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

[universal_101]

I recently read Griffiths paper on Hidden momentum, and still didn't found it complete. Following is the short summary.

The usual setup of current carrying loop and a charge lying nearby, according to the paper, Shockley and James presented the problem, that is, when we let the current die down then according to Maxwell's Equations there is a force on the nearby charge due to the induced electric field, but the same Maxwell's Equations do not predict a reaction force on the loop, and therefore there is a problem.

Now Shockley and James suggested that there is a hidden momentum in the current carrying loop so as to respect momentum conservation, and the Griffiths paper suggests that, since there is Electromagnetic momentum involved in the setup when the current is flowing, there must be some other momentum too in order to balance the EM momentum, otherwise the center of Energy Theorem from SR would be violated.

And Griffiths shows that this other momentum is hidden momentum from the Shockley and James Paper and is exactly equal and opposite to the EM Momentum. And shows that, this EM momentum is the same, which the charge would acquire when the current dies down.

And the hidden momentum is characterized as relativistic mechanical momentum, which is balanced by the EM Momentum, and when we let go the current, the EM momentum goes into the point charge whereas the hidden momentum comes into being and ends up moving the loop in opposite direction and therefore the reaction force. Therefore all the problems solved!

Well not quite, Remember that EM momentum is per unit volume, so we have non-zero EM momentum wherever E and B are perpendicular, in other words EM momentum is spread all over the volume, whereas hidden momentum is associated with the moving charges in the current carrying loop. So, how-come the momentum associated with moving point charges in presence of an E field is balanced by the momentum spread all over the volume.

Secondly, if hidden momentum is mechanical, how-come the loop is not moving already, that is, what kind of mechanical momentum does not produce motion, this is unacceptable physics.

Thirdly, we still don't have the solution to the original problem, that is, Maxwell's equations still do not predict back reaction force from the charge on the loop when current is changing. Ofcourse considering that the situation is well under the domain of the Maxwell's equations.

I think it is one thing to suggest that quasi-static fields carry momentum without moving anything, and entirely different and possibly wrong that mechanical momentum can also exist without having any net motion.

 

[Bill_K]

I recently read Griffiths paper on Hidden momentum, and still didn't found it complete.

This paper debunks the idea of "hidden momentum."

 

[universal_101]

This paper debunks the idea of "hidden momentum."

This is worse than the hidden momentum itself, that is neglecting the consequences of Center of Energy theorem for EM momentum. I mean it does not seem like a solution to the problem, rather it is a denial of the problem, that Maxwell's Equations don't predict the back reaction force on the changing current loop.

 

[Meir Achuz]

Maxwell's Equations don't predict the back reaction force on the changing current loop.

Yes...

 

[universal_101]

And that seems quite a big problem, and the problem becomes of utmost importance if Maxwell's equations are supposed to represent the classical electrodynamics, and it suggests that Maxwell's equations are necessarily incomplete.

 

[Meir Achuz]

it is a denial of the problem, that Maxwell's Equations don't predict the back reaction force on the changing current loop.

There is NO "back reaction force on the changing current loop."

 

[universal_101]

There is NO "back reaction force on the changing current loop."

Yes, but there is NO back reaction force experimentally or theoretically/principally?

That is, how do you get to upheld local momentum conservation if there is NO back reaction Force?

 

[Meir Achuz]

[URL="http://arxiv.org/abs/1302.3880" ]"

[/PLAIN]
"http://arxiv.org/abs/1302.3880"
This paper[/URL] debunks the idea of "hidden momentum.

 

[Meir Achuz]

Yes, but there is NO back reaction force experimentally or theoretically/principally?

That is, how do you get to upheld local momentum conservation if there is NO back reaction Force?

That is why and how EM momentum is introduced in textbooks.

 

[universal_101]

That is why and how EM momentum is introduced in textbooks.

Agreed, But you cannot use EM momentum for both conditions, that is, first when the charge is not moving there is EM momentum present, and second when this same EM momentum comes up in the changing current loop while charge is moving and experiencing Lorentz Force.

So basically, instead of having a back reaction force, we are expecting quasi-static EM fields to have momentum, and we are directly associating this momentum to the momentum that the loop would gain. Well then, that is NO more than a trick, for atleast we must know how this supposed EM Momentum is getting transferred to the loop, according to Maxwell's Equations, right? Because the whole setup is well under the domain of classical mechanics.

So I think, it does not matter from where we bring in the momentum for loop to have, but we must accommodate that in Maxwell's equations, to show that there is a back reaction Force.

 

[Meir Achuz]

So I think, it does not matter from where we bring in the momentum for loop to have, but we must accommodate that in Maxwell's equations, to show that there is a back reaction Force.

The loop has NO momentum, and Maxwell's equations show there is NO back reaction force.
The momentum is in the EM field, not the loop.
Did you read the reference Bill K. suggested?

 

[universal_101]

The loop has NO momentum, and Maxwell's equations show there is NO back reaction force.
The momentum is in the EM field, not the loop.
Did you read the reference Bill K. suggested?

Are you suggesting that EM momentum which can be present in a non-moving system, is the one that gets transferred to the moving charge ? Therefore all problems solved!

Yes I did read the reference, and it clearly says that, center of energy theorem does not apply to EM momentum, how convenient is that.

 

[Meir Achuz]

Are you suggesting that EM momentum which can be present in a non-moving system, is the one that gets transferred to the moving charge ? Therefore all problems solved!
Yes I did read the reference, and it clearly says that, center of energy theorem does not apply to EM momentum, how convenient is that.

1. In that reference, an external force holds the charge in place, and produces the EM momentum.
Without the external force the charge would acquire momentum, and the sum of its momentum and the EM momentum would be zero, conserving momentum. That is precisely how textbooks introduce EM momentum.
2. That reference SHOWS "that the center of energy theorem does not apply to EM momentum".

 

[universal_101]

1. In that reference, an external force holds the charge in place, and produces the EM momentum.
Without the external force the charge would acquire momentum, and the sum of its momentum and the EM momentum would be zero, conserving momentum. That is precisely how textbooks introduce EM momentum.
2. That reference SHOWS "that the center of energy theorem does not apply to EM momentum".

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 ?

I think it is time to make it little bit neat,

1.) The current carrying loop never experiences a back reaction force, all the momentum and force exchange is between 'EM Momentum' and the charge. Correct?

2.) If we use external forces to setup the static charge-current experiment, the system has non-zero net momentum in the form of EM Momentum. Correct?

3.) And when we let the current die down, this net EM momentum produces net momentum in the form of the Lorentz Force on the charge. Correct?

Answers to the above assertions would most certainly help me understanding your position on the matter. Thanks

 

[Meir Achuz]

"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.
The answers to 1, 2, and 3 are all yes.

 

[universal_101]

But how come you get to choose when to use 'conservation of angular momentum' and when to not use, that is, in one case you have net momentum(EM) and nothing is moving and nobody complains about the violation of conservation of momentum, whereas in the other situation, when the current dies down and charge starts moving, you chose to use conservation of momentum to cancel the EM momentum by the mechanical momentum of moving charge(due to Lorentz force).

 

[Meir Achuz]

External force = rate of change of momentum.
In your 1,2,3:
1. With no external force the mechanical momentum of the charge and the EM momentum add up to zero, conserving momentum.
2. With an external force holding the charge in place,tum the external force produces the EM momentum by
F=dP/dt.
3. If the current dies down (after (2) above), the EM momentum is transferred to the charge, conserving momentum.

 

[universal_101]

Well the point is, the system with net non-zero momentum has the non-moving center of energy. That is, there is NO way to detect the EM Momentum, it is abstract in a sense that, it is supposed to be there but nothing is moving.

 

[Meir Achuz]

Your question (3) detects EM momentum when it is transferred to the charge.
What is detection? Do you want to hold it in your hands?
That is how neutrinos are detected.
EM momentum is as real as neutrinos.

 

[universal_101]

Well, just do an experiment and show me, that while the charge is moving the loop is stationary, and then it can be considered as a detection of EM momentum.

Just do the simple experiment, and show us that a part of a charge-current system can move spontaneously when the current dies down, without moving any other part of the system, and may be it would be believable that center of energy theorem does not apply to EM Momentum.

 

[Jonathan Scott]

I'm puzzled by what's going on here, as it doesn't seem to be valid at all.

It should be well known by now that the standard definition of EM momentum as E x B can only be physically valid in cases where there are no charges present. See for example J W Butler's 1969 paper "A Proposed Electromagnetic Momentum-Energy 4-Vector for Charged Bodies", Am. J. Phys. 37, 1258 (1969). Even Butler's solution only covers limited cases, but it satisfies conservation laws in the usual way.

 

[pervect]

Well the point is, the system with net non-zero momentum has the non-moving center of energy. That is, there is NO way to detect the EM Momentum, it is abstract in a sense that, it is supposed to be there but nothing is moving.

Let's take the second point first - how sure are we that there IS a "center of energy" theorem? Griffiths claims there is, and Franklin claims there isn't. I'm of the mind to think that the conservative approach is to regard the theorem as questionable at this point.

Onto the first point - I don't agree. We know radiation caries momentum, this is one example of EM momentum. For instance, a directional radio antenna or a directional light beam will create thrust.

Franklin's paper isn't doing much for me, but, in spite of the lack of provenance

http://www.physics.usu.edu/Wheeler/EM/EMenergy.pdf

makes more sense to me (I haven't worked through all the details, it's funny that a semi-random paper from the internet seems clearer and more conventional than some of the peer-reviewed published papers I've found.).

Bottom line, it appears that Poynting vector represents the energy flux, and the Maxwell stress tensor represents the momentum flux.

Additonally we appear to have $\frac{1}{2}E \cdot D + \frac{1}{2}B \cdot H$ for the energy density, One thing that bothers me a bit is that "polarization ambiguity" means that D is a bit ambiguous. Also, I haven't seen any similar expression for the momentum density rather than it's flux - maybe this is where the controversy arises, in how to go from the flux to the density.

I'm also finding Wiki's discussion http://en.wikipedia.org/wiki/Polarization_density helpful, in sorting out the P's, D's, E's, M's which I've mostly forgotten :(. This is also where I ran across the idea of "Polarization ambiguity".

 

[universal_101]

Your question (3) detects EM momentum when it is transferred to the charge.
What is detection? Do you want to hold it in your hands?
That is how neutrinos are detected.
EM momentum is as real as neutrinos.

Neutrinos were historically expected as a consequence of experiments, whereas, EM momentum(of quasistatic fields) does not have experiments supporting them, it is all theoretical book-keeping, in order to avoid the conflict with momentum conservation.

I'm puzzled by what's going on here, as it doesn't seem to be valid at all.

It should be well known by now that the standard definition of EM momentum as E x B can only be physically valid in cases where there are no charges present. See for example J W Butler's 1969 paper "A Proposed Electromagnetic Momentum-Energy 4-Vector for Charged Bodies", Am. J. Phys. 37, 1258 (1969). Even Butler's solution only covers limited cases, but it satisfies conservation laws in the usual way.

Do you mean to say, quasistatic EM fields momentum is not physically valid ?

Let's take the second point first - how sure are we that there IS a "center of energy" theorem? Griffiths claims there is, and Franklin claims there isn't. I'm of the mind to think that the conservative approach is to regard the theorem as questionable at this point.

I think debunking the center of Energy theorem would allow the violation of conservation of momentum, in a sense that, only one part of the momentum is ever detected, the other part can be anywhere and in any-form. Which in-turn would inevitably make us doubt other experiments involving EM fields and momentum conservation(that is there would be chances of some undetectable missing momentum, because we NO longer respect the center of Energy theorem and therefore a perfectly stationary system can have net momentum).

Onto the first point - I don't agree. We know radiation caries momentum, this is one example of EM momentum. For instance, a directional radio antenna or a directional light beam will create thrust.

Yes, that is correct that EM radiation carries momentum, but it also carries energy, and this energy center is shifting at the speed of light, and therefore is perfectly compatible with classical mechanics laws. That is, the EM radiation(waves) respect center of Energy theorem, and the EM radiation is by all means detectable(in a sense that we can send signals across two stations).

Additionally we appear to have $\frac{1}{2}E \cdot D + \frac{1}{2}B \cdot H$ for the energy density, One thing that bothers me a bit is that "polarization ambiguity" means that D is a bit ambiguous. Also, I haven't seen any similar expression for the momentum density rather than it's flux - maybe this is where the controversy arises, in how to go from the flux to the density.

I'm also finding Wiki's discussion http://en.wikipedia.org/wiki/Polarization_density helpful, in sorting out the P's, D's, E's, M's which I've mostly forgotten :(. This is also where I ran across the idea of "Polarization ambiguity".

As pointed by Jonathan scott, the problem lies with the invalidity of the EM energy density and momentum flux for quasistatic scenarios.

 

[pervect]

As pointed by Jonathan scott, the problem lies with the invalidity of the EM energy density and momentum flux for quasistatic scenarios.

I didn't interpret his remarks that way. But I would like to know if he, or anyone else thinks there's a problem with interpreting the Poynting vector as energy flux, or the Maxwell stress tensor as momentum flux.

If we can agree on the fluxes, then we should also have the mometum and energy up to a constant factor. Which doesn't mean there aren't some arguments about the constant factor, of course.

One other thing I've notices (from another thread) is that I'd expect EM waves to cause dipoles to vibrate, and that this should involve some mechanical energy, but if we consider the problem of an EM wave entering a dielectric from a vacuum, there doesn't appear to be any source of energy to cause such mechanical vibrations of said dipoles.

Specifically http://farside.ph.utexas.edu/teaching/em/lectures/node104.html

Note that $R+T=1$. In other words, any wave energy which is not reflected at the boundary is transmitted, and vice versa.

 

[Jonathan Scott]

As Butler demonstrates in his paper, the Poynting vector and Maxwell energy density quantities do not transform correctly between frames of reference except in the case where no sources are present anywhere (such as for propagating electromagnetic waves).

This also led to the weird factor of 4/3 in the Abraham-Lorentz self-energy of a point charge, and the concept of "Poincaré stresses" to try to fix that. In my copy of Jackson "Classical Electrodynamics" (second edition) this is discussed in section 17.5, "Covariant Definitions of Electromagnetic Energy and Momentum".

If I remember correctly (I don't have my copy of the paper easily to hand) Butler suggests that for the energy and momentum associated with a source charge, one can take E^2/2 in the rest frame and transform that as the density of a four-vector energy-momentum quantity to get the effective energy and momentum density in any other frame, giving a physically consistent tensor quantity describing the flow of energy and momentum.

 

[universal_101]

I didn't interpret his remarks that way. But I would like to know if he, or anyone else thinks there's a problem with interpreting the Poynting vector as energy flux, or the Maxwell stress tensor as momentum flux.

A flux needs something measurable to be moving through(momentum, energy etc.), So no, you cannot use flux for quasistatic quantities, nothing is moving there. It can be easily seen by center of Energy theorem.

If we can agree on the fluxes, then we should also have the momentum and energy up to a constant factor. Which doesn't mean there aren't some arguments about the constant factor, of course.

Well, let me put it very simply, one can have net energy stored in his/her pocket, but one cannot have net momentum in his/her pocket, for it simply violates the center of Energy theorem and most probably it violates conservation of momentum.

One other thing I've notices (from another thread) is that I'd expect EM waves to cause dipoles to vibrate, and that this should involve some mechanical energy, but if we consider the problem of an EM wave entering a dielectric from a vacuum, there doesn't appear to be any source of energy to cause such mechanical vibrations of said dipoles.

Specifically http://farside.ph.utexas.edu/teaching/em/lectures/node104.html

Well ofcourse, charges vibrate, that is how we get reflection (speaking classically), but I suppose the equation R+T = 1, is expressing the relation after the electrons being setup to vibrate.

 

[Jonathan Scott]

There is no problem with having circulating momentum flux in a static situation, provided that the net overall momentum is zero.

However, the Poynting vector is not a physically consistent description of such a flow, as the Poynting vector in one frame of reference describes a different flow from that in some other frame of reference except when no sources are present anywhere (and despite its name, Poynting's vector is certainly not a vector for transformation purposes). Again, see Butler's paper and the relevant section of Jackson.

 

[pervect]

As Butler demonstrates in his paper, the Poynting vector and Maxwell energy density quantities do not transform correctly between frames of reference except in the case where no sources are present anywhere (such as for propagating electromagnetic waves).

I haven't found Butler's paper in a non-paywall version, but judging from the title, the issue is how to find a 4-vector that has the required property E^2 - p^2 = constant. It's fairly well known that to have a 4-vector for (E,P) you need to either have a system of zero volume , or a closed system. I don't have a handy reference at this moment alas :(, and I may not be wording the issue very sharply as a consquence. But it is utterly unsurprising then that if you have a system with interacting sources, that the system isn't isolated and that you'll find (E,P) doesn't transform as a 4-vector for the system - you'd need to include the sources.

This also led to the weird factor of 4/3 in the Abraham-Lorentz self-energy of a point charge, and the concept of "Poincaré stresses" to try to fix that. In my copy of Jackson "Classical Electrodynamics" (second edition) this is discussed in section 17.5, "Covariant Definitions of Electromagnetic Energy and Momentum".

I happen to have the third edition. In 16.5 Jackson gives "Covariant defintions of Electromagnetic Energy and Momentum"

As emphasized by Rohrlich, even if the electromagnetic stress tensor $\Theta^{ab}$ is not divergenceless, it is possible to give covariant definitions of the total electromagnetic energy and momentum of a system of fields.

Jackson then gives a few formulas, E being proportional to the integral of E^2 + B^2, P being proportional to the integral of E cross B, which can "be considered to define the energy and momentum at fixed time t in some particular inertial frame K to be specified shortly". These are later referred to as "the naieve expressions 16.41".

The issue about the electromagnetic part of the stress tensor having a non-zero divergence is the root of what Butler was talking about, I think. It has a non-zero divergence because the electromagnetic part is coupled to the mechanical part. The total stress tensor (electromagnetic + mechanical) will have zero divergence.

Jackson also gives a more general expression for a general frame.

Another remark that I think is important that I need to study more:

The definitions of the electromagnetic 4-momentum afford a covariant defintion starting fromt he naive expressions (16.41) in any frame K'. Different choices of the frame K' lead to different 4-vectors, of course, but that is no cause for alarm.

 

[pervect]

A flux needs something measurable to be moving through(momentum, energy etc.), So no, you cannot use flux for quasistatic quantities, nothing is moving there. It can be easily seen by center of Energy theorem.

I would regard this all as "wrangling about the constant factors". Which I'm happy to leave to the literature at this point, though I'm currently regarding Jackson 3'd edition as a good reference for what's commonly accepted.

Well ofcourse, charges vibrate, that is how we get reflection (speaking classically), but I suppose the equation R+T = 1, is expressing the relation after the electrons being setup to vibrate.

Well, as my understanding since the goal is to separate out mechanical energy from electromagnetic energy, I'm concerned about the book-keeping aspects of this, In particular, it seems to me that some idealization must be going on, since I expect dipole vibration, I expect it to be classified as mechanical, and I'm seeing a value of 0 for the amount of this energy.

[add]
I'd go so far as to say that the idea that the dipoles instantaneously respond to the electric field without any lag so that P is always directly proportional to E implies that they are idealized and have negligible mass, which means that as far as the standard book-keeping goes, they are assumed to have zero energy.

 

[Jonathan Scott]

...
I happen to have the third edition. In 16.5 Jackson gives "Covariant defintions of Electromagnetic Energy and Momentum"
...

Thanks for looking into this.

Jackson's argument seems to be that since the usual expressions don't transform correctly, we can fix that by taking them in some arbitrary frame and then transforming them correctly as a tensor, although this obviously doesn't give a unique solution, so it cannot be treated as a "physical" description of where the energy and momentum reside.

In certain cases there is a suitable rest frame from which to start. However, the fact that the solution is unique still doesn't mean it is physical.

It's something like 20 or more years since I looked into this, and I was particularly annoyed that the "Maxwell electromagnetic stress tensor" turned out not to be an actual tensor. I remember finding Butler's paper very helpful and surprisingly easy to follow.

I've never really understood how "Poincaré stresses" are supposed to fix this either, in that I found them very unsatisfactory, but I can't remember the details enough to say why.

Anyway, the one thing I got from it was a huge mistrust of the Poynting vector outside the original context of the energy in an electromagnetic wave (for which I was able to confirm that it transformed consistently). There are various paradoxes in electromagnetism which definitely require the field to have some circulating momentum (and hence angular momentum) in certain cases, and the Poynting vector can be used to understand some of the qualitative aspects, but it is not a consistent description of the physical flow of momentum when sources are present.

 

[universal_101]

There is no problem with having circulating momentum flux in a static situation, provided that the net overall momentum is zero.

I think there will be a problem if you consider the flux locally, in static situation, for static situation implicitly suggest everything is static even locally(everywhere).

However, the Poynting vector is not a physically consistent description of such a flow, as the Poynting vector in one frame of reference describes a different flow from that in some other frame of reference except when no sources are present anywhere (and despite its name, Poynting's vector is certainly not a vector for transformation purposes). Again, see Butler's paper and the relevant section of Jackson.

Agreed, but we must find solutions, or the source of problems, in order to gain insight of where it all went wrong.

 

[Jonathan Scott]

I think there will be a problem if you consider the flux locally, in static situation, for static situation implicitly suggest everything is static even locally(everywhere).

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.

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.

Note of course that when charge density is present (which is another form of derivative of the field), it is perfectly normal to assume an energy density equal to the potential (which is again a form of integral of the field) times the charge density, so assuming that time derivatives might participate in a similar way is not wildly speculative.

I've not yet managed to work out a consistent scheme for that which satisfies the requirements and equals the usual established overall energy density expressions both for a wave and for the electrostatic case. If anyone knows of any existing papers which explore that alternative way of looking at the energy density I'd be very interested to know about them.

 

[TrickyDicky]

Neutrinos were historically expected as a consequence of experiments, whereas, EM momentum(of quasistatic fields) does not have experiments supporting them, it is all theoretical book-keeping, in order to avoid the conflict with momentum conservation.

No, historically neutrinos were not expected at all, they were Pauli's solution to the theoretical problem of conservation of energy of continuous beta radiation, so until many years later they were exactly that, a theoretical book-keeping device to avoid conflict with conservation laws. The analogy Meir Achuz made was in that sense well chosen, as a matter of fact some of the theorists that are not convinced by the accepted solutions to the "hidden momentum" issues (solutions that in fact date back a century but are ignored and the issues revisited periodically) propose the existence of new particles to account for the putative inconsistencies.

 

[universal_101]

No, historically neutrinos were not expected at all, they were Pauli's solution to the theoretical problem of conservation of energy of continuous beta radiation, so until many years later they were exactly that, a theoretical book-keeping device to avoid conflict with conservation laws. The analogy Meir Achuz made was in that sense well chosen, as a matter of fact some of the theorists that are not convinced by the accepted solutions to the "hidden momentum" issues (solutions that in fact date back a century but are ignored and the issues revisited periodically) propose the existence of new particles to account for the putative inconsistencies.

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.

So, it is EM momentum which is there, purely on theoretical grounds, whereas, neutrino(or something) is required to account for the missing momentum/energy(deduced experimentally).

 

[Meir Achuz]

I think you missed the point made, I'm saying there is NO experiment that demands the Introduction of EM momentum,...
So, it is EM momentum which is there, purely on theoretical grounds, whereas, neutrino(or something) is required to account for the missing momentum/energy(deduced experimentally).

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

 

[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/