EM Faraday tensor transformation for conductor carrying current

[cianfa72]

TL;DR

Analysis of EM Faraday tensor transformation in the context of conductor carrying current.

Hi, starting from a recent thread, I'd like to discuss a point related to the Lorentz transformation of EM Faraday tensor field between inertial reference frame.

As explained in this video at minute 11:20, in the Lab inertial frame there is only a magnetic field B in the region surrounding the wire due to the current of electrons flowing inside the wire. There is not electric field since, inside each volume/section in the wire, the net charge is zero.

The video explains that instead, in the inertial frame where the electrons are at rest, on a external positive charge (ione proton) at rest in such frame actually acts a Lorentz force due to an Electric field E. This because the Lorentz contraction charges inside the wire are not longer balanced.

My doubt is whether there is or not a symmetry in the analysis done in the two frames.

Namely if in the Lab frame there is a balance of electrons and positive metal ions inside the wire, why it is not longer the case in the other frame where electrons are at rest and positive ions move in the opposite direction with the same speed ?

 

[Ibix]

If there's only a single long straight wire then there must be a pool of charge at each end and if one frame says charge started flowing from both ends simultaneously then another frame won't agree. So they won't agree about the net charge on the wire.

If you have a circuit with a battery then you can't have a single straight wire, you must have a loop. If you have an excess of charge on one side of the loop you'll have a deficit on the other. Again, the difference between frames will be traceable to different ideas about necessary electron density and velocity to keep supplying electrons to the other side of the loop.

 

[cianfa72]

If there's only a single long straight wire then there must be a pool of charge at each end and if one frame says charge started flowing from both ends simultaneously then another frame won't agree. So they won't agree about the net charge on the wire.

Sorry, do you mean that a deficiency of electrons at a side and an excess of them at the other side are actually required to start a current inside the wire ?

 

[Ibix]

Sorry, do you mean that a deficiency of electrons at a side and an excess of them at the other side are actually required to start a current inside the wire ?

If the current is flowing only in one direction then there has to be a source of electrons at one end.

 

[Paul Colby]

For the infinite wire consider two frames. The first is at rest wrt the positive charges with the negative ones drifting slowly to the right. In this frame there is only a magnetic field and the wire is neutral. Now boost along the wire at the drift rate putting the negative charges at rest. The negative charge per unit length in this second frame will decrease because it’s no longer contracted while the positive charge density will increase because it’s now contracted in this new frame. This results in the wire being charged in the second frame.

So, the real question is, why is the wire neutral in the first frame?

 

[cianfa72]

So, the real question is, why is the wire neutral in the first frame?

Exactly, but sorry I don't know why....

 

[Paul Colby]

well the charge of the wire in the first (or any) frame is a choice. One is permitted to charge the wire or not. People who came up with this problem chose neutral wires in the rest frame.

 

[cianfa72]

well the charge of the wire in the first (or any) frame is a choice. One is permitted to charge the wire or not. People who came up with this problem chose neutral wires in the rest frame.

Ah ok, so in a sense choosing zero for the net charge of the wire in Lab frame, already takes in account the contraction of electrons moving with drift velocity.

 

[cianfa72]

If the current is flowing only in one direction then there has to be a source of electrons at one end.

Ok, so in this case your point is that two different inertial frames won't agree on when electrons on both wire's end sides simultaneously start to move. Hence at any point in time (w.r.t. a given frame respectively) the net charge in the wire won't be the same.

Therefore if the net charge is assumed to be zero in the Lab frame, then it will not be the case in the frame in which moving electrons are at rest.

 

[Paul Colby]

Ok, so in this case your point is that two different inertial frames won't agree on when electrons on both wire's end sides simultaneously start to move. Hence at any point in time (w.r.t. a given frame respectively) the net charge in the wire won't be the same.

So, relativity only connects what two observers will see of the same physical situation. The situation here are the fields about the wire. I've assumed these fields are static (not time dependent) in either frame. There is no "start moving" unless one is discussing how these fields are established in the first place. Relativity doesn't dictate how these fields are established. The potential of the wire in the lab frame relative to infinity may be chosen at will. Each different potential yields a different radial electric field in the Lab.