Briefly, the question is:All popular expositions (e.g. these ones) of relativistic electromagnetism claims claim univocally that electrons in motion become more dense due to the speed, so you get more negative charge per unit. They teach that Lorentz contraction of charges causes charge imbalance and wire length than there is positivewith current charged. This immediately impliesThereby, no teacher says that the wire ceases to be neutral. Is it right?
The answer is:becomes negatively charged the length contraction compensates the density increase!
== Prehistory ==
I don't hope too much to get the answer to(because electrons move) in the electric current density Ehrenfest paradox. Butlab frame, let me ask a simple part ofwhere we had it originally neutral, when no current was flowing. Why discussions of relativistic origin of the magnetic field never discuss the simplest case:That is a neutralquestion: is wire withneutrality conserved, once we create a current and test charge at rest? How it is possible in Feynman lectures that positive and negative densities are equal thoughof electrons are moving while nuclei are stillin it?
They say that moving charge relativistically gets extra charge density and, thereforeIt must be the case because, wire becomes electrically charged (negativelyas Feynman points out, becausethe lighter electrons are moving). It will attract or repel electrical test charges. This is what should happen if only one polarity charges are moving inbetter react on temperature changes and would charge the wire when heated. Right? To stay neutralBut, the relativistic chargethen answer how neutrality is persisted after you say that electron contraction takes place and it increases their density should? Might be balanced by the current of"some light" positive charge! Positive ions must movecharges start moving in opposite direction, to compensate the motiongrowth of free electrons!negative charge?
Yet, the relativistic magnetismI see that all discussions of "magnetism as electricity+relativism" avoid discussingconcerning this curios fact, most basic and most interesting case. TheyInstead, they jump immediately to the case when there iswhere you have an electric balance for a test charge in motion (e.g. here, here and here). The second also makes a confession that
In the frame in which the wire is at rest, the positive and negative
charge densities exactly balance, otherwise there will be extra
electrostatic fields
It considers the lab frame as a case where positive and negative charges move in opposite directions so that net current is not zero. But, it forgets to mention that in the normal lab, the positive charges create a solid structure of the frame and, therefore, cannot move. And that is the contradiction, I am paying attention on.
I have revised the Feynman lectures. He considers test charge and electrons at rest, telling that wire is charged positively. This seems to confirm my idea that current of electrons charges the neutral wire negatively. But, he adds immediately thatOnce finished with this does not happen because if it would be true then, temperature would charge the bodies since electronsyou are much lighter and gain more speed when body heated. But, we observe no charge variations as temperature varies. This is another contradiction that I askencouraged to resolve along with the issue why nobody can say explicitly whether current makes wire charged or not.answer how it is possible that density of charges is increased whereas volume of the loop is intact?