Abstract - This article illustrates the violations of the action-reaction principle which can be foreseen in Maxwell-Lorentz electromagnetism, and investigates the possibility of conceiving upon them "perpetual motion" machines - of the kind for instance of the famous Testatika machine, extensively studied by Stefan Marinov. In conclusion, it is shown that, in spite of these theoretical violations, no possibility seems to exist for their practical utilization, since in a closed cycle the forecast effects exactly balance one another. In this way, it seems to be confirmed the opinion that any attempt to build these machines must rely on a theoretical set up which is different from the Maxwell-Lorentz electrodynamics.


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This paper is dedicated to the memory of the very dear Stefan Marinov, with the regret that even his best friends could not help him to overcome his "tiredness".


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1 -Introduction


I was preparing myself for the conference ''Physics as a Science'', anticipating the pleasure of the possibility of discussing many interesting questions with people free from all the 'absurdities' of modern Physics, when I was deeply shocked by the news concerning the sad end of Stefan Marinov's life. My happiness suddenly turned out into deep sorrow, since the friend who taught me so much, and shared with me the criticism against 'official Physics', had not maintained the promise we made each to the other, to live enough to see the end of the realm of relativity and the other connected subsequent theories(1).

At the time I was trying to write something concerning the so-called inverse Rowland experiment, to which Marinov dedicated theoretical and experimental work(2), since I felt that this experiment could be one of the possible crucial ones against the empirical validity of the principle of relativity in electromagnetism, which was too rashly claimed by Einstein. In a few words, Rowland showed that the motion of a 'discrete' set of macroscopic electric charges affected a magnetic needle, so opening the way for conceiving a current as a motion of 'small' electric charges; Marinov proposed to reciprocally move the needle with respect to the electric charges, and look whether there was or not any effect on it. Since a magnetic needle is something like a circular stationary current, this study is connected with the famous problem of the interaction between a charge and a circuit: first move the charge, and let the circuit at rest, then move the circuit leaving the charge at rest, and finally move together the charge and the circuit. Of course this kind of arguments leads to a comparison between classical and relativistic electromagnetism, to which an extensive study had already been devoted by Marco Mamone Capria and the present writer(''Symmetries and Asymmetries in Classical and Relativistic Electrodynamics'', Foundations of Physics, 21, 7, 1991, pp. 787-801): as a matter of fact in this study we regarded in any of the three cases only at the effect on the charge, rather than on the circuit, since this last one is more difficult even in principle - the problem of the so-called ''missing torque''. The tragic death of Stefan forced me to leave this new more deep examination of that question, and to dedicate instead to his memory the present study, which is in some sense connected to the previous one - since both rely on the interaction's law between moving electric charges - but more directly concerns the problem that obsessed most of Stefan's scientific activity, that is to say the search for perpetual motion machines, of the kind for instance of the famous Testatika, which Stefan extensively studied ("The Thorny Way of Truth", Part V) [see point 13 in the page dedicated to the History of Science].

I shall try to explain in the next as this idea has indeed a precise theoretical starting point in ordinary Maxwell-Lorentz electromagnetism, namely the well known violations of the action-reaction principle in the aforesaid interaction's laws, but moreover that these are not enough in order to theorize any their practical exploitation (and then the Testatika machine remains an unsolved mystery, if it is not just an hoax(3)).

In this way, one would conclude that from one side the idea of building up electromagnetic perpetual motion machines is not so fool as it is today commonly thought, and from the other side that every attempt to build them should instead be figured in a theoretical background different from the ordinary Maxwell-Lorentz electromagnetism. Reading this conclusion in the negative, it would then seem that any failure of building these machines is an argument in favour of this theory, which is the opposite opinion of the one that George Galeczki expressed to this regard: ''...every failure of Marinov to build this wonderful machine is automatically an argument against the Maxwell-Lorentz electrodynamics''(4).

This contrast shows that this problem is still far from a definitive solution, and I agree a priori with people who would think that what I have calculated in the next just refers to a special case, and does not take into account most interesting situations: at the end of this Introduction I cannot but repeat the words that Stefan loved, and that are even contained in his scientific testament: feci quod potui, faciant meliora potentes.



2 - The interaction between two moving electric charges


Let us calculate the force F1 which acts on a charged particle q1, endowed with a (uniform) velocity v1, due to the effect of another charged particle q2, endowed with an analogous velocity v2 (we suppose to do computations in some reference frame in which Maxwell equations hold, that is to say an 'aether frame' in classically conceived electromagnetism, any inertial frame in relativistic treatment). Exchanging roles between q1and q2, we have also an analogous force F2, and after computations we will provide to a comparison of these two forces.


As is well known, in Maxwell-Lorentz electrodynamics the first force has the expression:


(1) F1 = q1*(E2 + v1B2)


where the two new vectors in the right hand side (RHS) of (1) are respectively the electric and magnetic field generated by the charge q2, defined by:


(2) E2 = -ÑF2 - A2/t ;


(3) B2 = curl(A2)


when one has introduced the electric and magnetic potentials F2 , A2 generated by the charge q2.


One can get the expression of these potentials (the so-called Wienard-Liechert potentials) just integrating the relative set of Maxwell equations, so getting for instance, when the motion's law of the generating charge q is


x = vt , y = 0 , z = 0 :


(4) F(x,y,z,t) = q/(4p e0*sqr[(x-vt)2+(1-b2)*(y2+z2)]) ,


(5) A(x,y,z,t) = m0qv/(4p*sqr[(x-vt)2+(1-b2)*(y2+z2)]) .


Let us observe that the "retarded distance" which appears in the denominator of (4) and (5) does not concern at all the relativistic interpretation of the electromagnetism, since in the case of a single charge both classical and relativistic point of view give the same result(5); moreover, that it is indeed in the "aether interpretation", and not in the relativistic one, that this retardation receives an obvious physical meaning. In any case, it is easy to prove that, at least for the applications we have in mind (discrete point charges, small velocities and distances), one can ignore this retardation, and then write more simply for these potentials the widely used approximated expressions(6):


(6) F(x,y,z,t) = q/4pe0R ,


(7) A(x,y,z,t) = m0qv/4pR ,


where R = sqr[(x-at)2+(y-bt)2+(z-ct)2)] is just the ordinary distance from the point (x,y,z) and for instance the point (at,bt,ct) in which the charge q (endowed with the velocity (a,b,c) ) is supposed to be in the instant t .


Let us now insert (6) and (7) into (2) and (3). We get


(8) E2 = (q2/4p)*[-e0-1Ñ(R-1) - m0(R-1v2)/t] ,


(9) B2 = (m0q2/4p)*curl(R-1v2) =

= (m0q2/4p)*[R-1curl(v2) + Ñ(R-1)v2]



whence, since actually


Ñ(R-1) = -R-3*R21


(R-1)/t = -R-2R/t = R-3*<R21,v2> ,


where R is now the distance between q1 and q2, and R21 the vector going from q2 to q1:


(10) F1 = (q1q2/4p)*[e0-1R-3*R21 - m0R-1*v2/t +

-m 0R-3*<R21,v2>*v2 - v1(R21v2)] ,


which can be even written in the more expressive form


(11) F1 = (m0q1q2/4p R3)*[c2*R21 - <R21,v2>*v2 - v1(R21v2)] +

-(m0q1q2/4p R)*v2/t .


Let us observe that we have left in the RHS of (11) the term v2/t because we shall drop very soon the hypothesis that the considered motions are uniform ones, in the assumption that (11) will continue to give, at least in first approximation, the correct indication for the required interaction formula. Moreover, that one can find the previous formula in the analogous expression for the interaction between two current elements, after the substitution


qv = I*ds


and the other natural adjustments(7), which is necessary for the calculation of the interaction between two current carrying circuits after integration (but, as everybody who knows Marinov's work is well acquainted to, with integrations which are difficult in the "corners" of the circuits, and the trouble to decide where the obtained forces have to be applied).


Anyway, let us go on with our argumentation, and remark that it is quite obvious that the two forces F1 and the analogous F2 do not satisfy the action-reaction principle (of course, with the exception of particular cases, for instance the one in which the two charges move with the same velocity along parallel lines), and that this result does not concern the difference between the classical and the relativistic treatment of the question: in other words, this "violation" is a simple consequence of the specific analytical form of these two forces, and has nothing to do with the relativity of simultaneity(8). But of course, this violation is not dramatic for relativity, which denies the absolute meaning of simultaneity, and so does not worry in principle if these two forces, calculated in the same instant in some reference frame, but not in another!, are not equal and opposite. Moreover, that this kind of violation is very well known, but that it is dealt off either with the aforesaid relativistic arguments concernig the meaning of simultaneity, or with remarks which regard the "infinitesimal" expression of (11) in the case of current elements; this expression is then considered just a mathematical artifice, which receives a physical meaning only after integration along closed circuits(9).

In the next section we shall show how one could think instead of making use of (11) in the general case of a discrete set of single point charges, in this way avoiding the argument concerning closed circuits; but we shall also prove that, the observed violation notwithstanding, one cannot hope to make use of this discrepancy in order to get a "perpetual motion" electromagnetic machine.



3 - About the interaction between electric charges which move around a circle


Everyone who has seen the Testatika machine at work, has observed a turning wheel, connected with two capacitors, and has thought that perhaps different parts of the wheel are (differently) charged, and that the wheel would maintain its motion just because these parts interact the one with the other in such a way to get a significative resultant force in the direction of the motion.

Let us now then suppose that the two charges q1 and q2 of the previous section are fixed on a wheel of radius D, and that this wheel is put in motion with some angular velocity w . In other words, that the motion's equations for the first charge q1 are for instance:


x = Dcos(wt), y = Dsin(wt) ,


while those of the second charge q2 are:


x = Dcos(wt+j ), y = Dsin(wt+j ) .


Well, we can calculate the interaction force F1 that in each instant t the charge q2 exerts on the charge q1, and the analogous force F2 that the charge q1 exerts on the charge q2. What we are interested in, for our purposes, is just to look at the projections of these two forces on the tangents t1 and t2 at the wheel in the points in which q1 and q2 have been placed, let us call these two projections pr1(F1) and pr2(F2) .

Leaving apart the constant coefficients which appear in (11), it is quite obvious that the two vectors pr1(R21) and pr2(R12) equilibrate each other, and that this is still true for the other two terms pr1(v2/t) and pr2(v1/t).

What is less obvious is that this compensation holds even for the two residual parts of the given forces, namely that also


pr1(<R21,v2>*v2 + v1(R21v2))




pr2(<R12,v1>*v1 + v2 (R12v1))


equilibrate each other.

Anyway, in conclusion one finds that pr1(F1) and pr2(F2) have equal intensities, and are directed one in the motion's sense and the other in the opposite one. Thus, as we have already announced, there is no hope to foresee in this way the theoretical possibility that the wheel would maintain its motion, at least as long as Maxwell-Lorentz electrodynamics gives the correct result for the studied electromagnetic interactions.


One could of course think to do some modifications in the previous set-up, for instance to put one of the two charges at a distance D' < D from the center of the wheel, but the conclusion would not change (of course, in this case the intensities of the projections are exactly in the ratio D'/D). In other words, even in this case the two projected forces would equilibrate one another, and so neither this configuration could work as a perpetual motion machine.


We can add that, if one is not satisfied with just one wheel(10), one could think to introduce two wheels, one standing up and the other moving, or both moving either in the same verse or in opposite ones, but again in all these cases does not appear any significative discrepancy in the computation of the projections of the forces which could suggest the theoretical possibility of a perpetual motion. In order to be more precise, one should say that in this case one gets in truth some discrepancy in the projections of the forces, but what happens is that these discrepancies change from point to point and from instant to instant, in such a way that their global balance is equal to zero!


One final remark could concern instead the possible distinction between the classical and the relativistic point of view, that is to say to introduce the hypothesis that the given reference frame is endowed with an "absolute velocity" w. As a matter of fact, it is clear that the computations performed above hold only in an aether frame, and that one should take into further account, from the classical point of view, the absolute velocity of the reference frame in which computations are made, in order to eventually get the correct expression for the required electromagnetic interactions. In other words, one could think to build up a sort of electromagnetic wind-mill, which could maintain its motion at expenses of the supposed absolute velocity w, and then exploit in practice the probable absolute velocity of which the Earth itself is endowed(11). Well, even in this new more complicated situation, the previous conclusion does not seem to change: more laborious computations would show in fact that the introduction of w would give in truth some discrepancy in the projections of the forces, but that these discrepancies would change, as before in the case of the two wheels, from point to point and from instant to instant, in such a way that again their global balance would be equal to zero.


I tried to think of other possible configurations, for instance to let change the sign of one of the two charges from one point to another, in order to get projections which are always favourable to the motion (this idea could work even in the case of the two wheels, is this maybe the Testatika's secret?!(12)), but even this possibility seems rather impracticable, firstly for technical reasons, but more important because any such device, which would work after all at expenses of the absolute velocity, would very likely produce in any case the unpleasant consequence of slowing down this velocity.


At this point, what to say more?, if not that the author would be pleased to see in the next future other contributions on this argument, either suggesting different configurations of the machines, or computations made in different electromagnetic theories(13), which could perhaps throw even a small gleam of light in the direction of the possibility of building up perpetual motion machines, which were for so long dreamed of and sought-after by the very dear friend Stefan...





(1) For instance, one could say that all the successive trouble of the physical interpretation of Quantum Physics was due to the cancellation of the concept of the aether, which indeed could have been supposed to be the probable responsible for the so-called duality, the interference of elementary particles, and so on.


(2) See for instance his "The Thorny Way of Truth", Part VII, p. 325.


(3) Like for instance seemed to think in 1988 the Editor of Modern Physics Letters A, who wrote in a letter to Stefan Marinov: ''It is a pity that the TESTATIKA machine is connected to the electrical network of Linden...'', "The Thorny Way of Truth", Part IV, p. 244.


(4) Apeiron, 4, 1997, p. 83. Of course, we are talking in this paper just of closed cycles, and not of open ones, in which cases the situation could change indeed.


(5) But this does not mean that classical and relativistic electromagnetism give always the same (physical) result: for instance, in the case of a circuit one cannot just make a simple addition of all the contributions of the single elementary charges (see for instance the already quoted ''Symmetries and Asymmetries...''). For this reason, any claim that a "classical" deduction of these expressions provides a classical foundation for the whole of electrodynamics (in the sense that even relativistic electrodynamics would then be provided of a classical foundation) seems wrong (for a different opinion see: ''A Classical Foundation for Electrodynamics'', T.G. Barnes et al., Creation Research Society Quarterly, 14, 1977, pp. 38-45).


(6) But let us explicitely remark that the already recalled distinction between classical and relativistic electrodynamics does not simply consist in a different degree of approximation, which would matter only in the high-velocity cases (or great distances), since it is in the very definitions of the charge density and of the density current - which can or cannot take into account space contractions and time dilatations - that this distinction strongly relies.


(7) See for instance Stefan Marinov's "Classical Physics", Part V, p. 73 and following.


(8) See also the article "Newton's Third Principle in Physics" of P. Cornille in this same volume ["Physics as a Science", Edited by G. Galeczki, P. Marquardt, J.P. Wesley, Proceedings of the International Meeting on Empiricaly Correct Science, Cologne, Germany, August 25-30, 1997, Hadronic Press, USA, 1998 - , pp. 93-130].


(9) See for instance D. Halliday and R. Resnick, "Physics for Students of Science and Engineering", II, Chap. 34, who remark in a footnote that the infinitesimal espression of the so-called Biot-Savart's law does not satisfy the action-reaction principle, but that this disadvantage disappears after integration along a closed circuit.


(10) As a matter of fact, the Testatika machine seems to have indeed at least two wheels.


(11) On this argument one could write a lenghty paper, but at least one thing should be said, and that it is quite unlikely that the Earth is endowed with a "relevant" absolute velocity. As a matter of fact, even if one could give ad hoc explanations for many of the experiments which have been thought to confirm relativity, like for instance the famous Michelson-Morley experiment, everybody who believes in the aether should agree that too many other experiments have been done in order to put in evidence possible effects of this absolute velocity, but that no one of them has given a significative result (think one for all of the famous Trouton-Noble experiment). In view of this fact, and of the recognized presence of only "small" effects, one should stop to look for ad hoc explanations for every single experiment, and begin to consider the possibility that this absolute velocity is very small, if not straight away zero, at least on the Earth's ground. This author believes that it is very likely near the truth the hypothesis that H.C. Hayden has put forward in his ''Experimentum crucis'', Galilean Electrodynamics, 1, 1989, pp.10-11, that is to say that the only relevant absolute velocity of the Earth is that of its diurnal rotation. This point of view agrees with some cartesian intuition, the Earth would be dragged by the aether in its motion around the Sun, but its diurnal rotation would have the meaning of an absolute velocity (and let us observe that we would be in presence of a dragging aether, and not of a dragged one, like in the well known Stoke's theory. This absolute velocity could be also the very cause of the Earth's magnetism, due to a charged sphere which is in absolute rotation in the aether (about this question see for instance Stefan Marinov's ''Earth's rotation is the cause for its magnetization'', Il Nuovo Cimento, 19C, 2, 1996, pp. 215-223).


(12) George Galeczki has expressed the opinion that the Testatika machine could be an open system, which could collect free ions from the atmosphere - a fact which could be supported by the claimed sensitivity of the machine to meteorological conditions. He then suggested to test the functioning of Testatika in vacuum (for example between 1 atm. and 10-6 torr.).


(13) For instance, the electrodynamics of Weber and Wesley [for more information, see the link N. 34] relies upon a force-law between moving charges which satisfies Newton's third law and energy conservation in the case of charges forming a closed system, and so even in this theory one could not foresee the possibility of electromagnetic perpetual motion machines.




Umberto Bartocci

September, 1997