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I was looking at Purcell's derivation of a current-carrying wire's magnetic field using the Lorentz transform, and noticed something which bothered me. It assumes that the distance between positive (moving) charges and negative (static) charges are equal in the lab frame, and then boosts to the moving particle's frame where distances are different which causes a net non-zero charge. Why are the frames different at all though? They seem the same, just mirrored and with flipped charges. Shouldn't the distance between the positive, moving charges be contracted in the negative particle's frame just as the distance between negative particles is contracted in the positive particle's frame? Why isn't there an electric field away from the wire in the lab frame?

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Shouldn't the distance between the positive, moving charges be contracted in the negative particle's frame just as the distance between negative particles is contracted in the positive particle's frame?

I think the confusion is that we are assuming the wire is electrically neutral in the negative particle's frame (static, lab frame) and therefore in this frame the average distance between positive and negative charges must be the same. Then under this assumption when we look at the moving positive charge frame the wire is no longer electrically neutral and there is an electric field away from the wire in this positive charge frame.

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  • $\begingroup$ I am asking more generally, shouldn't every current-carrying wire create an electric field due to length contraction of the currents? $\endgroup$
    – Iddo
    Jan 6, 2022 at 19:55
  • $\begingroup$ I guess my thought is, if the contraction causes the wire to become non-neutral because of length contraction, wouldn't charge flow into the wire to make it neutral again? I'm thinking of the case where you have a wire connected to a battery. Like, maybe when it first starts moving the wire will be non-neutral but will quickly return to being neutral again. $\endgroup$
    – Hunter
    Jan 6, 2022 at 21:55
  • $\begingroup$ It is interesting to think though about a conducting ring with a current flowing around it. Would the same length attraction apply if it were a circle instead of a strait line? If so, it would seem you could have an electrically neutral ring, then induce a current in it and cause it to not be neutral any longer. I'm not sure. $\endgroup$
    – Hunter
    Jan 7, 2022 at 2:46

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