The Clausius force in electricity

[1] the Clausius force in electricity George mpantes mathematics teacher I have read Electromagnetic theory O’Rahily (1965) Concepts and methods of theoretical physics R.B.Lindsay (1969) A history of the theories of ether and electricity E.Whittaker (1960) Theory of electron’s A.Lorentz (1909) To φαινόμενο του ηλεκτρισμοφ Γιώργοσ Μπαντέσ www.mpantes.gr Maxwell treats electricity as an incompressible fluid wich fills all space…Clausius Clausius force between two electrons, does not refer to electric or magnetic fields, it is a force from a distance, without an ether, and was proposed in 1897. Having the f Neumann potential V= –ii΄dsds΄/r (1) [2] for two elements ids, i΄ds΄ and with the relation υe=cids between charge and current with υ the velocity of the charge e along ds and c the number of units of static electricity that is transferred of current in the unit if time, the Clausius potentials are V=- ee ΄υυ΄/c2r and adding the electrostatic U=ee΄/r (2) we have 𝑀= 𝑒𝑒΄ 𝜐𝜐 ΄ 𝑟 1− 2 𝑐 the electrodynamics potential of their mutual action … 𝟑 Now we may express the force in Lagrangian form Fi  M d M ( )  xi dt  i or finally Fx  ee' [ ( rx)(1    x  ' x / c 2)   ' x ( r   ' r ) / c 2  r f ' x / c 2] 2 r (4) This is the Clausius force giving the force on e. So simple, it’s foundations are in Ampere’ force. We have thus found an expression for the law of force between two particles, without an intermediate agent, an ether, involving their simultaneous distance and velocities and the acceleration f of the particle exerting the force. When the velocities and acceleration are zero, the formula reduces to Coulomb’s law. It will be observed that υ and υ΄ do not occur symmetrically. Thus , even when the accelerations are negligible so that there is no radiation, action and reaction, are unequal. Previously to Clausius the accepted formula , that of Weber or that of Riemann, involved only the relative velocity of the two particles. The innovation of Clausius consisted in importing absolute velocities, i.e. in conjoining the eelectron theory and the ether theory (electron plus aether). The Clausius force is instantaneous, and needs an absolute reference system. "Equations derived purely empirically are not necessarily the exact expressions of actual physical law. ...and it will always be possible to give the equations another form, always provided that these changes do not bring about a noticeable change in the results extracted from the experiment... L. Lorenz" [3] That is, by expressing the E and H fields as a function of the potentials, Maxwell's equations become (4). Conversely in 1897 Levi-Civita derived from (4) the Maxwell equations without reference to the displacement current at all. (O'Rahily) Thus the equivalence of Maxwell's equations (in vacuum) and (4) is mathematically complete! But the deep problem of physics which has always been the physical relationship of ether with matter is now becoming more mathematical. However defective, this formula of Clausius is the pioneer and model of the LiénardSchwarzschild force –formula which, implicitly and explicitly, is the condensed statement of the form of the electron theory which is almost universally accepted today, which in particular is accepted by all followers of Lorentz and Einstein. (O’Rahily) Propagation of action The idea of this article is to connect the Clausius and Lorentz assumptions for investigating the role of the aether, through the contribution of Lorentz theory to Clausius force and conversely. The Lorentz theory is based on the four aetherial equations of Maxwell together with the equation that determines the ponderomotive force on a charged particle. For us here, the two forces, Clausius and Lorentz is the essence of electrodynamics, as these are measured and verified. Their fundamental difference is the aether that is present in Lorentz theory and force, but no in Clausius theory and force. How can we reconcile the two forces, the one force to guarantee the other passing over the theory of ether? First we investigate the variation of Clausius’ force under the influence of Lorentz theory of ether. We shall now reject Clausius’ assumption that electrons act instantaneously at a distance, and replace it by the assumption that they act on each other through the mediation of an ether, which fills all the space and satisfies the equations o Maxwell. This [4] modification can be effected in Clausius theory without difficulty (Whittaker). For as we know, if the state of Maxwell’s ether at any point is defined by the electric vector E and the magnetic vector H these vectors may be expressed in terms of potentials A and φ by the equations 1 𝜕𝐴 𝜕𝑡 𝐸 = 𝑔𝑟𝑎𝑑 𝜑 − 𝑐 𝑎𝑛𝑑 𝐻 = 𝑐𝑢𝑟𝑙𝐴, And it is known that the potential of Lorentz force exerted from the electron e’ on the electron e moving at a velocity υ in field E and B is U  e(  .A c )......... ......( 5) Now the functions A and φ for the force on e will be produced from the Wiechert propagated potentials of charge e΄ and may in turn be expressed by the equations, (where *r+ the distance of e’ and *υ΄ + it’s velocity in that refers to the instant t΄=t-r/c A e'[ '] c[r ](1[ 'r ]/c  e' [r ](1[  'r ]/c ) So U of (5) is L ee' .[ '] ]......... .......... .......... ..(6) [1  [r ](1[ ' r ]) c2 where the brackets are the retarded values of the symbols, including in time t΄ =t-r/c . This result is due to Schwatzschild who call L the ‘electrokinetic potential’. It corresponds to the electrodynamic potential U of Clausius (3). Comparing these formulae with those given above for Clausius potentials (2,3), we see that the only change which is necessary to make in Clausius’ theory is that of retarded potentials in square brackets in the way indicated by L. Lorenz (βιβλίο μου «το φαινόμενο του ηλεκτρισμοφ , σελ.171) so, to consider Clausius action as propagated and not instantaneous. Then the existence of ether of Lorentz theory does not affect the equations of potential, viz. the force. [5] Ritz makes the following comment: “..this expression L, reduces in first approximation, to the law of the inverse square of the distance. …in this formulae the notion of field does not intervene…” This because this electrokinetic potential of Clausius force will derive the Lorentz force, the contribution of Clausius theory to Lorentz theory. Thus, through Lagrange as formalism, the relation (6) reduces the Lorentz force to the Clausius force (4) by adding the propagated view of the latter's action. The force from a distance, assuming it is propagated at a finite speed, is the same as the contact force! The identity of the two faculties is attained by the view that the electrical action exerted at a distance propagates at a finite speed. Lorentz Conversely If we consider now Clausius force as propagated and not instantaneous, then it’s electrodynamic potential M, becomes electrokinetic L, which will produce, through the Lagrangian formalism, the Lorentz’s force! The Lagrangian equations of motion of the electron e are Fi  L d L ( ) dt  i  x i where the potentials φ and A are taken as retarded ( Wiechert )and L denotes the total electrokinetic potential due to all causes, electric and mechanical. Then we have Fx e( L d L ( )  x dt  x 1  Ax 1 Ay 1  Az  e d Ax x y z ) c x c x x c x c dt And as d / dt  / t . (το φαινόμενο του ηλεκτριςμοφ ςελ. 144) we have for the force Fx on e [6] and this is the known Lorentz formula e F  eE  (  H ) 1 c “…..this remarkable result, due to Schwatzschild, shows that Lorentz’s theory resembles the older theories much more than we could at first sight believe…Ritz” So these results are described by Wiechert: “…It is characteristic of the electron theory that it assumes a propagation of the electrodynamic disturbances with the velocity of light in free ether. Hence arises the conjecture that it must be possible to represent the disturbance at any point as the consequence of processes which occurred elsewhere at such previous times as correspond to this velocity of propagation. Since we also assume that all ether disturbances originate in electric particles, we surmise that it is also possible to refer the process to these particles alone as did old theories…” epilogue So the aether is a mathematical aether as the displacement current was a mathematical current. The effect of a disturbance at a point A travels out from it in spherical waves , arriving at a point P in the time AP/c. The nature of the field is such as would arise if each portion of it were constantly emitting disturbances which were propagated from it in all directions with the velocity of light. The ether simply denotes a finite propagation. The final fact in the aether’s existence is the Michelson’s experiment, and the mathematization of physics of electromagnetism is emphasized in the following quotation: “…..It is important to remember that electric and magnetic fields are not directly observed in any experiments; only phenomena in material bodies are observed. The existence of fields in the space surrounding charges is assumed because phenomena can be conveniently described or explained by means of this assumption. In the electron theory we 1 Whittaker. Aether and electricity pag. 395. [7] assume the existence of these fields, but we do not attempt to explain how they are produced or what they consist of. We may if we like regard them as merely auxiliary mathematical quantities introduced into the theory for convenience in attempting to describe phenomena. Most physicists, however, believe that these fields really exist. It has been suggested that electric and magnetic fields are modifications of the ether, a medium filling all space. The hypothesis is not of much use; it is sufficient to suppose that the charges excite the fields in the surrounding space…H.A.Wilson Modern Physics 1928 (O’ Rahily p.225)” George Mpantes mathematics teacher Serres Greece 13/6/17