Yes, there are details needing to be considered. But these details do not overthrow longitudinal force. For invalidating longitudinal force one has to prove that it is zero. Without doing so, these details are only equally possible details needing to be considered, but not the correct and only cause that kills others. By ad hoc I mean for specific situation. Arcing works for multi-arc generator but not for Nasi owski's wire fragmentation. Arcing is specially for this situation or ad hoc for this situation. And all details you mentioned are of this sort. If particular complexities pleases you better, yes they are. Since you formally forbid to consider Lorentz forces from part circuit, Grassmann's expression is compatible with SR only in this particular situation, that is, a closed circuit acting on one current element. Ampere’s and my expressions are equally compatible with SR since they give the same result than Grassmann's expression for the case of closed circuit acting on one current element. Rejecting longitudinal force appears to you straightforward, but it is only your personal judgment. Yes, most people reject longitudinal force. So, this is not your personal judgment but the judgment of all physicists, so what? A judgment is from human mind, not physical reality. 100% people were against Copernicus, but their judgment was wrong and the theory of Copernicus was right. Occam's razor is far more powerful than human mind to shave off non necessary details needing to be considered from physical reality. My formulation is different from theirs. Again, Grassmann's expression is compatible with SR only for closed circuit and not compatible with SR otherwise. When considering Lorentz force between one current element and another, will it still be compatible with SR? That is, you can derive Lorentz forces using SR from the B field of a closed circuit computed using Biot-Savart’s law. But can you derive Lorentz forces using SR from the B field from a part circuit equally computed using Biot-Savart’s law? Obviously not. Then, how can you be so sure that Grassmann's expression is the unique and only law compatible with SR? I know that length contraction of electron flow is questionable for some. Someone else has made the same remark and said that if this were true, then one should see electric field from a coil. I have tried to do experiment in this direction but still not succeeded. It seems to me that length contraction was used to explain the distance between electrons in cathode ray. Was this accepted in physics? Here, http://www.iflscience.com/physics/4-examples-relativity-everyday-life/ they explain that engineers had to account for length contraction when designing the magnets that directed the electrons to form an image on the screen. Without accounting for these effects, the electron beam's aim would be off and create unintelligible images. In the same page, they also explained that if this charged object moves alongside the wire, then it starts to feel the effects of length contraction in the moving electrons. This means that the density of stationary protons becomes larger than the flowing electrons and the metal exhibits a positive charge, causing the object to be attracted or repelled. So, length contraction of electron flow is accepted by at least some physicists and cathode TV set. No.