**4. Conclusion**

The two respective currents have been thoroughly analyzed, using Coulomb's law, taking into account the effects of propagation delay and the Special Relativity Theory. The way the effects of the propagation delay have been derived is that of this author in a paper 1997 [1], which differs fundamentally from the traditional interpretation, as that of Feynman [21] and Jackson [22]. This author has been successful in showing what the fallacies are [2]. Basically, Feynman committed a mathematical fault with respect to the calculation of the propagation delay, when deriving the Liénard-Wiechert potentials [21]. In the 1997 paper by this author [1] it was crucial to the success of Coulomb's law that the effects of propagation delay had been correctly derived, both with respect to the "sending charges" of the "first conductor" and to the "receiving charges" of the "second conductor". The first effect gives account for the depend‐ ence of the first current in the expression for the electromagnetic force, the second one for the

second current in that same expression, but of course, it is arbitrary, which one is treated as sending or receiving current. This treatment makes it possible to see a product between two currents in an application of Coulomb's law, and, hence, there will be no need for the Lorentz force. When that analysis has been done, it remains to take into account to the effects of the Special relativity theory, especially the Lorentz transformation of lengths. Since this effect is related only to the relative movements of the two coordinate axes, and has nothing to do with the propagation delay an observer faces, it may be multiplied straightforwardly to the effect of propagation delay. The expression for the force between the two currents are compared to the expression that Ampère arrive at and to Lorentz' force law. Thereafter follows a discussion of the pro et contra of respective model. The result in this article is based on two wellcorroborated natural laws: Coulomb's law [23] and the Special relativity theory. Ampére in turn, derives his law in a strictly empirical sense [32], searching for similarities with Coulomb's law. However, since in his time, the individual electron had not yet been discovered and, secondly, the Special relativity theory had not been defined. Hence, Ampère had no other choice than to establish a fairly good empirical law. Lorentz (or first: Grassmann) faced the same problem, but his formula was derived through evident mathematical faults [15]. Nb. This term 'Ampère's force law´ is not the same law as that Jackson denotes Ampère's law. Please cf. the original paper by Ampère [13] and Jackson [24]. This would make it possible to create a continuous, logical chain, from the findings by Ampère to the established Maxwell electro‐ dynamics. Assis has made an effort to prove that both Ampère's law and Grassmann's law produce the same result, when the forces within Ampère's bridge are being derived [15]. Admittedly, he concedes that they are not equal at every point, but in the integral sense, when a complete, closed electric circuit is taken into account. From a strictly mathematical pint of view, however, if two functions are not equal at every point, they don't express equal functions. This is taught in the most basic undergraduate courses. Anyhow, stating that all electric circuits are necessarily closed, he arrives at the conclusion that both laws are equally applicable on electric circuits.

To conclude, all three of them: Coulomb's law, Lorentz' force law and Ampère's force law can account for the attractive force exerted between two parallel electric conductors, carrying a current in the same direction. On the mere basis of the shape of the functions, it is not possible to decide, which one is best expressing physical reality, since the very measurements of currents involves a theory for the force between currents in the context of traditional meas‐ urement instruments. Hence, for every choice of model, there will necessarily appear a coupling constant that makes the measurements fit with the theory. Therefore, it remains to make a qualitative analysis of the three models. Above it has already been explored that Coulomb's has been used in a very strict manner, applying only the effects of propagation delay and the Special Relativity theory, whereas Ampère's force law is only expressing an empirical estimation of the force and the Lorentz force has been fallaciously derived, using Ampère's force law.

Hence, the conclusion to be drawn is that Coulomb's law gives the most comprehensive explanation to the force.
