EM Momentum,Hidden Momentum,Centre of Energy Theorem and Lorentz Force

[pervect]

I've been trying to do some reading on this, off and on. Right now, I'm looking for a formal proof of the center-of-energy theorem in the context of a complete stress energy tensor $T_{ab}$.

Griffiths claims: http://www.science.unitn.it/~traini/didattica/fis3/Pdf2.pdf

If the center of energy of a closed system is at rest ( which I understand as the volume integral of $\vec{r} \, T_{00}$ being independent of time) then the total momentum is zero (which I understand as the volume integral of $T_{01}$, $T_{02}$, and $T_{03}$ all being zero)

The paper Griffiths cites as having a formal proof is Coleman and Van Vleck, “Origin of ‘hidden momentum forces’ on magnets,” which seems to be paywalled.

 

[Meir Achuz]

This is the C & VV paper.

 

[universal_101]

As static magnetic fields are related to loops of current, I don't have any problem with static situations containing loops of momentum flow. By a static situation, I mean one where the description of the state at any point in space does not change with time.

Consider a laminar flow, every state is static in time, but there is net flux(locally), nonetheless it respects the center of Energy theorem. But, for EM momentum(ExB), being static represents nothing is happening at all, anywhere, (i.e there is no quantity that is changing w.r.t time) and it does not respect the center of Energy theorem.

I get the feeling that the standard description of electromagnetic energy is currently missing something important. In a simple harmonic oscillator or a transverse mechanical wave, the total energy at a given point is essentially constant throughout the cycle, being converted between kinetic and potential energy. In an electromagnetic wave, the standard description implies that the energy is bunched up into the peaks. I feel it would make more sense if the energy in an electromagnetic wave was split into the equivalents of the "kinetic" and "potential" forms, where one of the forms is proportional to the square of the field and the other form is proportional to the time integral of the field (a form of the potential) times the time derivative of the field. The latter quantity would be pi/2 out of phase but equal in amplitude to the square of the field, so the total would be constant, and overall it would be similar to a mechanical transverse wave.

Well, classical description of EM radiation alone seems experimentally consistent, it is the merger of the classical electrodynamics with EM radiation which is problematic(self-force etc.), whereas, EM momentum is not represented as something moving like radiation, it is supposed to represent net momentum in a stationary box! that's the problem!

 

[universal_101]

Consider the EM forces between two moving charges. Explain that without EM momentum.

Again, this is all on theoretical grounds,(like onoochin's paradox etc.), where charges are supposed to feel a force spontaneously without back reaction on other charge, and all these are speculated examples, because nobody has ever seen anything moving on its own, which in turn means there is NO experimental evidence. And it is curious, that EM momentum is present only when we need to resolve a theoretical paradox.

 

[TrickyDicky]

"The external force that keep charge stationary w.r.t the loop, supposedly produces the EM momentum when setting up the static charge-current setup. So, does it mean that, if there were NO external force there would not be any EM momentum?"

No. In the absence of an external force to hold the charge in place, the charge would acquire momentum and the EM field would acquire equal and opposite momentum. The total momentum would then be zero, conserving momentum.

There appears to be an important dose of arbitrariness in Franklin's paper when it comes to the distinction between "EM momentum" and "hidden momentum", and also between mechanical and electromagnetic momentum of charges, so that in the end his conclusion could be interpreted as more semantic than physical(look at his insistence that what other authors call hidden momentum in equation 49 is in fact the good EM momentum) despite his effort to base it in physical considerations.
That he has to declare the center of energy theorem not valid for EM to make his point is a bad sign too.
It is using a prerelativistic theorem(Poynting's) to invalidate a relativistic one.

 

[TrickyDicky]

I think you missed the point made, I'm saying there is NO experiment that demands the Introduction of EM momentum, whereas, in the case of the neutrinos there were Experiments which suggested something is missing(when conserving energy/momentum).

Therefore, if you can show me an Experiment in which the stationary charge starts moving spontaneously without any back reaction force on the current loop, I would be willing to accept that something may be missing and is presently undetectable.

Let's see, that was the original expectation, but then came Trouton-Noble experiment and others similar that contrary to that expectation had a null result, which prompted the introduction by Lorentz and later Laue of the mechanical hidden momentum, so hidden momentum was indeed a response to experimental evidence.
EM field momentum had been introduced much earlier by Maxwell and Lorentz to salvage Newton's third law, do you agree with EM momentum for EM radiation but not for quasi-static charges?

 

[pervect]

The Trouton-Noble experiment is the one case where I have a good feeling about the origin and presence of something that could be called hidden momentum. The momentum in that case is in the matter stress-energy tensor. Stress in the rod doesn't contribute to the momentum or energy flux in the rest frame, but, when the stress is transformed, it does contribute to the momentum and energy flux in a moving frame. This is a simple consequence of the way the stress-energy tensor transforms. Inclusion of the transformed stress terms is necessary to describe the mechanical part of the momentum in the moving frame.

I don't think that the other cases of "hidden momentum" discussed in the literature involve momentum derivable from the matter-stress energy tensor, though. It all seems terribly murky , still :-(

 

[vanhees71]

In most cases, "hidden momentum" is in fact simply momentum. E.g., in problems where there are currents in a conductor at rest, in fact you have moving charges (e.g., electrons moving relative to the crystal lattice of a metal wire which is at rest in the considered frame). You can treat this with a simple model of an electron fluid moving relative to the crystal lattice which defines the restframe of the wire. If you write everything out in a covariant way (including using Ohm's Law in the correct form, i.e., $\vec{j}=\sigma (\vec{E}+\vec{v} \times \vec{B}/c)$, no trouble with "hidden momenta" appear.

Usually that's the most save recipy to treat such problems: Go to a preferred frame of reference of the given situation, i.e., one, where you can evaluate everything in a simple way and then do the appropriate Lorentz transformation to the frame of reference you are interested in.

There's also the famous example concerning the mass of a charged capacitor, including the energy of the electric field between the plates. Just calculate everything in the restframe of the plates and then transform to an arbitrary frame, where the capacitor is moving. No problems whatsoever appear in the energy-momentum relation, because everything is covariant by construction. The apparent problems come from not considering the forces needed to keep the capacitor plates at constant distance against the electrostatic attraction between them when they are oppositely charged.

 

[universal_101]

Let's see, that was the original expectation, but then came Trouton-Noble experiment and others similar that contrary to that expectation had a null result, which prompted the introduction by Lorentz and later Laue of the mechanical hidden momentum, so hidden momentum was indeed a response to experimental evidence.
EM field momentum had been introduced much earlier by Maxwell and Lorentz to salvage Newton's third law, do you agree with EM momentum for EM radiation but not for quasi-static charges?

I don't know how an experiment debunking aether and the associated EM momentum with the aether(Maxwell and Lorentz), is related to the EM momentum associated with the quasistatic fields.

Ofcourse, the discussion is about the EM momentum of quasistatic fields, for we don't need to introduce hidden momentum for EM momentum of radiation fields. It is the EM momentum associated with static fields which is paradoxical. So, No, your assertion of there being experimental evidence supporting EM momentum of static fields is not correct.

 

[TrickyDicky]

I don't know how an experiment debunking aether and the associated EM momentum with the aether(Maxwell and Lorentz), is related to the EM momentum associated with the quasistatic fields.

Ofcourse, the discussion is about the EM momentum of quasistatic fields, for we don't need to introduce hidden momentum for EM momentum of radiation fields. It is the EM momentum associated with static fields which is paradoxical. So, No, your assertion of there being experimental evidence supporting EM momentum of static fields is not correct.

I didn't mention any experiment debunking aether and didn't assert anything about experimental evidence about EM momentum in the static case. I talked about experiments in relation with hidden momentum because from post #2 linked paper much of the discussion was about that paper.
I take issue with that paper, and wonder if someone else finds the way it supposedly "shows" that the center of energy theorem doesn't apply to EM momentum is not valid at all, he uses the Poynting theorem for electrodynamics instead of its generalization$\frac{\partial}{\partial t}\left(u_e + u_m\right) + \nabla\cdot \left( \mathbf{S}_e + \mathbf{S}_m\right) = 0$ that includes the mechanical kinetic energy density and mechanical Poynting vector.

Most of the debate arises from the arbitrariness of the separation between mechanical and electromagnetic momentum, which allows each author to arrange terms as they see more fit.

I'm still not sure if you are implying that in (quasi)static situations momentum conservation doesn't apply, that I don't think is acceptable.

 

[universal_101]

I didn't mention any experiment debunking aether and didn't assert anything about experimental evidence about EM momentum in the static case. I talked about experiments in relation with hidden momentum because from post #2 linked paper much of the discussion was about that paper.

Since you quoted me, I thought you were responding to the problem I was referring to, its OK though.

I take issue with that paper, and wonder if someone else finds the way it supposedly "shows" that the center of energy theorem doesn't apply to EM momentum is not valid at all, he uses the Poynting theorem for electrodynamics instead of its generalization$\frac{\partial}{\partial t}\left(u_e + u_m\right) + \nabla\cdot \left( \mathbf{S}_e + \mathbf{S}_m\right) = 0$ that includes the mechanical kinetic energy density and mechanical Poynting vector.

Most of the debate arises from the arbitrariness of the separation between mechanical and electromagnetic momentum, which allows each author to arrange terms as they see more fit.

Well, it is fairly easy to see that center of Energy theorem was supposedly invalidated in that paper, so as to get rid of the so called hidden momentum, which adds to the confusion already in place because of the EM momentum in static fields case.

I'm still not sure if you are implying that in (quasi)static situations momentum conservation doesn't apply, that I don't think is acceptable.

Just look at the original starting post by me for detailed overview of the problem.

 

[TrickyDicky]

Since you quoted me, I thought you were responding to the problem I was referring to, its OK though.

Just look at the original starting post by me for detailed overview of the problem.

The OP mentions both hidden momentum and EM momentum, but your main concern appeared to be hidden momentum since you start your argument with the paper on hidden momentum by Griffith so it wasn't so clear to me what problem you referred to.

Regarding EM momentum, in a static situation the net momentum is zero, how you manage the book-keeping to combine momenta to get that vanishing net momentum is to some extent arbitrary and theory-dependent and as shown by the last century fighting over this without a real agreement among the different experts.

 

[universal_101]

Regarding EM momentum, in a static situation the net momentum is zero, how you manage the book-keeping to combine momenta to get that vanishing net momentum is to some extent arbitrary and theory-dependent and as shown by the last century fighting over this without a real agreement among the different experts.

Well, thanks for sharing your view, but same problem resides with hidden momentum too. Consider the same setup of a loop carrying current and a charge near by, this static situation has zero net momentum, according to you(EM Momentum{ExB} -Hidden momentum{due to current in a E Field} = 0). Now, when the current dies down, the charge starts to move without any back reaction on loop, implying there is net momentum in this situation. So how did we end up with net momentum, from zero momentum without applying any external force ?

 

[TrickyDicky]

Well, thanks for sharing your view, but same problem resides with hidden momentum too. Consider the same setup of a loop carrying current and a charge near by, this static situation has zero net momentum, according to you(EM Momentum{ExB} -Hidden momentum{due to current in a E Field} = 0). Now, when the current dies down, the charge starts to move without any back reaction on loop, implying there is net momentum in this situation. So how did we end up with net momentum, from zero momentum without applying any external force ?

IMO it all depends on how one analyzes the scenario and distributes internal and external forces and wether the initial set up system is treated as a closed or open loop, for instance it is different considering your setup as a closed loop or as an open loop with an external force keeping the charge in place. I'm not claiming that there are no difficulties or that everything is clear and solved, but I certainly don't regard as solutions Mansuripur's nonsense about Lorentz force law not being relativistic or non-peer reviewed papers like Franklin's dismissing relativity basic principles like inertia of energy that verge on the crackpotty.

To address your point more directly, when the current dies if the charge starts moving, i.e. it accelerates, it must radiate with energy flux S and momentum flux S/c^2 so taking into account this EM field momentum one should be able to conserve total momentum.

 

[vanhees71]

Mansuripur's nonsense is repaired in articles in several reviewed papers. What I never unerstood is, how this nonsense could pass the peer-review barrier, which should have prevented it from being published in the first place. It's a shame for PRL! Here are some papers, where the wrong claims are revealed and the apparent paradox clearly solved:

D. J. Griffiths and V. Hnizdo. Mansuripur's paradox. Am. Jour. Phys., 81:570-574, 2013.
http://arxiv.org/abs/1303.0732

A particularly clear exposition can be found here:

http://www.physics.princeton.edu/~mcdonald/examples/current.pdf

You find tons of other great writeups about interesting problems with classical physics, particularly electromagnetics and hidden momentum, at MacDonald's website:

http://www.physics.princeton.edu/~mcdonald/examples/

 

[universal_101]

I'm not claiming that there are no difficulties or that everything is clear and solved, but I certainly don't regard as solutions Mansuripur's nonsense about Lorentz force law not being relativistic or non-peer reviewed papers like Franklin's dismissing relativity basic principles like inertia of energy that verge on the crackpotty.

Well, even if you are sure about you being correct, there is No need to call anyone anything, I mean, just because if one is among the "commonly accepted solution/theory" people, he/she does not have any right to call others name.

To address your point more directly, when the current dies if the charge starts moving, i.e. it accelerates, it must radiate with energy flux S and momentum flux S/c^2 so taking into account this EM field momentum one should be able to conserve total momentum.

I don't think, radiation is going to help, because radiation 'experimentally' comes with back reaction force, it's a whole package with zero net momentum. Therefore you still need to explain, the 'apparent' violation of momentum conservation.

 

[universal_101]

Mansuripur's nonsense is repaired in articles in several reviewed papers. What I never unerstood is, how this nonsense could pass the peer-review barrier, which should have prevented it from being published in the first place. It's a shame for PRL! Here are some papers, where the wrong claims are revealed and the apparent paradox clearly solved:

D. J. Griffiths and V. Hnizdo. Mansuripur's paradox. Am. Jour. Phys., 81:570-574, 2013.
http://arxiv.org/abs/1303.0732

A particularly clear exposition can be found here:

http://www.physics.princeton.edu/~mcdonald/examples/current.pdf
[/url]

May be someone can repair this 'nonsense' too, the upper setup is with loop carrying current and a stationary charge w.r.t loop, in the lower setup the current dies down and according to Maxwell, the charge starts to move, without any back reaction force.

https://www.physicsforums.com/attachment.php?attachmentid=68924&stc=1&d=1398184742​

So, what is the solution to this 'apparent' violation of conservation of momentum.

 

[TrickyDicky]

Well, even if you are sure about you being correct, there is No need to call anyone anything, I mean, just because if one is among the "commonly accepted solution/theory" people, he/she does not have any right to call others name.

Nothing I said is usually considered name-calling.

I don't think, radiation is going to help, because radiation 'experimentally' comes with back reaction force, it's a whole package with zero net momentum. Therefore you still need to explain, the 'apparent' violation of momentum conservation.

That back reaction of the field in the form of EM momentum that opposes the charge's momentum was what you were looking for, no?

 

[universal_101]

Nothing I said is usually considered name-calling.

I don't want to lecture but I would like to put across what I feel. It seems you are labeling an author for his/her controversial mistakes, and the term you used is derogatory to say the least, especially when we are talking about scientific people, who are supposed to be wrong automatically if they are not correct. So, considering the highly controversial nature of the problem, I think it is alright for authors to go wrong or be not correct.

That back reaction of the field in the form of EM momentum that opposes the charge's momentum was what you were looking for, no?

Here it is again, the upper part of the image is the static situation with current, the lower part represents the situation after the current dies down. The question is clear, why are we ending up with net momentum, when current dies down, if we started with net zero momentum and there is No external force involved ? This is a serious problem with classical electrodynamics represented by Maxwell's equations.

 

[vanhees71]

Could you point me to a clear statement of the problem? The only thing wrong I can make out in the figures is the assignment of the momentum in non-relativistic (Newtonian) terms. If it comes to "hidden momentum" it is very important to use relativistic expressions everywhere.

Further, it's clear that in both situations there are forces acting on the loop and the byflying charge. It's not a priori clear to me, whether you really can neglect the radiation reaction on the whole sysetm. Then it becomes a pretty complicated problem.

 

[universal_101]

Could you point me to a clear statement of the problem? The only thing wrong I can make out in the figures is the assignment of the momentum in non-relativistic (Newtonian) terms. If it comes to "hidden momentum" it is very important to use relativistic expressions everywhere.
Further, it's clear that in both situations there are forces acting on the loop and the byflying charge. It's not a priori clear to me, whether you really can neglect the radiation reaction on the whole sysetm. Then it becomes a pretty complicated problem.

Yes, we can neglect the radiation.

Further, seems like you are missing the point, first, ofcourse the hidden momentum in the figure is relativistic in nature, but still it comes under the domain of classical electrodynamics. And yes, we can use relativistic expressions everywhere, but that's not going to change anything.

And here is the overview of the problem again,

There is a current carrying loop and a charge at rest w.r.t each other, now, according to Griffiths, there is an EM momentum(ExB) density around the setup and a relativistic mechanical momentum residing in the current carrying loop. And this make the whole setup stationary and the situation does not violate center of Energy theorem. Do you agree till now?

If yes, then respecting the conservation of momentum, we should also have zero momentum when the current in the loop dies down, but Maxwell's equations predict that only the charge will experience the force ($\textbf{F} = q\textbf{E} = -q\frac{∂\textbf{A}}{∂t}$), whereas, we don't have anything in Maxwell's Equations that says there will be an equal and opposite force on the loop.

Therefore, we end up with net momentum, in the form of moving charge, which violates conservation of momentum.

I hope this is a clear representation of the problem, I was referring to.

Thanks.

 

[TrickyDicky]

Nope, there is no violation of momentum. Admittedly it is much simpler to neglect radiation here.
When the current dies there is no hidden momentum or anything relativistic, but anyway...
It is quite easy to see that there is going to be an opposite momentum in the electromagnetic field within the loop(let's consider it a thin solenoid to be more specific).
Think about the electric field due to the charge inside the solenoid where previously was the magnetic field when there was current. To this field it corresponds a momentum density within the solenoid that can be integrated to a momentum that is equal and opposite to the linear momentum of the charge.

Basically where we had net momentum zero with the EM and the hidden momentum, now we just substitute hidden momentum by momentum of the charge, the EM momentum is the same.

 

[universal_101]

Basically where we had net momentum zero with the EM and the hidden momentum, now we just substitute hidden momentum by momentum of the charge, the EM momentum is the same.

How can there be EM momentum when there is NO current and therefore NO magnetic field, i.e. $\textbf{E}\times \textbf{B} = 0$ everywhere, for, $\textbf{B} = 0$ as current $\textbf{I} = 0$.

 

[TrickyDicky]

How can there be EM momentum when there is NO current and therefore NO magnetic field, i.e. $\textbf{E}\times \textbf{B} = 0$ everywhere, for, $\textbf{B} = 0$ as current $\textbf{I} = 0$.

If there were no magnetic field initially and no charge, there would be no momentum imparted to the charge, the momentum density is EXB, it is the changing magnetic field that imparts that momentum to the charge.
The vector product refers to 3D location of the fields rather than their temporal sequence.

 

[universal_101]

If there were no magnetic field initially and no charge, there would be no momentum imparted to the charge, the momentum density is EXB, it is the changing magnetic field that imparts that momentum to the charge.
The vector product refers to 3D location of the fields rather than their temporal sequence.

There is magnetic field initially, since there is current initially, the charge starts to move when we let the current die down. It is a simple Setup, I don't know what is so difficult to understand.

That is, a current carrying loop and a charge nearby, we let the current die down and that makes the charge move. It is that simple.

 

[TrickyDicky]

There is magnetic field initially, since there is current initially, the charge starts to move when we let the current die down. It is a simple Setup, I don't know what is so difficult to understand.

That is, a current carrying loop and a charge nearby, we let the current die down and that makes the charge move. It is that simple.

Really, this doesn't have anything to do with relativity anymore, you should take it to classical physics if you don't understand the answer given which is simple too.

 

[universal_101]

Really, this doesn't have anything to do with relativity anymore, you should take it to classical physics if you don't understand the answer given which is simple too.

Well, Maxwell's equations and Lorentz Force being Lorentz invariant, and hidden momentum being relativistic, the above problem very well comes under the domain of SR. Even the center of Energy theorem originates from SR.

And I don't see any answer in any of your posts, and anyone who had any theoretical background with classical electrodynamics acknowledges that there is a long lived problem with mechanical conservation laws, and this not only resulted in EM momentum(static fields) but also in hidden momentum(relativistic) because according to SR the EM momentum must also respect the center of Energy theorem.

Now, ignoring a problem is not a solution, that is, pointing me to the classical forum because there aren't enough gammas involved.

 

[Jonathan Scott]

According to some notes I have from 1993 (which I wrote myself, so they are not necessarily authoritative), when the current (assumed to be made up of an overall neutral mixture of charges) is changing around a ring, there is an imbalance of charge carriers as seen from points outside the ring, with the "coming" and "going" sides of the current loop having excesses of charge in either direction, with the sense being determined by the rate of change of the current. This results in an electric dipole as seen by a charge positioned near to the ring, which is presumably the same as the electric field predicted by Maxwell's equations.

If this is correct, then it is clear that this is effectively equivalent to a simple electrostatic field and the back-reaction works in the usual way.

 

[universal_101]

According to some notes I have from 1993 (which I wrote myself, so they are not necessarily authoritative), when the current (assumed to be made up of an overall neutral mixture of charges) is changing around a ring, there is an imbalance of charge carriers as seen from points outside the ring, with the "coming" and "going" sides of the current loop having excesses of charge in either direction, with the sense being determined by the rate of change of the current. This results in an electric dipole as seen by a charge positioned near to the ring, which is presumably the same as the electric field predicted by Maxwell's equations.

If this is correct, then it is clear that this is effectively equivalent to a simple electrostatic field and the back-reaction works in the usual way.

This is an insightful approach, additionally we would not need static EM momentum and therefore no need for hidden momentum either. But we are assuming the current to be ideal(Maxwell's), and in according to Maxwell's equations changing current does not produce any charge densities. So even the small suggeested change would require the Maxwell's equations to be changed.

 

[vanhees71]

Yes, we can neglect the radiation.

Further, seems like you are missing the point, first, ofcourse the hidden momentum in the figure is relativistic in nature, but still it comes under the domain of classical electrodynamics. And yes, we can use relativistic expressions everywhere, but that's not going to change anything.

Thanks.

I still don't know exactly which problem you are really discssing, but the use of relativistic expressions for all terms is essential when it comes to "hidden momentum", which is a relativistic effect to begin with. A very illuminating example can be found in Griffiths Electrodynamics textbook 3rd. edition Example 12.12 (p. 520).

Most apparent paradoxes of this kind (like the Trouton Noble experiment) come from the use of non-relativistic expressions for the mechanics part of the problem.

 

[Jonathan Scott]

This is an insightful approach, additionally we would not need static EM momentum and therefore no need for hidden momentum either. But we are assuming the current to be ideal(Maxwell's), and in according to Maxwell's equations changing current does not produce any charge densities. So even the small suggeested change would require the Maxwell's equations to be changed.

I don't have the detailed workings which led me to that conclusion, but I remember it was very closely related to another paradox, which involves a current in a straight conductor and a charge moving parallel to the current in the conductor. In the initial frame, the charge bends towards the conductor because of the magnetic field, but if you switch to the frame of the charge it is not immediately obvious why the charge, initially at rest, should accelerate towards the conductor, which is moving in the opposite direction.

If I remember correctly, it turns out (rather counter-intuitively) that when you apply a Lorentz transformation to a segment of the wire, you end up with more charges going one way than the other in a fixed length, essentially because of the change of simultaneity at the ends, so there is now a net charge per unit length, and the charge at rest experiences an electrostatic force.

I must admit it's not immediately obvious to me how something similar applies to the ring case for a changing current but it seems plausible. I had so many old physics notes that I used to pick out "gems" which seemed interesting and store them separately so I could find them easily, but the trouble with that is that in some cases I don't have enough of the background material to understand them now!

At the time I was also studying an alternative way of looking at electromagnetic forces which is mostly in terms of four-vectors, avoiding explicit magnetic fields.
$$
\frac{d}{dt} \left( p^* + q A^* \right ) = q \frac{\partial}{\partial x} (A^* . v )
$$
In this notation, p, x, v and A are four-vectors, and the asterisk denotes switching the sign of the space part. If the time is replaced with the proper time, the right hand side becomes the four-gradient of the potential in the rest frame of the charge, which is an invariant scalar.

This equation is mathematically equivalent to the following more conventional equation for the rate of change of energy and momentum expressed in terms of the usual 3-vectors:
$$
\frac{d}{dt} \left( p_0 - \mathbf{p} \right) = q \mathbf{E}.\mathbf{v} - q \mathbf{E} - q \mathbf{v} \times \mathbf{B}
$$
However, the four-vector form clearly does not involve any cross-product terms and seems more straightforward.

 

[universal_101]

I still don't know exactly which problem you are really discussing, but the use of relativistic expressions for all terms is essential when it comes to "hidden momentum", which is a relativistic effect to begin with. A very illuminating example can be found in Griffiths Electrodynamics textbook 3rd. edition Example 12.12 (p. 520).

Most apparent paradoxes of this kind (like the Trouton Noble experiment) come from the use of non-relativistic expressions for the mechanics part of the problem.

Defining the hidden momentum is not the problem here, so everything you said is very well known. The point is, introducing/injecting the hidden momentum does not save classical electrodynamics to contradict with the classical mechanical laws. Jonathan Scott recognizes the problem, therefore all the articles/papers written on hidden momentum are invalid, until we get to resolve the basic contradiction. That is, the property of hidden momentum being relativistic or other properties, are not important, if these momentum violates the conservation of momentum theorem.

Remembering that it is the conservation of momentum theorem which lead to the invention of static EM Momentum, which in turn (by another form of the momentum conservation theorem) lead to the invention of hidden momentum. But sadly, introducing momentum after momentum, does not solve the redundancy of the original missing momentum, and it can be easily seen that redundancy can only be solved by introducing a back reaction force.

 

[Jonathan Scott]

I think I now have a qualitative explanation for the charge displacement effect.

According to the exact Lienard-Wiechert potentials, the potential due to a moving charge seen at retarded time is effectively from the location at which the charge would be at the current time if it kept moving in a straight line at constant speed.

For purposes of our ring, with constant non-relativistic charge velocity, a straight line simply means that the effective source of the potential is where the charge would be now. If the charges are evenly distributed around the ring, the potential is then also effectively due to an even distribution of charges around the ring.

However, if the charge is accelerating or decelerating, the extrapolated positions will not be correct, and will reflect an earlier value of the speed. Points on the far side of the ring will be further back in time, so if the flow is decelerating, the charge positions will be further ahead of their true positions on the far side of the ring than on the near side. This causes an apparent net displacement of the charge towards one side of the ring.

The effect is proportional to the rate of change of current, and is also proportional to the difference in distance to the near and far sides of the ring and to the width of the ring, so I think that it works from a dimensional point of view.

I'll leave sorting out the details (and checking the signs against Lenz's Law) as an exercise for the student, mainly because I seem to be too rusty to sort it out myself.

 

[vanhees71]

Defining the hidden momentum is not the problem here, so everything you said is very well known. The point is, introducing/injecting the hidden momentum does not save classical electrodynamics to contradict with the classical mechanical laws. Jonathan Scott recognizes the problem, therefore all the articles/papers written on hidden momentum are invalid, until we get to resolve the basic contradiction. That is, the property of hidden momentum being relativistic or other properties, are not important, if these momentum violates the conservation of momentum theorem.

Remembering that it is the conservation of momentum theorem which lead to the invention of static EM Momentum, which in turn (by another form of the momentum conservation theorem) lead to the invention of hidden momentum. But sadly, introducing momentum after momentum, does not solve the redundancy of the original missing momentum, and it can be easily seen that redundancy can only be solved by introducing a back reaction force.

Obviously you don't want to define the problem properly or hint to the posting/source, where this is done. So I cannot analyze this specific example. One thing is, however, very safe to say: There is not contradiction with the conservation law of momentum within special relativity, and electrodynamics is a theory that is consistent special relativity.

Minkowski space is Poincare symmetric, and Poincare symmetry includes spatial translation invariance. The generator of this symmetry is by definition called (canonical) momentum and this is a conserved quantity due to Noeter's theorem. From this it is very clear that the total momentum of any closed system is conserved for the system of electromagnetic fields + charges).

 

[Jonathan Scott]

Obviously you don't want to define the problem properly or hint to the posting/source, where this is done. So I cannot analyze this specific example. One thing is, however, very safe to say: There is not contradiction with the conservation law of momentum within special relativity, and electrodynamics is a theory that is consistent special relativity.

Minkowski space is Poincare symmetric, and Poincare symmetry includes spatial translation invariance. The generator of this symmetry is by definition called (canonical) momentum and this is a conserved quantity due to Noeter's theorem. From this it is very clear that the total momentum of any closed system is conserved for the system of electromagnetic fields + charges).

The problem is the Shockley-James paradox, which can be found in many places on the web.

It's often defined in quite a complicated way to eliminate spurious explanations, but basically it's that if you have a changing current in a ring, causing changing magnetic flux, standard E/M equations show that there is an electric field around the ring which can act on a free charged particle near the ring. However, there is no obvious equation which shows how the particle can act back on the ring to conserve momentum. As we fully expect momentum to be conserved, we want to know the mechanism by which the particle acts back on the ring.

I had not previously looked at the Shockley-James paradox, but back in 1993 I had previously looked at the effect of a changing current in a loop and concluded that the electric field around the ring is effectively due to an apparent charge density imbalance when the current is changing, causing an electric dipole effect. If this is correct, it seems to provide a possible basis for the back-reaction (noting that the retarded potential due to the free particle from the point of view of the rest frame of a charged particle in the ring follows a similar pattern).