cuent within a conducto, in spite of an oveall chage neutality, will give ise to a foce upon anothe such cuent, due to Coulomb s aw, theeby ejecting the claims of the oentz Foce. Hee it is shown that the same Coulomb foce can also account fo electomagnetic induction. compaison between the pedicted phase shift fom the pimay to the seconday loop within a tansfome accoding to this theoy and accoding to the Induction aw gives cedit to the fome, while the latte fails. This esult follows as a consequence of the discovey that any electic cuent though a esistive cicuit must be popotional to the time deivative of the applied voltage, not pimaily the voltage itself, as usually has been infeed fom Ohm s aw. It is shown that it is only a coincidence with the fact that the time deivative of an exponential function is popotional to that same exponential function, which gives this esult, usually undestood as Ohm s aw. The exponentially decaying natue of a cuent though a D.C. cicuit is due to the continous loss of excesss chages at the poles, as the cuent flows, just an analog to what happens, when a chaged capacito is connected to a esistive loading. Keywods: electomagnetic induction, Coulomb s aw, Ohm s aw, Maxwell s equations.. Intoduction The autho has ealie succeeded in showing that the electomagnetic foce between two cuent caying conductos can satisfactoily be explained by using Coulomb s oiginal foce law of 785. The oentz Foce is no moe needed. Futhemoe, it even fails to pedict the qualitative behaviou of the foce within mpee s Bidge, as measued by Pappas and Moyssides in the ealy 98 s. How Coulomb s aw can be used in ode to explain the electomagnetic foce between cuents is thooughly explained. Shotly, thee appeas a diffeence in the stength of the electic foce, which the positive and negative chages give ise to upon the second cuent, due to the velocity dependent effects of etaded action. This diffeence can account fo the foce, usually explained by using oentz s Foce aw.

2 In this pape the consequences of the above mentioned discovey is futhe developed in the case of electomagnetic induction and Ohm s aw. It is shown that in both cases thee is a time dependent change in the stength of the electic field within the conducto, thus causing a cuent to flow as a consequence of the Continuity Equation of Electicity and the aw of Electic Displacement. The diffeence is that in the fist case the electic field is extenally induced, in the latte it comes fom within, the field lines being aligned along the conducto, going between the both poles.. nalysis of a D.C. Cicuit If fist looking at a taditional D.C. cicuit, thee is a D.C. voltage souce, e.g. a battey, with an excess of chages at one of the poles, a deficit at the othe. If they ae connected by a esistive loading, a cuent will flow unde the influence of the electic field between the poles. It ought to be noticed that in the case of a battey, with no continuous suppot of new chages, the stength of the cuent must continuously decease, until the battey has become completely dischaged. This pocess would best be descibed mathematically by using the exponential function, as in the case of a capacito, at least when studying the steady state case. The chemical o physical popeties of a battey would best be neglected hee. Othewise it is bette to think of a sufficiently lage capacito such that it is capable of deliveing an almost constant cuent, in the same way as a battey. The elevant popety is the chage content of the D.C. voltage souce, often expessed in ampeehous. It is limited, and as the cuent flows between the poles, it is simultaneously deceasing slowly, a pocess best descibed by the exponential function, as when a capacito is being dischaged though a esistance. The mathematical analysis could pefeably be pefomed as follows: j + ρ t = (). This is the so-called Continuity Equation of Electicity, one of Maxwell s Equations. Theeafte is chosen D = ρ (), also this one of Maxwell s Equations. If those two ae combined, follows: ( j + D ) (3). Integation of Eq.(3) with espect to t gives: j = - D t + f(t) (4).

3 The funcion f(t) can be detemined if ealizing that within a closed D.C. cicuit, without any extenally applied electomagnetic fields, the electic displacement D will appoach zeo, as the cuent density j does. Of couse, by mathematical easons, any suitable function f(t) could be added to D, but it does not make any sense. when egading the elationship between cause and action in this special case. The fundamental physical pocess to be egaded is the tanspot of excess chages fom one pole to the othe, thus causing a continuous decease in the stength of the electic field, just as Coulomb s aw pescibes. When all the excess chages have been annihilated, by mediation of the cuent, the cuent too becomes zeo. Hence, in this case must be the pope choice, and accodingly f(t) = (5), j = - D t (6). ppaently, the cuent (hee defined in tems of the cuent density) must be popotional to the time deivative of the voltage (hee defined in tems of the electic displacement)... Consequences fo Ohm s aw If assuming a esistive loading between the two poles of the D.C. souce, with length, coss section aea, using the elation and theeby defining diections coectly, gives D = ε ε E (7) E = - V (8), I = ε ε V (9). Since ε ε is the coupling constant between D and E within the conducting medium, hee the esistive loading betweeen the both poles of a battey, it can be infeed that the facto ε ε befoe V in Eq.(9) by vitue of dimension analysis means a capacitance. The consequence is that the elevant coupling constant between cuent and voltage is a capacitance, povided fist the voltage is diffeentiated with espect to time. ssuming an exponential dependence at both the cuent and the voltage, as in the factual D.C. case, the time deivative any of the function happens to be popotional to that vey function and hence, the cuent and

4 the voltage ae popotional to each othe in that case, i.e. nothing else than what Ohm s aw claims. 3. Simple Tansfome Fom a conducto, caying a cuent I, thee will aise an electic field accoding to Coulomb s aw, popotional to this cuent. This has not ealie been popely undestood, but in a pape by this autho it is thooughly shown, how two cuents affect each othe with an electic foce, accoding to Coulomb s aw (785). No magnetic field no oentz foce is needed. In this pape, howeve, the inteesting vaiable is the electic field, since it has above been established that a cuent will flow in a cicuit, due to a changing electic field. This is assumed to be the cause behind both Ohm s aw and the Induction aw by Neumann. Theefoe, the expession fo the Coulomb foce between cuents in the mentioned pape must be divided by the effective - o vitual - chage density of the second cuent. Eq. (5) gives the foce on diffeential fom. Figue is also given hee fo sake of claity. d F d x d x = ( µ I I cos ϕ cos ψ ) u (). 4 π In ode to attain an expession fo the electic field at a specific point, one has to divide Eq.() with the amount of chages at the seconday conducto the fist cuent affects by this foce. This amount must be equal to the effective -i.e. vitual- chage density of the second conducto times the length element, thus using Eq.() and (3) of the just mentioned pape. Hence, d E = - d x ( I cos ϕ ) 4 π ε ε c u (). Still being on diffeential fom, it is necessay to finally integate Eq.() along whole the pimay cicuit, i.e. with espect to. This gives the esult: x E = - d E d x d x (). x The details of the integation ae delibeately omitted hee. The impotant esult above is Eq.(), which clealy shows that the electic field is popotional to both the cuent of the pimay cicuit and to the invese squae of the distance, popeties it shaes with the magnetic field. If succeeding in falsifying the Induction aw, which is based upon the usage of the magnetic field, the model of this autho emains a eliable candidate to giving the coect explanation to the vey phenomenon of electomagnetic induction. ate in this pape it will be shown that the new model is capable of pedicting the obseved phase shift between the pimay and the seconday cicuit of a tansfome. If now the pimay cuent is allowed to vay with time, so does the electic field, accoding to the esults above. Hemce, a cuent will be induced accoding to Eq.(5) and (8), theeby taking into account Eq.(6) and (7). Hence,

5 I = - ε ε ( I cosφ cos ψ ) dx dx (3) 4 π ε ε c x x whee the integal coesponds to the induced voltage (electomotoic foce, emf) of the seconday cicuit. Theefoe, one may shote wite: I = - ε ε V (4) Figue. Evidently, the tem ε ε coesponds to a capacitance. This was aleady discussed in connection with Eq.(8). Fo convenience, define theefoe ε ε = (5). It thus appeas vey clealy that the esult concening Ohm s aw is valid also in this case. The fundamental cause behind any cuents appaently is a vaying electic field, independently of how this electic field is bought into an actual cicuit; an intenal souce, as in the case of Ohm s aw, o extenal, as in the case of the Induction aw. In the following chapte it will be shown that the Induction aw accoding to Neumann is false and that the new model can account fo the chaacteistic phase shift between the pimay and the seconday cicuit of a tansfome.

6 4. nalysis of Voltage Measuements at the Output of the Seconday Cicuit of a Tansfome 4.. The New Theoy If now connecting e.g. a volt mete to the output of a tansfome, it is necessay to fist make an analysis of what the instument eally is feeling. In fact, a volt mete will feel the same measuable as an ampee mete, and independently of the method one pefes to use, the measuement esult will be popotional to the cuent. In the case of an amplifie volt mete, o an oscilloscope, it is the chages, i.e. the cuent, which is enteing the gate of the input tansisto, theeby detemining the output level of the sceen o the volt mete scale. Hence, only a eal coupling constant is multiplied with the cuent in ode to ead voltage. It would be easily ealized that if only a cuent thus can be detected by using odinay measuement instuments, it is not meaningful to discuss any phase shift between cuent and voltage, due to e.g. an oscilloscope figue. ny measued phase shift must namely be due to a phase shift between cuents. But since seveal existing laws deal with both of them simultaneously, they must be dealt with, though caefully. In the actual case, with a tansfome, the applied volt mete at the output may best be descibed as a polonged seconday loop, obeying basically Eq.(4). must only be eplaced by an expession fo the equivalent capacitance of the total cicuit, including the seconday loop and the volt mete. That expession must obey the law of seies connection fo capacitances, o: = ' + C v (6). ccodingly, the voltmete will show: ' V v = R v ( - V ) (7). Figue. ssuming a sine diving cuent of the pimay cicuit, as is the common paxis in case of a tansfome, thee will appea a -9 phase shift fom V to V v, as will easliy be ealized fom Eq.(4)above. Howeve, of most inteest is the displayed phase shift on a typical oscilloscope. This must be between the pimay cuent and

7 the seconday one, accoding to the theses of this aticle, i.e. thee ae cuents which ae measued o felt by the instuments. ccoding to Eq.(3) thee must be a -9 phase shift fom pimay to seconday cuent and this is well in accodance with measuements. 4..The Induction aw In ode to show the failue of the Induction aw, one must fist claify, what quantity is eally being measued in connection with this model. Fo convenience, the Induction aw may be witten: V = - B d s (8). s Since the magnetic field B is popotional to the pimay cuent - and in phase with it as well - µ I d x B = 4 π (9), 3 x the induced voltage lags 9 behind the pimay cuent I, and hence thee is a total 8 phase shift between the pimay and the seconday cuent, and it is these two cuents, which ae felt by the two oscilloscope inputs, accoding to the theses of this aticle. This, howeve, is not in accodance with measuements, and thus the Induction law has failed. 5. Conclusions The esult of the analysis above is vey clea. Both Ohm s aw and the Induction law can be explained as oiginating fom two of Maxwell s Equations: the Continuity Equation of Electicity and the aw of Electic Displacement. The fist equation tells us, how a cuent aises, wheneve an electic field is vaying with time, thus acting in ode to eliminate an excess of chages somewhee. The second equation expesses the elationship between the chage density and the electic field. It has been shown that these two laws give an appopiate and adequate desciption of what is acting behind Ohm s aw and the Induction aw. Finally, the cuently widespead concepts concening these ae finally being satisfactoilly ejected. In doing so, it appeas that fundamental electodynamics must once again become the cente of inteest among scientists. Refeences () J.O.Jonson, Chinese Jounal of Physics, VO.35, NO., pil 997, pp () P.G.Moyssides and P.T.Pappas, J.ppl.Phys. 59,9 (986).

8 Units Thoughout the aticle MKS units have been used. The vaiables used ae listed below: s B c ' o an C v D E F f(t) I I I j R R v aea of the coss section of a conducto aea vecto of the seconday loop of a tansfome magnetic field due to thecuent of the pimay cicuit of a tansfome velocity of light equivalent capacitance of the seconday cicuit of a tansfome equivalent capacitance of the total seconday cicuit, including a connected volt mete oscilloscope equivalent capacitance of the volt mete electic displacement electic field electostatic foce accoding to Coulomb s aw abitay function of time cuent pimay cuent seconday cuent cuent density pe unit volume length of applied esistive loading esistance esistance of the volt mete distance vecto, defined fom a cuent element of the pimay loop to a cuent element of the seconday loop t time u V V V v loop x x unit vecto along a distance vecto geneal potential dop induced voltage (electomotoic foce, emf) of the seconday loop egisteed voltage of an applied volt mete o oscilloscope, connected to the seconday length vaiable along the pimay loop length vaiable along the seconday loop Geek symbols ε ε ρ φ ψ dielecticity constant of vacuum elative dielecticity constant of the medium being used cgahe density pe unit length angle between the distance vecto and the pimay cuent angle between the distance vecto and the seconday cuent µ pemeability of vacuum Jounal Home Page Jounal of Theoetics, Inc. 3