In ≪Relativistic length contraction and magnetic force≫ I have explained the mechanism of creation of magnetic force from Coulomb force and relativistic length contraction. For facilitating the understanding of this mechanism I used parallel current elements because the lengths are contracted in the direction of the currents. But real currents are rarely parallel, for example, dIa and dIb of the two circuits in Figure 1. For correctly applying length contraction on currents in any direction, we will consider conductor wires in their volume and apply length contraction on volume elements of the wires. Please read the article at PDF Length-contraction-magnetic-force between arbitrary currents http://pengkuanem.blogspot.com/2017/05/length-contraction-magnetic-force.html or Word https://www.academia.edu/32815401/Length-contraction-magnetic-force_between_arbitrary_currents

"Volume elements". Argh; Euclidean geometry strikes yet again! Good luck with that. https://en.wikipedia.org/wiki/Banach-Tarski_paradox Pure mathematics cannot define a volume because it cannot unambiguously define an infinitessimal element of volume, much less a relativistic one for this problem. rpenner here "cleaned my clock" a very long time ago about a similar treatment given in the first chapters Purcell's "Berkeley Physics Series, vol 3 Electricity and Magnetism". It is much more difficult than you think to "really" define the right volume element to represent electrical current density. From rpenner's very valid objections, I can tell you, what you propose will not be easy. The Lorentz roadbed (wire, in this case) has inertia, but shinks and stretches at whim, depending on relative velocities of the charge carriers. The electrons don't necessarily travel at uniform straight line velocities either. You would have to know more details about the process of electrical conduction than anyone who currently thinks they do. Two parallel, current carrying wires is a nice example for demonstration of the power of Special Relativity as it applies to explaining the effect of Lorentz contracted charge densities vs. what we at rest perceive as a magnetic field. But after that, you are best advised to rely on Faraday's notes and Ohm's law whenever working with electric currents.

The basic approach of working out the purely electric forces acting in the proper frame of one current element owing to differential charge densities seen in another current element in motion relative to the first, is ok in principle for slow motions where the fully relativistic retarded field eq'ns are not required. However, your starting assumption that: "When a current circulates, the free electrons move at velocity v and their charge density ρ_ increases according to equation (6) and becomes ρ'_ of equation (7)." is wrong. In the conductor rest frame, conduction charge density is independent of charge drift velocity v, and always equal to the lattice charge density. Otherwise the conductor would acquire a net charge. To make your rather cumbersome approach consistent, it's important to always define velocities relative to the proper frames of both the lattice charges and that of the conduction charges. In the latter case, proper charge density is seen to be *less* than in the conductor rest frame, by the inverse factor to that in your eqn (7). I haven't bothered to plow through the rest of your document after spotting that conceptual error. It may be that one error cancels another in at least some of your findings, but either way best to start again.

Thanks for explaining that. Rpenner once left me hanging on another forum, another thread about that. I've done physics graduate lab work on the Hall effect in semiconductors (charge carriers were "holes"). I understand, the problem is not an easy one. Is a "hole" in a p-doped semiconductor at rest or not? Holes actually may drift as well. Just trying to help.

Thanks. But it seems that physics theories using volume elements give correct laws. Pure mathematical notions may not be suitable for physics.

Richard Feynman has explained in page 13-8 of his ≪The Feynman Lectures on Physics, Volume II≫ why this assumption and he derived correctly the magnetic field of an infinitely long current. See also Steve Adams in page 266 in his ≪Relativity: An Introduction to Spacetime Physic≫.

Referring to the freely available online version of the Feynman lectures on physics vol II, section 13: http://www.feynmanlectures.caltech.edu/II_13.html The expression for lattice charge density ρ'+ in the moving frame S', in terms of the proper lattice charge density ρ+, given in (13.24) is correct. The following on bit leading to (13.25) and (13.26) has to be read rather carefully. You will find my criticism is correct and accords with those expressions. Incidentally, while strictly technically correct, Feynman made a poor and misleading decision leading up to (13.21) there. It suggests a magnetic Lorentz force on a charge moving at velocity v wrt a current carrying wire will be proportional to v² not v, which is nonsense. As I said, the expression is technically correct but an artifact of Feynman's decision to 'simplify' by making conduction charge drift velocity and test charge velocity exactly equal - a poor choice when trying to illustrate a generality. When first realizing this and pointed it out in another forum now defunct, the admin/mod who started the particular thread extolling that expression, said admin/mod went ballistic and basically shouted at me "Feynman is right in all things physics!!" It ended with my asking and, after a shocked silence of several days, being granted deletion from that forum which went belly up a few months later. Such is life.

You win (by finding fault with a Feynman lecture) Try as I might, I could never find fault with Feynman, but I'm pretty certain, he must have made an error in analysis somewhere, sometime. I was never able to find even one. Possibly the closest I ever came to finding what must have been a mistake was Feynman's path (line) integral formalism, which I read somewhere involved integrating paths "around all of the moons of Jupiter", and around each of his toes and his bongo drums as well, no doubt. Or at least, the idea seemed silly the first time I read it, at least.

No-one is infallible but there are many who uncritically accept every word of authority figures as gospel Truth. Most times it works ok but a bad outlook to think most times = always. Feynman was of course an outstanding genius and any slip-ups extremely rare. Yes that formalism has a certain weirdness to it. Allowing even paths back in time. Feynman afaik when pressed as to whether particles 'really' took all possible paths, was always noncommittal. A wise position to take. Another famous thought experiment attributed to Feynman but actually originating much earlier with iirc J.J.Thompson was his so-called 'Feynman disk paradox' (Google it). I disagree with his conclusion, but cannot prove it wrong. Only delicate experiment could decide. The notion that independent static fields generate a physically real field momentum density g ~ E x B is imo absurd. He along with nearly all physicists make the assumption conservation of angular momentum is sacrosanct - therefore crossed static fields 'must' carry an actual Poynting type momentum flow. But this is a digression.

Feynman uses this example to illustrate the notion that magnetic field is a moving electric field. So, he has not to be rigorous. But the idea that magnetic field is a moving electric field is correct. I even find a Youtube that explains magnetism this way What you do not agree is that the distance between moving electrons do not shrink. This is a prediction of relativity. A moving ruler's length shrinks. If the ruler has one electron at each end, the distance between the electrons shrinks too. Why would they stay at the same distance if relativity makes them approaching?

'Moving electric field' in the context of currents in conductors, is a misnomer. Properly, there is motion wrt a source, or motion of the source. In a given frame, an E field simply exists. The same for B fields. Very old textbooks on EM or electrical engineering referred to 'moving lines of force'. Especially wrt 'flux-cutting' by 'expanding' and 'shrinking' lines of force in transformer action etc. It was early on realized such a concept led to paradoxes and justly went out of favour long ago. The rest of your post is garbled reasoning. As per first para in #7, my critique is correct and exactly accords with eqn's (13.25), (13.26) in the cited article. Why can you not concede that? Too much personal investment in a pet idea I guess. Be prepared to let go and start from scratch. PS: That YT vid exactly backs what I wrote in #3. You cannot see that??!! PS 2: The narrator made a common error in claiming an extremely low speed of conduction charge motion. Anyone clued up on solid state physics knows better. The current is carried by a very small fraction of 'free electrons' moving at around the Fermi speed of ~ 10^6 m/s. Which discrepancy doesn't matter in this case - only the product of actual conduction charge density and speed matters and the radically different pictures effectively give the same overall result.

You said that the density of negative charge equals that of the positive charge. Correct. In this case, the mean distance between 2 electrons equals that between 2 protons.

In agreement with both Feynman and that YT vid, I wrote the two densities were equal in the rest frame of the conductor i.e. of the lattice charges. Independent of conduction charge velocity. Which is no more profound than demanding conservation of charge i.e. that the total count of charge carriers in a circuit must be independent of drift velocity. A corollary is that the conduction charge density in it's rest frame i.e. proper charge density has to be *less* than in the conductor rest frame.

You mean, in the rest frame of the charge carriers there are *less* electrons than in the conductor rest frame. When no current exists, the quantity of positive and negative charges in the conductor are equal, both charges are in the rest frame of the lattice charges. When current is on, the charge carriers get the velocity v progressively, the frame of the charge carriers moves with them. Since there are *less* electrons in this frame, for an observer fixed in this frame, he see the number of electrons diminishes progressively. Is this correct?

Just as what Feynman wrote and what that YT vid displayed, the proper conduction charge density is necessarily - to accord with SR - lower than in the lattice rest frame by the factor 1/γ. The exact inverse of what you claim in your article. BUT - get this - that is true as locally measured. More on that below. Of course. As all agree. Think this is a paradox that invalidates Feynman, that YT vid, and myself? No doubt. Well it's only an apparent paradox and here's the resolution: As I wrote above, the locally determined conduction charge proper density is indeed lower. And that will hold true everywhere around the circuit. However, a proper application of SR must respect the ground rule that an evaluation between frames must be an inclusive one between two inertial i.e. constant velocity frames. Which requires an observer drifting with the local conduction charges to sum current densities around the entire circuit - in relative motion wrt that observer's instantaneous drift velocity. Suppose the circuit is a circular loop current. What do you suppose the local observer will determine the conduction charge density will then be for a location on the opposite side of the loop? It will be greater than for the lattice charges. The entire sum must and will yield the necessary result total charge number is invariant. If you search long enough online using e.g. 'electric dipole moment of a moving loop current' a link to a non paywall article should come. I have wasted too much time looking. Anyway an illustration in any such will likely show exactly what I have described above - but seen at a glance. On p10 onward, the following article discusses the relativistic electric polarization of a moving magnetic moment, but is mathematically based: http://www.physics.princeton.edu/~mcdonald/examples/movingdipole.pdf The point is - a relativistically induced electric dipole moment carries no net excess or deficit of charge, but does have regions of higher and lower charge density.

Not just "constant velocity" frames, but frames in which no other forces are acting. In a wire conductor of any practical gauge and cross-sectional area, there will also be repulsive forces between charges, but in no case will an electron ever be stationary anywhere in a conductor carrying a current, or even on the surface of a conductor or non-conductor holding a static electric charge. A current carrying wire also produces a magnetic field. If this magnetic field changed at all as a result of electrons taking different paths in a conductor, you would be able to exploit the induction to induce currents in other wires. DC transformers would work just like AC ones, because the path the current takes would fluctuate. This never happens. So, why is it that static charges build up on the surface of the sphere in a Van de Graf generator, but not on the outsides of conductors of a current-carrying wire? No wonder we still don't have a model of conduction that is easy to visualize.

Irrelevant. The key is to evaluate the whole system wrt the instantaneous rest frame of any given observation point. Your point being? The trivial resolution of first case involves nothing more than that any excess charges in a conductor are free to move so as to create an equipotential electrostatic distribution. Surely you know that much. As for the second case, actually there is a very small surface charge generated when a current flows. Easily seen from the pov of conduction charges experiencing a transverse magnetic Lorentz force owing to the circular B field set up in the conductor. Which exactly accords with the pov based on locally determined relativistic charge densities - in keeping with the main OP topic. Search Kirk McDonald's massive site and you will eventually find the relevant article: http://www.hep.princeton.edu/~mcdonald/ PS; found it: https://puhep1.princeton.edu/~kirkmcd/examples/wire.pdf

What does mean inclusive frame? Is the locally determined conduction charge proper density in a inclusive frame? Where is it lower ? Suppose a circuit is a circular loop current. Then, at every point the density is the same. Is the density of conduction charge lower or greater? I agree that the total conduction charge stays the same. But how is relativity law respected? The charge density can neither be greater nor lower if the same quantity is in the same volume. You said "As I wrote above, the locally determined conduction charge proper density is indeed lower. And that will hold true everywhere around the circuit. " So, there are less electrons in the circuit.

I'm afraid you simply don't understand how to apply SR consistently. And given this is essentially a dialogue between just two, very few others here do either. Pick any conduction electron anywhere in the closed circuit. It will have some particular velocity v. Make that the origin of one instantaneous rest frame S. Then the rest of the circuit is instantaneously moving at uniform velocity -v wrt S in an instantaneous rest frame S'. I said circuit - meaning the conductor i.e all the positive lattice charges. All the other conduction electrons by contrast will in general be moving at various instantaneous velocities not equal to -v. And in those parts of the circuit, net Lorentz contraction OR dilation (relative to lattice charge density) of conduction charge densities will be quite different than at the origin of S where our particular reference electron resides. That is the only fair and consistent way to evaluate net conduction charge. Pick any other conduction electron somewhere else in the circuit, and repeat above. Local to that charge, conduction charge density is always lower than for the lattice, while elsewhere it will be seen as greater than locally determined. Hell, your own article acknowledges that, but in an inconsistent way. Wish I could find that illustration of a moving current loop. Once a real feel for how SR actually works is obtained, all this falls out easily, but presents paradox's until then. Such as the apparent paradox of locally lower charge density for a moving conduction charge there. It is simply wrong use of SR to then extrapolate the local result as implying a 'disappearance' of net circuit charge. If you are still confused after this, well sorry but I suggest the next port of call should then be a lecturer in relativity at some university department.

Erratum: Too hasty writing #19. Where I wrote: "Then the rest of the circuit is instantaneously moving at uniform velocity -v wrt S in an instantaneous rest frame S'." , that in red should not have been there. Apologies for any confusion caused.