Length contraction in a current carrying wire?

[Hans de Vries]

@Hans:
From your paper "The simplest, and the full derivation of Magnetism as a Relativistic side effect of electroStatics" after equation 8: "We see that the field of the moving electrons is the same as the field for the electrons at rest. The result is independent of the speed of the electrons."

Don't get me wrong, I'm just trying to understand, but how does this match with the fact that a magnetic force can be transformed into an electric force in the correct reference frame, i.e. the frame of the observer?

The electron density is only equal to the ion density in one
single reference frame. If the test charge is at rest in this
reference frame then it feels no force. The speed of the
electrons and ions is not important, They can move both
in this reference frame but the only criteria is that the charge
densities are equal.

The proof that the field of the wire is only dependent on the
densities and not the speed, even though the individual charge
fields are velocity dependent, is given in section 2 here.
http://physics-quest.org/Magnetism_from_ElectroStatics_and_SR.pdf

The reason that the densities of the electrons and ions are
different in all other reference frames is non-simultaneity.
For instance: more electrons have entered the wire if Δt
is positive there, and consequently less electrons have left
the wire at the other end because Δt is negative at that
end of the wire.

Than what is the correct reference frame for a charge particle to feel no force? Is it when it is stationary with the electrons or is it when it is stationary with the protons?

See DaleSpam's post.

Regards, Hans

 

[maartenrvd]

The wire is only uncharged in a single reference frame. If the test charge is stationary in that reference frame then it will experience no force.

I agree. Which frame is it in the mentioned example?

Therefore, in any other frame moving at velocity v wrt the uncharged frame the net force on the test charge must also be 0 if the charge is moving with velocity v.

I disagree: A current carrying wire produces a magnetic field. When a charge is moving true that magnetic field it feels a force.
The thing I am discussing here (and I hope you are too) is that the magnetic force is transformed into an electrostatic force when all coordinates and velocities are transformed to the reference frame of the particle (see http://www.phys.ufl.edu/~rfield/PHY2061/images/Lectures_all.pdf" page 78 to 80)

 

[Per Oni]

Everything we have delt with so far is low speed. Why is DrGreg then correct?
Why do you refer to articles with Lorentz contraction in at the first place?

Hans?

 

[Dale]

I agree. Which frame is it in the mentioned example?

In this example the wire is uncharged in the rest frame of the wire (stationary protons). I described it the way I did because in addition to passing a current through the wire you could also charge it (by applying a high overall voltage) in the lab frame. If you did that then some other frame would be the uncharged frame.

I disagree: A current carrying wire produces a magnetic field. When a charge is moving true that magnetic field it feels a force.

Yes, but it also experiences a force due to the electric field. These two forces must cancel each other out (no net force) in every reference frame if they are zero in one frame.

The thing I am discussing here (and I hope you are too) is that the magnetic force is transformed into an electrostatic force when all coordinates and velocities are transformed to the reference frame of the particle

Yes, it should be clear that this must be the case since in that frame the magnetic force will be 0 regardless of the current.

 

[Per Oni]

In this example the wire is uncharged in the rest frame of the wire (stationary protons). I described it the way I did because in addition to passing a current through the wire you could also charge it

Completely and totally wrong.

 

[Dale]

Completely and totally wrong.

That's pretty amusing since the first sentence said the same thing you did in post 35. Also, it is customary to support such assertions with some evidence or logic.

 

[Per Oni]

That's pretty amusing since the first sentence said the same thing you did in post 35.

This sentence doesn’t make much sense to me.

Also, it is customary to support such assertions with some evidence or logic.

This makes much more sense.

After all that has been said in this thread you are still of the opinion that by sending a current through a wire that wire will become charged.
That is totally wrong.
However I cannot give references of a fact that doesn’t exist. The onus is on you to provide us with independent evidence in which is indicated the sign, location and magnitude of that charge.

 

[maartenrvd]

After all that has been said in this thread you are still of the opinion that by sending a current through a wire that wire will become charged.

No, the wire will exert an EM field: depending on the inertial reference frame of the observer this field is pure magnetic (this is supposed to be the wire frame) or pure electric (the moving charge frame) or anything in between. This is an observational effect and does not change anything to the actual state of the wire, so the wire remains uncharged.

Charging the wire due to connecting the wire to + and - poles is not relevant because that is not what we are talking about: we are talking about a hypothetical infinite long straight wire that has exact the same amount of positive as negative charge for every dx (can I be more specific?).

maartenrvd: I disagree: A current carrying wire produces a magnetic field. When a charge is moving true that magnetic field it feels a force.

Dalespam: Yes, but it also experiences a force due to the electric field. These two forces must cancel each other out (no net force) in every reference frame if they are zero in one frame.

I agree when you mean that "every reference frame" is not the frame of the test-charge but a random other frame (http://en.wikipedia.org/wiki/Lorentz_force should still be correct though).
The reason that I started this thread was that the usual assumption is that the force is zero when the test-charge is in the same frame as the positive charge in the wire, whereas I reasoned that that would only be the case when the positive charge is moving at the same speed as the negative charge (relative to the test charge).
This is easy to see for a wire without an electric current. Now figure out for yourself in which inertial reference frame this is for a current carrying wire (straight, infinite long, uncharged ...).

 

[Per Oni]

No, the wire will exert an EM field: depending on the inertial reference frame of the observer this field is pure magnetic (this is supposed to be the wire frame) or pure electric (the moving charge frame) or anything in between. This is an observational effect and does not change anything to the actual state of the wire, so the wire remains uncharged.

If the wire exert an EM field there must be charges on the wire responsible for that field. This a logic result of Gauss's law.

Hey maartenrvd you wrongly attributed some words to me which are from DaleSpam.
Its an easily done but can you still rectify them?

 

[maartenrvd]

@Per oni: Sorry about that. I corrected it in post 8.

 

[Dale]

That's pretty amusing since the first sentence said the same thing you did in post 35.

This sentence doesn’t make much sense to me.

OK, the first sentence was:

In this example the wire is uncharged in the rest frame of the wire (stationary protons).

Which is exactly what you said in post 35:

For a normal copper conductor: when it's stationary with the protons ie visible wire.

Surely you can understand my amusement. You and I say the same thing, but when I say it you call it "completely and totally wrong".

After all that has been said in this thread you are still of the opinion that by sending a current through a wire that wire will become charged.
That is totally wrong.
However I cannot give references of a fact that doesn’t exist. The onus is on you to provide us with independent evidence in which is indicated the sign, location and magnitude of that charge.

I will try to be as clear as possible. Any wire has some resistance, capacitance, and inductance. If there is an overall voltage applied then it will be charged (C=Q/V). If a voltage difference is applied across it will have a current (dV=IR). So if you have an arrangement like this:
(V+dV/2) -/\/\/\/\/\/- (V-dV/2)
then V determines the charge and dV determines the current.

Again, there is no relativistic reason for the charge or the current. They are under experimental control and must be supplied as boundary conditions (ie nature does not set them). Once you have specified the charge and the current in one reference frame then you can use relativity to determine the charge and the current in any other frame.

 

[Per Oni]

I will try to be as clear as possible. Any wire has some resistance, capacitance, and inductance. If there is an overall voltage applied then it will be charged (C=Q/V). If a voltage difference is applied across it will have a current (dV=IR). So if you have an arrangement like this:
(V+dV/2) -/\/\/\/\/\/- (V-dV/2)
then V determines the charge and dV determines the current.

I’ve got no problem with this part.

The problem is your statement that you can charge a wire by sending a current through that wire.
What is the relation/dependency between that charge and current?

 

[Dale]

The problem is your statement that you can charge a wire by sending a current through that wire.

When did I say that? In a given reference frame the charge and the current are independent. That is why in one reference frame they must both be supplied as boundary conditions.

Once you have independently specified the charge and the current in one reference frame then you can use the Lorentz transforms to determine the charge and the current in any other reference frame.

 

[Per Oni]

When did I say that?

Post #39

in addition to passing a current through the wire you could also charge it

 

[Dale]

I said you could charge the wire in addition to passing a current through it. I did not say that you could charge a wire by passing a current through it.

Again, the charge and the current are independent boundary conditions determined by the setup, not nature (as I showed earlier and which you apparently agree with). I think the rest was simply a miscommunication.

EDIT: I just noticed that my post 39 was completely explicit on this point if it had not been quoted out of context:

in addition to passing a current through the wire you could also charge it (by applying a high overall voltage) in the lab frame.

 

[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.