It is also often claimed that the Lorentz force law is consistent with Amp?re's force law. An obstacle to achieving agreement on this point has been the fact that the two laws have different shape as well as properties. While the Amp?re force law obeys Newton's third law, so does not the Grassmann law. The problem is avoided by claiming that they are equivalent provided a complete circuit is being studied. A strong argument for this assumption is that all circuits are closed.
It may easily be raised objections against the above claims. At first, if studying the very basic paper by Grassmann that is claimed to contain the derivation of Bi?t-Savart's law, that proof does not appear to be very clear to the reader. If regarding the result of Grassmann as a rudimentary form of Bi?t-Savart's law, that claim may be acceptable. However, if following the proof chain by Grassmann, it is full of elementary mathematical mistakes.
Assis and Bueno have made an effort to prove that both Amp?re's law and Grassmann's law lead to the same result, when a whole closed electric circuit is being regarded, as for example Amp?re?s bridge. However, in this paper the claim by Assis and Bueno is questioned through a rigorous control of the derivations by Assis and Bueno. Regrettably, it appears that they were not correct. A rigorous check of the derivation by Assis and Bueno with respect to one part of the circuit shows that his calculation has not been correctly performed. Hence, the whole claim by him can be discarded. All the steps of the derivation of the force within a selected section of Amp?re's bridge is shown, the selection originally being made by Assis. Another objection against the claims by Assis and Bueno lies on a more principal level. Firstly, no two formulae are equal if not being equal in every point. Secondly, what is usually regarded as a closed circuit contains a mathematical singularity at the point of the voltage source. There, the voltage steps Heaviside-like once at every revolution. To conclude, the claim that Amp?re's law and the Lorentz force are mathematically connected must be definitely rejected.