Length contraction in a current carrying wire?

[Per Oni]

Ok, no problem.

The op stated this :

would think that a current carrying wire would only appear electrically neutral when the observer is moving along the wire with half the speed of the electrons, and not when it is stationary relative to the protons in that wire.

At this point you have moved a good deal away from that. We have now a problem with 3 components.
1 A radial field because of voltage V say Er
2 A an axial field generated by the voltage difference, say Ea
3 The current in the wire.

To me, points 1 and 2 only add to the complication. But that’s my opinion. Maybe maartenrvd is ok with that.
Now you still need to solve his problem.

 

[Dale]

Now you still need to solve his problem.

I did that back in post #4:

Again, the fact that the wire is neutral and has no E field in the lab frame is observed. It is a fact. It is under experimental control. Consider it like an initial condition or a "given" in the problem. It has nothing to do with relativity.

What relativity explains is: Given the fact that a wire has a current in the lab frame and given the fact that a wire is neutrally charged in the lab frame, then what does it look like in other reference frames?

The rest of this thread has been simply to establish the fact that the neutrality of the wire in the lab frame is indeed a boundary condition.

Do you have any remaining disagreement with the above response now that the charge has been established as being under experimental control? Perhaps the analogy with the projectile problem makes more sense now.

 

[Per Oni]

Do you have any remaining disagreement with the above response now that the charge has been established as being under experimental control? Perhaps the analogy with the projectile problem makes more sense now.

Er ….well yes there’s just one more problem to be solved.

Suppose I am this guy in the lab. To investigate the forces on a test charge by a current in a wire, I make sure that V=0 so that Er=0 this way I avoid seeing results which are of no use.
After running a current in a long straight conductor, I do a couple of experiments with a +ve test charge having various velocities parallel with this wire. I come to the conclusion that when the velocity is zero there’s no force towards this wire even if I change the current and/or the potential difference.

Knowing that moving electrons must be affected by Lorenz contraction can you explain this result?

 

[Dale]

Suppose I am this guy in the lab. To investigate the forces on a test charge by a current in a wire, I make sure that V=0 so that Er=0 this way I avoid seeing results which are of no use.
After running a current in a long straight conductor, I do a couple of experiments with a +ve test charge having various velocities parallel with this wire. I come to the conclusion that when the velocity is zero there’s no force towards this wire even if I change the current and/or the potential difference.

Knowing that moving electrons must be affected by Lorenz contraction can you explain this result?

I don't understand the question. Lorentz contraction is a comparison of lengths in two different reference frames, so what effect are you considering in which frames that seems to contradict Lorentz contraction?

 

[Per Oni]

The frame of importance is my lab frame, the frame in which the wire is at rest. Wire here means the visible part of the wire, ie the +ve ions or as people in previous posts said the protons (although I don’t like the word protons in this context).
In this rest frame the conduction electrons are moving and therefore a stationary +ve test charge (v=0) sees these conduction electrons moving with the drift speed. The spaces between the electrons are Lorentz contracted as viewed in the lab frame. Therefore the –ve charge density has increased as viewed from the test charge and it should (theoretically) experience a force towards the wire. The question is: why is that force not there in practice?

 

[Dale]

The frame of importance is my lab frame, the frame in which the wire is at rest. Wire here means the visible part of the wire, ie the +ve ions or as people in previous posts said the protons (although I don’t like the word protons in this context).

I understand that. How about "lattice"?

In this rest frame the conduction electrons are moving and therefore a stationary +ve test charge (v=0) sees these conduction electrons moving with the drift speed. The spaces between the electrons are Lorentz contracted as viewed in the lab frame. Therefore the –ve charge density has increased as viewed from the test charge and it should (theoretically) experience a force towards the wire. The question is: why is that force not there in practice?

Remember, we have (exhaustively) established the fact that the wire is neutral in the lab frame. This is a given boundary condition under experimental control. Because the wire is neutral in the lab frame we know that the spacing between the conduction electrons is equal to the spacing between the lattice charges in this frame. All of this is due to Maxwell's equations, not relativity.

Now, once we have this complete description in one frame you can use relativity to find the description in another frame. When you do so you will indeed find that the distance between the conduction electrons is Lorentz contracted in the lab frame relative to the electron frame. So what contradiction do you think exists here?

 

[DrGreg]

Per Oni

I suspect you may still have a misunderstanding about what Lorentz contraction really is. One distance is smaller than another distance.

Spell out, clearly and unambiguously, which two distances you think you are comparing, who is measuring each of them and when, and why you think it's a problem. Then maybe we'll get somewhere.

 

[Per Oni]

I understand that. How about "lattice"?

OK, but don't the conduction electrons make some sort of lattice as well?

Remember, we have (exhaustively) established the fact that the wire is neutral in the lab frame. This is a given boundary condition under experimental control. Because the wire is neutral in the lab frame we know that the spacing between the conduction electrons is equal to the spacing between the lattice charges in this frame.

This spacing in the lab frame is also equal when I=0.

All of this is due to Maxwell's equations, not relativity.

In that case you will have to explain why those equations keep the distance equal when a current flows.

 

[Per Oni]

Per Oni

I suspect you may still have a misunderstanding about what Lorentz contraction really is. One distance is smaller than another distance.

Spell out, clearly and unambiguously, which two distances you think you are comparing, who is measuring each of them and when, and why you think it's a problem. Then maybe we'll get somewhere.

I think I've just come to the central problem in my previous post. I'll see how that goes, then there might be no reason to answer you. Thanks for now.

 

[Dale]

OK, but don't the conduction electrons make some sort of lattice as well?

No. In a lattice there is a potential well with a minimum at the lattice spacing. This is what keeps the lattice rigid. For the conduction electrons by themselves there is no potential well, the potential is strictly increasing.

This spacing in the lab frame is also equal when I=0.

Yes, the spacing is independent of the current, it depends only on the net charge.

In that case you will have to explain why those equations keep the distance equal when a current flows.

Since V=0 there is no E-field in the radial direction, and since dV is non-zero there is at most a small E-field in the longitudinal direction. If we use Gauss' law and look at the electric flux across a cylindrical surface around the wire we see that there is equal and opposite flux across the ends and no flux across the middle, so there is no net flux across the cylinder and therefore no net charge inside. Therefore the charge density of the conduction electrons exactly equals the charge density of the lattice. Any more or less spacing would not satisfy Gauss' law.

 

[Per Oni]

Which is roughly the same answer I’ve given in post #44 .

So what’s the difference between us? I want to know why the –ve charge density doesn’t increase in a non moving lab frame, not even for the most massive currents. You are apparently happy to accept the facts and to get on with life. (perhaps not a bad idea).

 

[Dale]

I still don't understand your confusion on this point. Again, there is no lattice for the electrons so they are free to take any spacing that satisfies the laws and the boundary conditions.

You seem to understand that everything in the lab frame follows Maxwell's equations, and you seem to understand that when you Lorentz transform into the "drift" frame you get correct results in that frame also. Do you possibly think that Maxwell's equations should be violated in the lab frame for large currents?

I don't think it is a matter of me being happy to accept the facts. The facts are empirical data and there is never any question of accepting them or not; you must accept them or you are not doing science. In my mind the point is that the theories fit the facts and that is what makes me happy to accept the theories and get on with life.

 

[Per Oni]

there is no lattice for the electrons so they are free to take any spacing that satisfies the laws and the boundary conditions.

You seem to understand that everything in the lab frame follows Maxwell's equations, and you seem to understand that when you Lorentz transform into the "drift" frame you get correct results in that frame also.

That almost sounds that you could go along with the statement that in a drift frame conduction electrons are spread out.

 

[Dale]

Certainly. The distance between conduction electrons in the drift frame is larger than the distance between electrons in the lab frame. In fact, the distance between electrons in the drift frame is larger than the distance between electrons in any other frame since the proper distance is always greater than or equal to the coordinate distance.

 

[Per Oni]

Just some thoughts.

In their rest frame the conduction electrons must all move apart a little and therefore end up forming a longer length than the lattice. In the lab frame this expansion is not observed but it must be there in the drift frame to start with. These electrons see the lattice contracted but they still feel a force somehow which moves them apart. How do the electrons know how far to move? What force drives them apart?

 

[Dale]

How do the electrons know how far to move? What force drives them apart?

The EM-field (Maxwell's equations, especially Gauss' law). Just like in the lab frame.