B Stationary charge next to a current-carrying wire

[lightlightsup]

TL;DR Summary

I understand how relativistic length contraction leads to electrostatic forces on a moving charge next to a current-carrying wire. But, shouldn't this lead to forces on a static charge too?

(1) https://www.youtube.com/results?search_query=relativity+and+electromagnetism

(2)

If I understand this correctly, then, well, shouldn't all current-carrying wires exhibit a small amount of positive charge?

 

[PeroK]

TL;DR Summary: I understand how relativistic length contraction leads to electrostatic forces on a moving charge next to a current-carrying wire. But, shouldn't this lead to forces on a static charge too?

(1) https://www.youtube.com/results?search_query=relativity+and+electromagnetism

(2)

If I understand this correctly, then, well, shouldn't all current-carrying wires exhibit a small amount of positive charge?

I think the guy who posted that video is something of a crackpot. You need better source material if you plan on studying relativity.

 

[lightlightsup]

I think the guy who posted that video is something of a crackpot. You need better source material if you plan on studying relativity.

Are you able to explain this phenomenon, then?

 

[PeroK]

Are you able to explain this phenomenon, then?

Yes. It's covered in the SR textbook (Helliwell) and Electrodynamics (Griffiths) that I have. It's also covered here:

https://phys.libretexts.org/Bookshelves/University_Physics/Book:_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/24:_The_Theory_of_Special_Relativity/24.05:_Electric_and_magnetic_fields_and_Special_Relativity

 

[Dale]

I think the guy who posted that video is something of a crackpot. You need better source material if you plan on studying relativity.

I haven’t watched anything else from him, but he doesn’t seem crackpot-ish to me. I think he is wrong here, but not a crackpot.

As far as I know holes require a specific band structure that metals don’t have. You see holes in semi-conductors where the valence and conduction bands are separate, but not in metals where they overlap.

 

[PeroK]

I haven’t watched anything else from him, but he doesn’t seem crackpot-ish to me. I think he is wrong here, but not a crackpot.

I may be confusing him with someone else. I thought I'd seen that video before and he was rejecting SR. But, perhaps you are correct and he was just wrong in his calculations.

 

[PeroK]

Here's something of an explanation of what's going on. We start with a very large rectangular loop of wire. The wire is neutral, with the positive and negative changes uniformly distributed throughout the wire.

By means of a battery, we then set up a steady current in the wire, whereby the negative charges are moving. Now, since we have not changed the total charge in the wire, the negative charges must remain equally spaced (in the rest frame of the wire). Note that this argument is more difficult to make if we consider a single, infinitely long wire. So, making the practical assumption of a large loop of wire is a good way to see what is happening.

The result of this analysis is that there can be no net charge in the wire.

Now, let's take a linear segment of the wire and consider the situation in a frame of reference where the electrons are at rest.

The first thing to note is that if two objects initially at rest accelerate until they reach the same speed and remain the same distance apart in the original rest frame, then (by length contraction) they must be further apart in their rest frame. So, we see that relative to each other the moving electrons are less dense than when there was no current.

Note that this is where the asymmetry between the positive and negative charges is seen. The scenario is not symmetrical because the positive charges remain at rest in the inertial frame of the wire; whereas, the negative charges are moving round an inertial loop of wire. Locally the electrons are moving inertially, but globally (when we take the full loop of wire into account), the electrons' rest frame is not inertial.

Finally, the positive charges are moving in the electron rest frame, so the distance between them is length contracted. That's why there is a double case of length contraction in the electron frame: electrons themselves are further apart and the positve charges are closer together. The wire is not locally neutral in that frame (although globally the wire is neutral).

That's why Veritasium is correct and the guy in the video is wrong.

 

[Dale]

The way I think of it is that a wire has a certain amount of self capacitance. So if we want to make it charged we can just attach it to a high voltage source. If we want it neutral we ground it. So the statement that it is uncharged in the lab frame simply means that it is grounded.

 

[lightlightsup]

Yes. It's covered in the SR textbook (Helliwell) and Electrodynamics (Griffiths) that I have. It's also covered here:

https://phys.libretexts.org/Bookshelves/University_Physics/Book:_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/24:_The_Theory_of_Special_Relativity/24.05:_Electric_and_magnetic_fields_and_Special_Relativity

These don't carry the explanation I'm looking for.
Why doesn't the positive test charge (cat, here) experience an attraction to the wire if the electrons are getting length contracted?

 

[Dale]

Why doesn't the positive test charge (cat, here) experience an attraction to the wire

Because the wire is electrically neutral in that frame.

You are under the mistaken impression that the spacing between the electrons is fixed, like the protons. It is not. If you want to pack the electrons in closely you can simply charge the wire to a strongly negative voltage. If you want to spread them apart you charge it to a positive voltage. If you ground it then they will be spaced the same as the protons in the wires frame.

Length contraction happens, but is not relevant to the spacing in the wires frame. That spacing is set by the circuit. The spacing in other frames is then determined by the Lorentz transform from the spacing set by the circuit.

 

[Ibix]

Why doesn't the positive test charge (cat, here) experience an attraction to the wire if the electrons are getting length contracted?

It isn't the particles that matter here, it's their spacing. By the way the problem is set up, the spacing of the electrons and the spacing of the protons are the same in the wire frame; thus it is not charged because there the number densities of the particles are equal.

Viewed in the electron rest frame, however, the electrons are now stationary so their spacing is larger because length contraction is no longer a factor. The protons, on the other hand, are now noving so length contraction means that they are closer together. Hence there is a net positive charge.

Stating all that the other way around, the wire is positively charged in the electron rest frame. When we switch to the wire frame length contraction means that the electrons are closer together and the protons are further apart, and the wire is neutral.

 

[PeroK]

These don't carry the explanation I'm looking for.
Why doesn't the positive test charge (cat, here) experience an attraction to the wire if the electrons are getting length contracted?

I already gave a full explanation of the scenario in post #7.

 

[PeroK]

It isn't the particles that matter here, it's their spacing. By the way the problem is set up, the spacing of the electrons and the spacing of the protons are the same in the wire frame; thus it is not charged because there the number densities of the particles are equal.

Note, however, that if you consider a single, infinite length of wire, then the scenario is ambiguous - depending on how the electrons are accelerated. We could have a symmetrical scenario where the negative charges retain their proper density.

Considering the more realistic large loop of wire highlights that the scenario cannot be symmetrical in terms of the positive and negative charges.

 

[vanhees71]

Yes. It's covered in the SR textbook (Helliwell) and Electrodynamics (Griffiths) that I have. It's also covered here:

https://phys.libretexts.org/Bookshelves/University_Physics/Book:_Introductory_Physics_-_Building_Models_to_Describe_Our_World_(Martin_Neary_Rinaldo_and_Woodman)/24:_The_Theory_of_Special_Relativity/24.05:_Electric_and_magnetic_fields_and_Special_Relativity

Unfortunately this gives the usual wrong description of the DC carrying wire. It's not neutral in the rest frame of the wire but in the rest frame of the electrons because of the Hall effect. For a full analysis, see

https://itp.uni-frankfurt.de/~hees/pf-faq/relativistic-dc.pdf

 

[PeroK]

Unfortunately this gives the usual wrong description of the DC carrying wire. It's not neutral in the rest frame of the wire but in the rest frame of the electrons because of the Hall effect.

If the wire is not neutral in its rest frame, then are there equal numbers of positive and negative charges in total in the wire?

 

[vanhees71]

There is some charge from the battery necessary to charge the wire. Of course, overall there's charge conservation. The wire appears neutral in the sense the $\rho=0$ in the rest frame of the electrons. For details, see

https://itp.uni-frankfurt.de/~hees/pf-faq/relativistic-dc.pdf

 

[PeroK]

There is some charge from the battery necessary to charge the wire. Of course, overall there's charge conservation. The wire appears neutral in the sense the $\rho=0$ in the rest frame of the electrons. For details, see

https://itp.uni-frankfurt.de/~hees/pf-faq/relativistic-dc.pdf

I'll look at your analysis. But, an infinite wire is unphysical and subject to paradoxes of the countably infinite. Like increasing the charge density without "adding" any changes, but simply by moving them all closer together.

PS more pointedly, charge conservation is not a valid concept for an infinite number of charges.

 

[Sagittarius A-Star]

If I understand this correctly, then, well, shouldn't all current-carrying wires exhibit a small amount of positive charge?

The line charge density is frame-dependent and definitely not positive in the lab frame.

Unfortunately this gives the usual wrong description of the DC carrying wire. It's not neutral in the rest frame of the wire but in the rest frame of the electrons because of the Hall effect. For a full analysis, see
https://itp.uni-frankfurt.de/~hees/pf-faq/relativistic-dc.pdf

I find (not very convincing) contradicting conclusions in different papers.

1. In the lab frame the line charge density is negative because of the self-induced Hall effect and "Here, we will follow Clausius (joining his many followers) in ignoring the effect of the surface charge density on the external charge", according to
   http://kirkmcd.princeton.edu/examples/wire.pdf
   ( via: http://kirkmcd.princeton.edu/examples/ )
2. In the lab frame the line charge density is zero because "It is a well-known fact that a current-carrying wire is neutral ... in the lab frame" and therefore a positive surface charge density must compensate the negative volume charge density, according to
   http://web.mit.edu/wangfire/misc/AJP000360.pdf

 

[vanhees71]

That's interesting. In my derivation, I assumed that there is no surface-charge on the wire, because the electric field should be continuous, i.e., no jump of the radial component.

 

[Dale]

Surface charges on conductors are essential for the functioning of a circuit. Any analysis that ignores them is unlikely to be applicable in real circuits.

It is a simple observed fact that conductors have self capacitance. The self capacitance means that the total charge on the wire can be varied. An analysis that “proves” that the charge cannot be varied is therefore contradicted by observation. The claim that the wire is necessarily uncharged in the frame of the charge carriers is such a claim.

I suspect that the issue is precisely neglecting the surface charges. That is where capacitative charge is stored. So neglecting them neglects precisely the part of the wire that allows the net charge to vary.

 

[vanhees71]

That's true for the non-relativistic treatment, where the charge density inside the wire is taken to be 0, and the constitutive equation is simplified to $\vec{j}=\sigma \vec{E}$. In the relativistic case you have $\vec{j}=\sigma (\vec{E}+\vec{\beta} \times \vec{B})$ and a charge density inside the wire due to the so described Hall effect.

I'm not certain about the correct boundary condition for $\vec{E}$ though. In the two papers quoted in #18 the assumptions are different. McDonald seems to have the same assumptions as in my writeup, i.e., there's only a charge density, not a surface-charge density and thus $\vec{E}$ (particularly the radial component along the straight wire) is continuous. In the AJP article Gabuzda assumes the wire as a whole to be neutral. Since the (negative) charge density is necessary due to the Hall effect this implies that there must be a corresponding positive surface charge along the wire compensating the negative charge inside.

It's not so clear to me, which is the correct assumption. The argument for the charge neutrality of the wire as a whole in the AJP paper, i.e., that the free electrons in the "source and sink of the battery", however, sounds a bit strange since of course the charge carriers also must flow through the battery.

 

[Dale]

It's not so clear to me, which is the correct assumption.

IMO, neither assumption is correct. Both imply a fixed net charge, which is experimentally falsified.

The surface charge is not fixed by nature in any frame. The surface charge is the location of the self capacitance. Through that self capacitance, the amount of surface charge is under experimental control and may be changed by the experimenter through an appropriate circuit design.

 

[Sagittarius A-Star]

Through that self capacitance, the amount of surface charge is under experimental control and may be changed by the experimenter through an appropriate circuit design.

To my understanding, both papers describe a wire, that is connected only to the two poles of one battery. The whole system of wire and battery contains an equal number of protons ans electrons. The assumptions differ in the questions, if one straight segment of the wire has a net negative line charge density in it's rest-frame, or not because of an additional positive surface charge density.

 

[Dale]

a wire, that is connected only to the two poles of one battery.

Such a circuit is ambiguous since it has a floating ground. The self capacitance forms a capacitative connection to ground. If you don’t control the voltage between the circuit and ground then the surface charge may fluctuate easily. The assumption of zero net charge is less likely to be true in such a circuit than in one designed for the purpose of making it zero.

IMO, especially for that circuit, the surface charge should be considered unknown, not assumed to have a specific value. Any such assumption makes the resulting analysis inapplicable to many circuits.

 

[PeroK]

This thread is tackling two very different questions. The first is whether there is an elementary explanation for the magnetic force of a steady current on a moving charge. This is covered, for example, in section 12.3.1 Magnetism as a Relativistic Phenomenon of Introduction to Electrodynamics by Griffiths.

The second question is what happens in a real wire attached to a real battery? We have complications like the Hall effect, self-capacitance and the battery itself. This is a different question. Griffiths, for example, considers the current to be an equal but opposite flow of positive and negative charges. Which is not the rest frame of any circuit.

Perhaps we have to accept the elementary explanation in Griffiths etc. as merely a thought experiment. Then we have the added problem of explaining why a magnetic force of the appropriate magnitude is seen in a real wire attached to a real battery, assuming that the hypotheses of the thought experiment fail in this scenario.

 

[Dale]

Then we have the added problem of explaining why a magnetic force of the appropriate magnitude is seen in a real wire attached to a real battery, assuming that the hypotheses of the thought experiment fail in this scenario.

I don’t think this last part is much of a problem. A typical wire has a very small self capacitance. The force on a nearby charge at rest may not be exactly zero, but it will be small. Easily unnoticed if the experiment is not designed very carefully.

I agree with you that these are two distinct questions. What Griffiths, Purcell, Feynman, and everyone else does show is that a magnetic force in one frame is an electric force in another frame. The OP’s objection is irrelevant to that, as is all of the discussion about surface charge and which assumptions are most accurate.

 

[pervect]

TL;DR Summary: I understand how relativistic length contraction leads to electrostatic forces on a moving charge next to a current-carrying wire. But, shouldn't this lead to forces on a static charge too?

If I understand this correctly, then, well, shouldn't all current-carrying wires exhibit a small amount of positive charge?

It depends. Typically, a current carrying wire is set up to be uncharged (electrically neutral) in the lab frame. There are an equal number of positive and negative charges overall, and the charge density of positive and negative charges is uniform, so any small piece of wire will be uncharged.

In a frame moving near a charged wire, though, this needs to be modified. It's best to consider a current loop, because charge needs to flow in a closed loop. While the total number of charges will be constant, independent of the frame of reference, the same is not true for the distribution of the charges and the charge density.

So - while total charge of a system is frame- invariant, the density of charge, the amount of charge contained in a unit volume, is not frame-invariant.

If you have a pair of wires carrying current in opposite directions in a current loop, for instance, a current loop that is presumed to be electrically neutral in the lab frame, one side of the loop will have a higher density of positive charges in the moving frame, so a small volume of the wire will have a positive charge density in that frame, while the other side of the loop will have a higher density of negative charges, in the moving frame. The total charge remains constant as you change frames, it is just distributed differently.

This can be regarded a a consequence of the relativity of simultaneity in Special Relativity. This causes much confusion, unfortunately, but it's internally self-consistent. It's just a bit weird.

This general treatment is usually attributed to Purcell, and there are various people who have various things they don't like about Purcell's original treatment, including the fact that Purcell treats an idealized wire which doesn't get into the precise details of how the charges in an actual wire are distributed, so Purcell's treatment is over-simplified.

Mathematical treatments of the problem can vary, but they usually point out that charge density and current density, usualy symbolized by $\rho$ and j form a mathematical entity which is known as a 4-vector. Then there is an idenity that $c^2 \rho^2 - j^2$ is an invariant, much like $c^2 dt^2 - dx^2$. This may not be helpful if one is not already familiar with 4-vectors, however, which are usually introduced in the concept of space-time intervals, though they can also be applied to electromagnetics.

 

[Dale]

Why doesn't the positive test charge (cat, here) experience an attraction to the wire if the electrons are getting length contracted?

This concern doesn’t matter in the end. As long as the velocity of the positive and negative charges is different then they will undergo different amounts of length contraction. The different amount of length contraction in each frame will lead to different charge densities in each frame.

As long as there exists some frame where the wire is neutral then the remainder of the argument holds. In that frame the force on a moving test charge is purely magnetic. In the test charge’s frame the force is purely electric. What is a magnetic force in one frame is an electric force in another frame.

That is the point.

 

[vanhees71]

IMO, neither assumption is correct. Both imply a fixed net charge, which is experimentally falsified.

The surface charge is not fixed by nature in any frame. The surface charge is the location of the self capacitance. Through that self capacitance, the amount of surface charge is under experimental control and may be changed by the experimenter through an appropriate circuit design.

I think the problem is to consider just a single wire. To get a unique solution one must have a complete closed circuit. I think thus one has to solve the full problem of a coaxial cable. The infinite length makes it also somewhat critical, but at least one gets a formal solution, which seems to describe the problem for a finite wire not too close to the "ends".

I'll completely rewrite my manuscript within the next few days. I think now one should just solve the boundary-value problem with the correct relativistic equations for Ohm's Law.

 

[Sagittarius A-Star]

I think thus one has to solve the full problem of a coaxial cable.

A coaxial cable would be part of a completely different scenario. I think the best would be to consider a current loop, as described by @pervect in posting #27.

 

[vanhees71]

A current loop is pretty difficult to solve. I'm not aware of a closed solution. The infinite DC-carrying coax cable is the most simple case for a complete treatment. However, it's hard to find the complete solution of the magnetostatic problem including the electric field. For the non-relativistic approximation it's solved in Sommerfeld, Lectures on Theoretical Physics, vol. 3.

 

[vanhees71]

I have now done the calculation and also typed it. To treat the complete coaxial cable, i.e., having a conductor for a full look solves all problems. One does not need to impose any artificial surface charges to get global charge neutrality but only the usual continuity arguments on the electric an magnetic fields at the boundaries of the conductors. Thanks to @Dale for pointing out the mistake in the earlier manuscript, where I imposed a boundary condition for the single cylindrical wire, which obviously is wrong due to the unphysical idea to describe a single current-conducting wire without taking into account that one needs a back current too. As demonstrated in the new manuscript, just solving the entire DC circuit problem in the rest frame of the wire leads to the correct charge neutrality (assuming an ideal voltage source). Maybe it's worthwhile to send this as a manuscript to AJP. What do you think?

https://itp.uni-frankfurt.de/~hees/pf-faq/relativistic-dc.pdf

 

[Sagittarius A-Star]

What do you think?

https://itp.uni-frankfurt.de/~hees/pf-faq/relativistic-dc.pdf

Is equation (47) consistent to 2nd equation (43)?

 

[Dale]

Maybe it's worthwhile to send this as a manuscript to AJP. What do you think?

I have never published there, but it does seem like the sort of thing that they do publish.

 

[vanhees71]

I think, I'll try it. The only question is, whether this has been done already somewhere else. Since it's not too difficult, I'm really surprised that I couldn't find the solution of this problem anywhere. Even the non-relativsitic approximation is not presented completely in almost all textbooks, i.e., usually they calculate only the magnetic but never the electric field. The only exception is Sommerfeld's Lectures of Theoretical Physics, vol. III, which I cited. What's now missing is the discussion of the Poynting vector, i.e., the energy transfer along the cable, which is interesting in its own right (and also discussed for the non-relativistic approximation by Sommerfeld).

 

[Sagittarius A-Star]

The only exception is Sommerfeld's Lectures of Theoretical Physics, vol. III, which I cited. What's now missing is the discussion of the Poynting vector, i.e., the energy transfer along the cable, which is interesting in its own right (and also discussed for the non-relativistic approximation by Sommerfeld).

Do you mean that Sommerfeld didn't consider the self-induced Hall effect?
I'm not sure why this is called "non-relativistic".

He assumed that there is no radial E-field in the wire volume, because the volume charge density in the wire is homogeneously zero.

If you calculate in Sommerfeld's book, §17, equation (9a), the $\lim_{b \rightarrow \infty}$ of $E_r$ in the intermediate space $a<r<b$ of the coaxial cable, then you get $E_r=0$ and therefore no surface charge density on the inner wire.
Source:
https://archive.org/details/in.ernet.dli.2015.147970/page/n137/mode/2up

Calculation

I think this might be a good model for Purcell's infinite straight wire, when the outer return path is infinitely far away.

I can't find in your actual paper a proof for the following statement:

... in which reference frame the wire is uncharged. It is not the rest frame of the wire (i.e., the rest frame of the ions) but the restframe of the conduction electrons [Pet85].

In the AJP paper, which I linked in #18, a positive surface charge density compensates the negative volume charge density. Sommerfeld's assumption, that there is no radial E-field in the wire, is not fulfilled. But the positive surface charge density would compensate the $E_r$ outside of the surface to zero.

I see no indication in your paper (I don't have access to the full paper of Peters), what motivation the battery should have to provide extra electrons, if the wire is overall electrically neutral in it's rest frame.

 

[vanhees71]

Do you mean that Sommerfeld didn't consider the self-induced Hall effect?
I'm not sure why this is called "non-relativistic".

What I mean with "non-relativistic" is the use of the standard "constituent equation" (Ohm's Law) as $\vec{j}=\sigma \vec{E}$. This neglects the self-induced Hall effect and it's of course not relativistic, but nevertheless it's a damn good approximation. Note that the charge density inside the conductors due to the self-induced Hall effect is $\rho_{\text{wire}}=\beta^2 \rho_{\text{cond}}$, and $\beta=v/c$ with $v \simeq 1 \text{mm}/\text{s}$. For all practical purposes of electrical engineering the standard form of Ohm's Law is thus completely sufficient.

He assumed that there is no radial E-field in the wire volume, because the volume charge density in the wire is homogeneously zero.

Exactly.

If you calculate in Sommerfeld's book, §17, equation (9a), the $\lim_{b \rightarrow \infty}$ of $E_r$ in the intermediate space $a<r<b$ of the coaxial cable, then you get $E_r=0$ and therefore no surface charge density on the inner wire.
Source:
https://archive.org/details/in.ernet.dli.2015.147970/page/n137/mode/2up

Calculation

I think this might be a good model for Purcell's infinite straight wire, when the outer return path is infinitely far away.

That's of course right, and then you end up with the solution of Gabuzda's paper. If didn't find this argument so convincing, but now given that it's found as this limit of the complete calculation, where only the usual physically well-motivated boundary conditions are used, it seems all right.

I can't find in your actual paper a proof for the following statement:In the AJP paper, which I linked in #18, a positive surface charge density compensates the negative volume charge density. Sommerfeld's assumption, that there is no radial E-field in the wire, is not fulfilled. But the positive surface charge density would compensate the $E_r$ outside of the surface to zero.

But in my calculation, the overall charge neutrality comes out automatically from the standard boundary conditions, but that doesn't work with the single wire. That's why in this AJP paper Gabuzda adds the additional positive surface charge by hand. As the limit $a_2 \rightarrow \infty$, i.e., putting the return path to infinity, it makes sense.

That there's no radial field in Sommerfeld's "non-relativistic" treatment follows immediately from $\vec{j}=\sigma \vec{E}$, i.e., with $\vec{j}=j \vec{e}_z$ of course $\vec{E}=\vec{j}/\sigma=E \vec{e}_z$. Gabuzda assumes the relativistic version of Ohm's Law and thus there must be a radial electric field compensating $\vec{\beta} \times \vec{B}$. That's the same argument as in my writeup.

I'll add this limiting case as a model for a single DC conducing wire, including a citation of Gabuzda's paper.

I see no indication in your paper (I don't have access to the full paper of Peters), what motivation the battery should have to provide extra electrons, if the wire is overall electrically neutral in it's rest frame.

Indeed, the charge-neutrality assumption isn't needed anymore with the full calculation of the complete coax cable including the return path, but it comes out as the said limit, $a_2 \rightarrow \infty$. Everything that happens is a rearrangement of charges with the overall charge being 0 as it should be. That's, of course a frame-independent statement!

 

[Sagittarius A-Star]

That's of course right, and then you end up with the solution of Gabuzda's paper.

Yes. I saw, that you added his paper today to the references.

But in my calculation, the overall charge neutrality comes out automatically from the standard boundary conditions, but that doesn't work with the single wire. That's why in this AJP paper Gabuzda adds the additional positive surface charge by hand. As the limit $a_2 \rightarrow \infty$, i.e., putting the return path to infinity, it makes sense.

She didn't add the positive surface charge density by hand. The self-induced Hall effect shifts/compresses the electron current more towards the axis and then the surface has missing free electrons, which are not replaced from the battery.

Sommerfeld's calculation yields only an additional surface charge density.

I'll add this limiting case as a model for a single DC conducing wire, including a citation of Gabuzda's paper.

That's, of course a frame-independent statement!

Yes, if the "+" and "-" pole of the battery are defined to have the same $z$ coordinate, also while the ramp-up phase of the current.

 

[vanhees71]

The only quibble I still have with my calculation is the homogeneous part of the $\vec{E}$-field in $z$ direction in region IV (outside the coax cable), but it seems to be unavoidable from the continuity condition. On the other hand, intuitively, there shouldn't be a field outside the cable. Maybe this has to do with the somewhat artificial assumption of an infinitely long cable and the ignorance of the fact that except the ideal voltage source at $z=0$ (which is indeed assumed to be infinitely thin too) that you have to close the circuit at some finite point $z=L$. One could just put a conducting circle there, but then the problem seems not to be treatable analytically anymore.

Of course, I also didn't solve any transient state, i.e., "switching on the circuit" but only considered the final stationary DC state. That would also be an interesting task. One would have to solve the coax cable as a wave-guide problem. That's for sure much more complicated than the DC case, but it would also be worthwhile to study, because it would clearly show that the energy transfer is through the em. fields and not somehow along the wire as in this unfortunate analogy picture as if an electric circuit were analogous to a water pipe, where of course the energy (e.g., heat from hot water) is transferred through the flow of the water.

 

[A.T.]

Why doesn't the positive test charge (cat, here) experience an attraction to the wire if the electrons are getting length contracted?

As others explained, the distances between the electrons in the wire frame don't change when the current starts flowing, because unlike the positive ions, the electrons are not forced to keep constant proper distances. So the electron density in the wire frame doesn't change when the current starts flowing.

Here is a good explanation and diagram by @DrGreg :
https://www.physicsforums.com/threads/explanation-of-em-fields-using-sr.714635/post-4528480

This has been asked many times:
https://www.physicsforums.com/threads/electric-currents-and-length-contraction.930943/#post-5877453
https://www.physicsforums.com/threa...tion-of-electromagnetism.982883/#post-6284262
https://www.physicsforums.com/threa...magnetism-with-relativity.932270/post-5886984

 

[Sagittarius A-Star]

The only quibble I still have with my calculation is the homogeneous part of the $\vec{E}$-field in $z$ direction in region IV (outside the coax cable), but it seems to be unavoidable from the continuity condition. On the other hand, intuitively, there shouldn't be a field outside the cable.

Sommerfeld avoids this problem in $17 by defining "outer radius $c \rightarrow \infty$", from which follows his equation (3).

 

[vanhees71]

Yes, but that's also pretty artificial ;-).

 

[vanhees71]

I've uploaded a corrected version of the manuscript concerning the covariant treatment of Ohm's Law. I've gotten it wrong, because I've written down the equation too early in the special frame of reference, where the wire's at rest. Now everything is consistent in the sense that both transport coefficients, i.e., the Stokes-friction coefficient as well as the electric (or electrical?) conductivity are scalars and defined in terms of scalars (electron density as measured in their (local) rest frame):
$$\sigma=\frac{n_- e^2}{\alpha m},$$
where $n_-$ is the scalar number density of the electrons (i.e., the number density as measured in the rest frame of the electrons), $e$ the elementary charge, $\alpha$ the friction coefficient, and $m$ the electron mass.

I'm still somewhat puzzled by the result that there's a constant electric field along the wire outside of the coax cable, but it seems inevitable, if you don't make this somewhat artificial assumption as done in Sommerfeld by making $a_3 \rightarrow \infty$.

 

[Sagittarius A-Star]

I'm still somewhat puzzled by the result that there's a constant electric field along the wire outside of the coax cable, but it seems inevitable, if you don't make this somewhat artificial assumption as done in Sommerfeld by making $a_3 \rightarrow \infty$.

In the following paper, they don't make this somewhat artificial assumption. From their figure 3 follows the existence of longitudinal and radial components of the external electric field.

Surface charges and fields in a resistive coaxial cable carrying a constant current
Publisher: IEEE
A.K.T. Assis; J.I. Cisneros
...
Abstract:
We calculate the surface charges, potentials, and fields in a long cylindrical coaxial cable with inner and outer conductors of finite conductivities and finite areas carrying a constant current. It is shown that there is an electric field outside the return conductor.

Source:
https://ieeexplore.ieee.org/abstract/document/817391

 

[Sagittarius A-Star]

From the same author A.K.T. Assis exists also the following German book.
See on page 94, chapter 6.4 "Radialer Halleffect" and on page 105, chapter 7 "Koaxialkabel":
https://www.ifi.unicamp.br/~assis/Kraft.pdf

via:
https://www.ifi.unicamp.br/~assis/books.htm

Electrical field lines in the symmetrical case, according to the book:

Also, A.K.T. Assis mentions on page 8 of this book a here not yet discussed force of a wire on a charged test-particle at rest in the rest frame of the wire:

Das heißt, die externe Punktladung q induziert eine Verteilung von Ladungen längs der Oberfläche des Leiters und ergibt insgesamt eine elektrostatische Anziehung zwischen Leiter und q.

Google-translate:

That is, the external point charge q induces a distribution of charges along the surface of the conductor and results in an overall electrostatic attraction between the conductor and q.

From the publisher of this book, Apeiron, one can find also many dubious anti-SR books. However, that does not automatically mean, that every book of them is dubious.

 

[pervect]

This is not related to the offshoot question of what the charge distribution in a real wire is, but I hope it will be helpful to the OP if they are still around, or to other readers interested in the original question. It uses only Gauss' law to explore the issue.

A preliminary point is that Maxwell's equations are fully relativistic, so that we can use Gauss' law (one approach of one of Maxwell's equations) to determine the charge enclosed in a box (any box) at some instant in time. In this case, the box is a cylinder around an idealized wire with zero resistance.

While we have a pair of wires in a current loop, we only draw the Gauss' law cylinder around one wire.

We consider a pair of such cylindrical boxes, one that is stationary in the lab frame, one that is moving in the lab frame.

The charge enclosed by the box in the lab frame could, in principle, be anything. It's part of the experimental setup that current loop is at zero potential, and that there is no electric field around the wire. But we assume that the wire is uncharged, that it is an uncharged conductor in the lab frame, the experimental test of this is to measure the electric field around the neutral conductor in the lab frame and find no electric field. Then the Gauss' law surface integral is easy - the integral of zero is zero.

Now we consider a moving box around the wire. We know there is an induced electric field in the moving frame. Gauss' law now tells us, because of the induced electric field (proportional to v x B), that the material enclosed by the box contains a charge.

The next part I have to say is a little more advanced, but I don't know how to say it as simply as the first part. We don't expect that a stationary box will include no net charge, while a moving box will enclose net charge when we do a non-relativistic transformation. But when we do a proper relativistic transformation, we find that the box in the moving frame does contain a non-zero charge. The point is that for Maxwell's equations to work, the source must transform relativistically.

The result that the moving box contains a non-zero charge is unexpected, but it's not inconsistent, either internally or with the laws of relativity. The prediction that the moving box contains a net charge is not an error.

To anticipate an objection about the creation of charge. Where did the charge in the box "come from"?

The answer to this objection is that current must flow in a closed loop. To find a "total charge", we must consider a closed loop.

When we do, we find in the moving frame that one leg of the loop has acquires a positive charge, while the other leg of the loop has a negative charge. The total net charge is zero in both the lab frame and the moving frame. I suggest again, without an overly detailed explanation, that we can explain this by the relativity of simultaneity. The total charge remains constant at zero, but the distribution of charge at any given "instant of time" varies with the choice of frame. In the moving frame, we see a positive charge density on one wire, and a negative charge density on the other, but the integral of the charge density around the loop remains zero.

 

[Sagittarius A-Star]

A video of Veritasium about surface charges:

 

[vanhees71]

In the following paper, they don't make this somewhat artificial assumption. From their figure 3 follows the existence of longitudinal and radial components of the external electric field.Source:
https://ieeexplore.ieee.org/abstract/document/817391

Yes, that's exactly the solution I consider in my writeup (in the non-relativistic approximation, i.e., $\vec{j}=\sigma \vec{E}$. My quibble is with (12). I come to another solution though (my Eq. (41)) by assuming that there should be no divergent component for $R \rightarrow \infty$. That leaves a homogeneous electric field outside of the conductor. The math seems to be correct, but I'm a bit puzzled, whether this is the right physics.

I didn't understand their boundary condition at $\rho=L$. What has the longitudinal extent of a finite coax cable to do with the boundary conditions perpendicular to the conductors? There should be a boundary condition at $\rho \rightarrow \infty$ ($\rho$ is my $R$, i.e., the radial coordinate in cylinder coordinates).

Of course it would be desirable to have a solution for a conductor of finite length, but that seems to be too difficult for a closed, analytic solution.

Are their papers with a single wire (torus) around? Maybe that can be solved, but I guess it will involve some elliptic integrals or some other "higher functions", if it's solvable in closed form at all.

 

[Sagittarius A-Star]

I didn't understand their boundary condition at $\rho=L$.

I can't find a capital $L$ in their equation (12), only a small $l$. The $E_z$ fits there to a boundary condition at $\rho = c$.

Source:
https://www.google.de/search?q=file...ive+coaxial+cable+carrying+a+constant+current

 

[vanhees71]

Yes, sorry, it's $l$ rather than $L$. I don't understand the physics of their "fourth boundary condition", i.e., $\Phi(\rho=l,\varphi,z)=0$. Where does this come from? What has the length of the wire to do with the radial boundary conditions?

All I use is the continuity of $E_z$ (the standard continuity condition of the parallel component of the electric field at boundaries). As my forth boundary condition I use that for $\rho \rightarrow \infty$ (using the paper's notation) the electric field must not diverge, and this leaves me with the constant electric field $\vec{E}=-I/[\pi (c^2-b^2) \sigma]$ (in the non-relativistic approximation, where $\gamma=1+\mathcal{O}(\beta^2)$. In their Eq. (10) the electric field diverges logarithmically for $\rho \rightarrow \infty$, which I find even more worrysome than my constant electric field for $\rho>c$.