THE ELECTRODYNAMIC FORCE Introduction. The force between two long rectilinear parallel current-carrying conductors will now be described in a non orthodox way (ignoring by the time being the mainstream description based on the magnetic field or more generally on the Maxwell laws of electromagnetism).
Let X be the direction along which both conductors proceed. Let Y be the direction perpendicular to X joining the conductors. To simplify it will be supposed that the conductors are made just by static protons and by an equal number of moving conducting electrons. It will also be supposed that both conductors are ‘homogeneous’ meaning that in all microscopic segments of length Dl of a conductor there is (macroscopically speaking) the same number of electrons, the same number of protons and the same current intensity as in any other segment of length Dl of the same conductor. NOTE 11-A In a typical wire conductor, besides the current carrier electrons, there are of course a great number of other electrons that remain bound to the wire atoms and don’t participate in the current in the sense that their average velocity is zero. The electric influence of these bound electrons on an external charge q will be assumed to be cancelled by the influence of an equal number of static protons of the wire. Therefore, assuming that all pieces of a wire conductor have an equal number of electrons and protons, then to give a full account of the force produced by the conductor on an external charge q it is sufficient to add to the force of the moving "conducting" electrons (i.e. the current carriers) the force produced by an equal number of "static" protons. To avoid second order complications it will also be supposed that, in both conductors, the conducting electrons have small (non-relativistic) speeds. Let vE be the lab speed of all the conducting electrons of conductor #1 and ve the lab speed of all the conducting electrons of conductor #2. In what follows upper case letters will be used to characterize the charges of conductor #1 and lower case letters for those of the current element of conductor #2. Note: in a real wire conductor not all the pertinent electrons (the current-carriers) have the same speed. It can only be said that those pertinent electrons have an average speed v. But since it happens that the predicted force between these simplified lines of current has a linear dependence on the speed of the electrons, it is therefore licit to extend the results to more realistic conductors in which the current carrying electrons have a wider variety of speeds (by just considering that the original conductor is the addition of a big number of conductors each of which has electrons of a specific speed). Suppose by the time being that the lab (that is here the reference frame of description) is at rest in the aether. Let FPp be the y component of the total force exerted by all the protons of conductor 1 on the protons of the current element 2. Let FEp be the y component of the total force exerted by all the electrons of conductor 1 on the protons of the current element 2. Let FPe be the y component of the total force exerted by all the protons of conductor 1 on the electrons of the current element 2. Let FEe be the y component of the total force exerted by all the electrons of conductor 1 on the electrons of the current element 2. The y component of the total force exerted by conductor (1) on the current element (2) is therefore: [I-11-1] F = FPp + FEp + FPe + FEe Consider first the case in which the pertinent charges of the current element and the pertinent charges of the rectilinear conductor are all at rest in the lab. In mainstream electromagnetism, since Coulomb’s law (written for example in CGS units as) Q.q/r2 gives the force between two charged particles at rest, whose electric charges are respectively Q and q and whose distance is r, it is straightforward to calculate/integrate that for charges at rest in the frame of description: The total force that the Q-type charges of a uniform rectilinear infinitely long conductor exert on the q-type charges of a small current element of a second conductor whose distance to the first conductor is "y" can be expressed by: [I-11-2] FQq[0] = (2/y) lQ lq Dl where lQ is the linear density of charge of the Q-type charges of the first conductor, lq is the linear density of charge of the q-type charges of the current element at the second conductor, Dl is the length of that current element (target of the force being measured) and where it has been supposed that both the Q-type charges of the first conductor and the q-type charges of the second are at rest in the lab. (It is assumed that a linear density of charge l is of negative sign if its overall charge is of negative sign, e.g. for a line of electrons). The force between two long rectilinear parallel currents can be predicted if the following hypothesis is made: A uniform infinitely long rectilinear line of Q type charges of speed vQ (along the X axis) exerts on a group of q type charges that move at speed vq parallel to the first a force whose Y component (i.e. perpendicular to the straight line of the Q type charges) is given by: [I-11-3] FQq[vQ, vq] = FQq[0] (1 – ( k1 vQ2 + k2 vq2 + k3 vQ vq) ) where k1, k2 and k3 are constants (with dimensions of speed –2) to be determined by the model, and where FQq[0] is the force given in [I-11-2] corresponding to vQ = vq =0. As explained below the expression [I-11-3] must be considered only an approximation valid for speeds vQ and vq much smaller than the speed of light c. Therefore the total force expressed in [I-11-1] (considering both the protons and the electrons) acting on the current element of conductor (2) due to the indefinitely long conductor (1) is: [I-11-4] F = FPp + FEp + FPe + FEe = = FPp[0] (1 – ( k1 vP2 + k2 vp2 + k3 vP vp) ) + FEp[0] (1 – ( k1 vE2 + k2 vp2 + k3 vE vp) ) + + FPe[0] (1 – ( k1 vP2 + k2 ve2 + k3 vP ve) ) + FEe[0] (1 - ( k1 vE2 + k2 ve2 + k3 vE ve) ) = and since the protons of both conductors are at rest in the lab it is vP = vp = 0 and therefore [I-11-5] = FPp[0] + FEp[0] (1 - k1 vE2 ) + FPe[0] (1 - k2 ve2 ) + FEe[0] (1 - ( k1 vE2 + k2 ve2 + k3 vE ve) ) then according to [I-11-2] = (2/y) Dl [lP lp + lE lp (1 - k1 vE2 ) + lP le (1 - k2 ve2 ) + lE le (1 - ( k1 vE2 + k2 ve2 + k3 vE ve))] and recalling the assumed electric neutrality of both conductors implies that: l P = -lE and lp = -leand calling to simplify l1 = lP = -lE and l2 = lp = -le then F = (2/y) Dl [l1 l2 - l1 l2 (1 - k1 vE2 ) - l1 l2 (1 - k2 ve2 ) + l1 l2 (1 - ( k1 vE2 + k2 ve2 + k3 vE ve))] =
= - (2/y) Dl l1 l2 k3 ve vE = [I-11-6] = - (2/y) Dl k3 I2 I1 where it has been taken into account that I1 = l1 vE is the intensity of the current of the first conductor and I2 = l2 ve that of the second. When both currents I1 and I2 are of the same sign (semi direction) it is an experimental fact that the force is of attraction. That implies that the constant k3 introduced in the hypothesis [I-11-3] must be positive. But since (in CGS units) the force exerted by an infinitely long rectilinear conductor with current I1 on a unit length of a second parallel conductor with current I2 is known to be given by: [I-11-7] F = - (1/c2) (2/y) I2 I1 (where c is the speed of light) then the constant k3 must have the value: [I-11-8] k3 = 1/c2 NOTES 11-B .- It is interesting to notice that if both conductors are given a constant velocity V along the direction X and therefore the new speeds of the pertinent charges (including the protons) are all increased by V according to: vE à vE +V, vP à vP +V, ve à ve +V, vp à vp +V then, making those substitutions in [I-11-4] and making the same assumptions (l1 = lP = -lE and l2 = lp = -le) as before, all the terms containing a V cancel out and the same expression [I-11-6], independent of V, is obtained. - Since the force [I-11-6] between the conductors only depends on the constant k3 of the hypothesis [I-11-3] but not on k1 or k2 , it could be asked, why not assume k1=k2=0 and enounce the simpler hypothesis FQq[vQ, vq] = FQq[0] (1 –k3 vQ vq). The reason is that the hypothesis must be consistent with other developments of the model, and for example it has been calculated in an Annex to this section that, assuming the redistribution paradigms of the model, the y_component of the force that a uniform infinitely long rectilinear line of Q type charges at rest in the lab exerts on a q type charge that moves at a speed vq parallel to the line is not constant but behaves indeed according to FQq[vQ, vq] = FQq[0] (1 – k2 vq2). - It could similarly be argued that a simplified instance and more "reasonable" hypothesis for the force [I-11-3] would be to assume k1=k2=k and k3=-2k and therefore that: FQq[vQ, vq] = FQq[0] (1 – ( k vQ2 + k vq2 - 2 k vQ vq) ) = FQq[0] (1 – k (vq - vQ)2 ) which only depends on the relative speed |vq-vQ| of the pertinent charges. But this hypothesis (that was the one proposed in older versions of this work) requires that k<0 (to give a correct prediction of the sign of the force [I-11-6] between parallel conductors). But as said above (see the fore mentioned Annex), the more plausible paradigms of the model predict that, for vQ=0, such force is of the type FQq[0, vq] = FQq[0] (1 – k vq2) with k>0. Although a force that does not depend only on the relative velocity of the interacting bodies but also on the individual (absolute) velocities themselves of the bodies might be considered unacceptable by today's orthodox physics (that believes in a strict principle of relativity), from the point of view of an aether model it is the most natural result to be expected. Furthermore, according to the model the correct interpretation of the hypothesis [I-11-3] should be: It must be understood that the speeds vQ and vq (with their sign) appearing in the hypothesis [I-11-3] are the x-components of the absolute velocities of those particles in the aether, (the direction X being that of the straight line joining the Q-type charges of the long line of charges). - Finally, and most important, the hypothesis [I-11-3] must only be considered an approximation for speeds vQ and vq much smaller than c. Other studies of the model (e.g. the Annex A) suggest that the force, that a body at rest in the aether exerts on a body that moves in it at speed v, diminishes with v according to an exponential law of the type F[0] Exp[-k2 v2], rather than with a quadratic law of the type F[0] (1 – k2 v2). Intuition invites to extrapolate this exponential behaviour to the hypothesis [I-11-3] that, to make it valid also for high absolute speeds should be modified as follows: Generalisation of the hypothesis [I-11-3] proposed above: A uniform infinitely long rectilinear line of Q type charges of speed vQ exerts on a group of q type charges that move at speed vq parallel to the first a force whose component perpendicular to the straight line of the Q type charges is given by: [I-11-3b] FQq[vQ, vq] = FQq[0] Exp[- ( k1 vQ2 + k2 vq2 + k3 vQ vq) ] where vQ and vq are the components along the direction joining the Q-type charges of the absolute velocities of the corresponding particles k1, k2 and k3 are constants (with dimensions of speed –2) to be determined by the model, and where FQq[0] is the force given in [I-11-2] corresponding to vQ = vq =0. ----------------------- A controversial feature of this description is the prediction of a non-zero force between a long neutral rectilinear wire with current and a charge at rest in the lab. Suppose also that the lab is at rest in the aether. According to mainstream Electrodynamics the force is zero because the magnetic field due to the current produces (according to the Lorentz-force) a zero force on a charge at rest. But according instead to the description proposed above the force exerted by the long neutral conductor with current I1 on a charge e at rest relative to the conductor (i.e. to the protons of the wire) is no longer zero. Consider first the force of the long current-carrying wire on a charge e moving parallel to the wire with speed ve at a distance y from the wire.
but according to [I-11-3] and since the protons of the conductor are assumed to be at rest (vP=0) : [I-11-9] F = FPe[0] (1 - k2 ve2 ) + FEe[0] (1 - ( k1 vE2 + k2 ve2 + k3 vE ve) ) But the force that a rectilinear long line of protons at rest exerts on a single test charge +e also at rest can be written (in analogy with [I-11-2] but replacing le Dl by e) as [I-11-10] FPe(0) = (2/y) lP e and therefore the force [I-11-9] of both the protons and the electrons of the wire on the test charge e takes the form: F = (2/y) e [lP (1 - k2 ve2 ) + lE (1 - ( k1 vE2 + k2 ve2 + k3 vE ve) )] but the assumed electric neutrality of the conductor implies that lP = -lE and calling l1 = lP = -lE then
[I-11-11] = (2/y) e l1 [k1 vE2 + k3 vE ve] that for ve = 0 has the non-zero value: [I-11-14] F(0) = (2/y) e l1 k1 vE2 where the constant k1 has not yet been determined by the model. If (instead of CGS) MKS units had been used in the calculus, this force F(0) would adopt the expression: [I-11-14b] F(0) = (2/y) e l1 k1 vE2 /(4 p e0) It has recently been found that other authors support independently the existence of this force (that is non-orthodox in mainstream Electrodynamics) . See for example: A.K.T. Assis et al [1]. In an earlier paper A.K.T. Assis [2] deduced such force from Weber’s electrodynamics. ----------------------- It is recognized that, although basically correct, the prediction made by the model of the force between parallel conductors is by itself of limited interest since it corresponds to a very specific non-fundamental force. It is believed nevertheless that a description based in a unique fundamental "electro-dynamic force" between 2 charges moving relative to one another can describe all the interactions between moving electric charges using Galilean relativity and without the need to introduce the concept of the magnetic field. To be consistent with the other paradigms of this work such electro-dynamic force must be an aether-implemented force and is therefore expected to behave according to the features already described in this work. In particular, since the aetherinos carrying the information of the source charge Q to the target charge q do not travel at infinite speed but at a plurality of finite speeds, the force suffered by q at the epoch t due to the presence of Q will depend not so much on the state of Q at the epoch t but on the history of Q (previous to the epoch t). Since this history can be as varied as can be imagined, the expression of the so called "electro-dynamic force" will only be analysed for simple cases of practical interest and for which the source charge has had a simple history (e.g. has always been moving at constant velocity, or has been moving in a circle at constant speed, or has been oscillating in some simple way, etc). Coming next ... The next step of this study of "magnetism" will be to try to deduce the general expression of the force produced by a current element on a test charge that moves in a general way relative to the current element and not necessarily parallel to it. It is believed that this will lead to the right prediction of the "Ampere expression" that gives the force between 2 circuits of current of any shape. This "Ampere expression" is interpreted in official Physics as the consequence of applying both the Lorentz force and the Biot Savart Law. It is advanced that the thesis of this section is to show that the official interpretation is an unnecessary complication and that "the magnetic field is an unnecessary concept from the point of view of fundamental Physics since all the forces between moving charges can be described by a unique electro-dynamic force (explained by this aether model) that depends on the relative velocities of the interacting charges (and is therefore different from the electric force of official physics)." calculus based on the aether model home REFERENCES [1] A. K. T. Assis, W. A. Rodrigues Jr., and A. J. Mania, The Electric Field Outside a Stationary Resistive Wire Carrying a Constant Current, Foundations of Physics, Vol . 29, No. 5, 1999 See their equation (22) (the paper is available also at http://www.ifi.unicamp.br/~assis/Found-Phys-V29-p729-753(1999).pdf[2] A. K. T. Assis, Phys. Essays 4, 109 -114 (1991). |
