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Consider a current carrying wire. There is a stationary charge $q$ at a distance $r$ from wire.

In lab frame when the electrons in wire are moving at $v_1$ velocity we say that there is some linear charge density of poisitve ions equal to $\lambda_1$, and the linear charge density of electrons with the affect of length contraction is equal to $-\lambda$.

The electrons in wire were moving and therefore length contracted. Their linear charge density become equal to charge density of positive ions due to length contraction. Therefore the net charge on wire was $0$ and no electric force applied on $q$.

Now again in lab frame, we increase the current in wire and the velocity of electrons becomes $v_2$. The electrons are now more length contracted and their absolute linear charge density should increase from $|-\lambda|$. The density of positive charges is same as before. It means there would be a net charge density on wire and therefore a force on stationary $q$.

But it doesn't happen. Why? According to length contraction, more the velocity of electrons, more the contraction of electrons but it doesn't seem to work in case of current carrying wires. Why?

James Webb
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  • Electrons do not contract. Distance between electron does not change. Fields of electrons contract. At some point some electron's field becomes stronger and some other electron's field becomes weaker. Net change of net field is zero. Oh, maybe moving point particles are contracted, so forget the first sentence. – stuffu Mar 27 '22 at 11:34
  • @stuffu maybe? If space b/w electrons doesn't contract then there would be no force on $q$ in it's moving frame if $q$ was moving with some velocity – James Webb Mar 27 '22 at 11:41
  • Sigh ; ) https://youtu.be/xG91krXuxyw?t=145 Distances between runners does not change during first 0.1 seconds, or never if the guys are equally good. According to the bullet that flies off of the start pistol at speed 0.999999 c the stadium contracts to size of one millimeter. – stuffu Mar 27 '22 at 12:14

  • "Therefore the net charge on wire was 0 and no electric force applied on q" This does not follow. In addition to mobile charges that participate in the current, there are also static charges on the surface of the wire. These produce electric field both inside and outside the wire.

     – Ján Lalinský Mar 27 '22 at 16:16
  • @Jan Lalinsky I agree with you that there would be some charge on surface of wire. But the net electric field is 0. Cause we know it. It is found experimentally. The statement isn't wrong. To make it more clear we can add the statement- " the charge density, $-\lambda$ is given by charge of moving electrons and the static surface electrons" – Predaking Askboss Mar 27 '22 at 21:58

  • "But the net electric field is 0. Cause we know it. It is found experimentally." Not inside the wire. And because of boundary conditions electric field has to obey due to Maxwells equations, it is not 0 outside the wire just above the wire surface either. Electric field due to the wire decreases with distance so far away from the wire, it may be unmeasurable, but it is not zero.

     – Ján Lalinský Mar 28 '22 at 01:46
  • @Jan Lalinsky and Predaking this is not a point to discuss here. My question doesn't need to talk about them. The point is that length contraction increases as velocity of electrons increases. So with higher current there would be a large net negative charge over wire. – James Webb Mar 28 '22 at 08:36
  • Duplicate : Why don't stationary charge feel force from a current carrying wire?. – Frobenius Mar 29 '22 at 15:04

1 Answers1

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But it doesn't happen. Why?

The electrons don’t have a fixed separation in their rest frame. So when you change the current you also change the separation in the electron’s rest frame. Changing the current in the wire is not a boost.

Dale
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  • What if we replace such a current carrying system with two pulley systems where one pully belt carries positive charge and another carries negative charge. And put both belts closer to copy a current carrying wire as belt moves. Or we can have an alternate system to make sure that there is fixed distance between charges in their frame of rest. – Predaking Askboss Mar 27 '22 at 22:04
  • Now you need to have a better explanation for it cause you can't say that absolute charge separation changes. Also this answer is not satisfactory cause you are saying that the absolute spacing of electrons changes or increases as we increase current. – Predaking Askboss Mar 27 '22 at 22:08
  • @PredakingAskboss that would be a physically different question from the one JamesWebb asked. It is not an equivalent scenario. You should ask it separately so that we don’t clutter his question up with unrelated tangents – Dale Mar 28 '22 at 01:58