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

[vanhees71]

This was well sloved by Shockley and James themselves in their famous paper:

W. Jockley and R. P. James. “Try Simplest Cases” Discovery of “Hidden Momentum” Forces on “Magnetic Currents”. Phys. Rev. Lett., 18:876, 1967.
http://dx.doi.org/10.1103/PhysRevLett.18.876

The point is, as usual, to take into account all momenta in a relativistic way, even when the speeds involved are small against $c$:

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

 

[Jonathan Scott]

This was well sloved by Shockley and James themselves in their famous paper:

W. Jockley and R. P. James. “Try Simplest Cases” Discovery of “Hidden Momentum” Forces on “Magnetic Currents”. Phys. Rev. Lett., 18:876, 1967.
http://dx.doi.org/10.1103/PhysRevLett.18.876

The point is, as usual, to take into account all momenta in a relativistic way, even when the speeds involved are small against $c$:

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

The first of those is behind a paywall and the second appears to be about a different but related case which does not involve changing current.

I'm perfectly happy that there is an explanation, but the question is whether there is an easy way to understand it. I think that the effective charge imbalance does so in a way which I find helpful, and I hope it's correct. This lies in the same area as those who end up talking about a "moving magnetic field" or a "moving electric field", both of which are faulty concepts.

 

[pervect]

Using the suggested model of the magnet as a pair of charged disks which are initially counter-rotating and eventually come to a stop discussed in several of the papers, there isn't any reason I can see for there to be an electric dipole field to be generated around the magnet.

 

[universal_101]

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.

Alright, I think it is time to make it 'more' clear, and I would like to walk you through it. Just reply Yes or No, so that I can figure out why we are not on the same page.

1. A current carrying superconducting solenoid, and a charge are lying stationary beside each other, the temperature is slowly increased, and the current starts to die down, resulting in an electric field around the solenoid, which in turn put a force on the charge and the charge acquires momentum. (Yes/No) ?

2. Now since there is No back reaction force on the solenoid(according to Maxwell), people figured, this is a violation of momentum conservation. (Yes/No)?

(This concludes Shockley-James paradox, Feynman paradox etc.)

3. But ofcourse, if anything suggests the violation of momentum conservation is in need of a repair, therefore, we tried to solve the problem with static EM momentum density(a concept borrowed from EM waves). (Yes/No)?

4. So now, the initial setup of the solenoid and the charge, is supposed to contain the net momentum in the form of static EM momentum(ExB density), even when nothing was moving, and it is this net momentum which ended up in the charge once the current is switched off. (Yes/No) ?

5. But people again figured, that something stationary(as the initial setup of current carrying solenoid and a charge) cannot have a net momentum, if the center of Energy is stationary, which resulted in another form of momentum, the hidden momentum. So now, in the initial setup we not only have the static EM momentum but equal and opposite amount of mechanical relativistic hidden momentum(located in the solenoid) (Yes/No) ?

6. Now, the introduction of hidden momentum, made the initial net momentum zero i.e. HM - EM = 0, and there is No conflict with center of Energy theorem. (Yes/No) ?

(This concludes the stand of Griffiths of this situation)

7. But it seems very easy to recognize, that the whole exercise(from point 1 to 6 of this post), is just ended up being redundant. That is, after introducing two kinds of momentum, we are back to square one. That is, the net momentum in initial setup is zero(point 6), which ended up having net momentum in the form of moving charge(point 1 and 2). (Yes/No) ?

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 Noether'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).

Yes, Noether's theorem implies momentum conservation in inertial frames(which also include the closed system of charge and currents) for there being spatial symmetry for the closed system, but that, by far does not mean, there is NO problem with Maxwell's equations. That is, implying that momentum is always conserved does not make the wrong laws(which imply the violation of momentum conservation) correct!

 

[Jonathan Scott]

Using the suggested model of the magnet as a pair of charged disks which are initially counter-rotating and eventually come to a stop discussed in several of the papers, there isn't any reason I can see for there to be an electric dipole field to be generated around the magnet.

The model is equivalent to a current loop, but the disk model is used to ensure symmetry to make it clear that the net mechanical momentum and angular momentum are zero.

When magnetic flux through a current loop changes, an EMF around the loop is generated proportional to the rate of change. This is standard electromagnetism.

My suggestion is that at the microscopic level, this EMF can be seen to be due to the changing current at the far side of the loop effectively lagging behind that at the near side, creating a charge imbalance and hence an electric dipole as seen anywhere near the loop. This has the effect of accelerating a nearby charged particle in a tangential direction.

The effect on the potentials the other way (the effect of the nearby charge on the charges in the loop) is a little tricky but is presumably equal and opposite, so the loop (assuming it is rigid) is pushed in the opposite direction to the charge, by effectively being pulled by the charge on one side and pushed on the other, creating a sideways impulse.

So I expect the overall mechanical momentum to be conserved, without any need for anything hidden (although I don't have a copy of Griffiths, so I'm not sure what the "hidden" stuff is about anyway). The question of how it propagates through the field is separate from this issue.

 

[pervect]

The model is equivalent to a current loop, but the disk model is used to ensure symmetry to make it clear that the net mechanical momentum and angular momentum are zero.

When magnetic flux through a current loop changes, an EMF around the loop is generated proportional to the rate of change. This is standard electromagnetism.

I think I see at least part of your point. The of E around a loop must be equal to the rate of change of the magnetic flux passing through the loop. If we orient the loop perpendicular to the spin axis then there must be an induced E field in the loop in the lab frame as the magnet decays. (This is what causes the charge to move in the first place).

My mental picture of the electric fields of the spinning charged wheel doesn't allow for the E field in the lab frame, therefore it must be incomplete or wrong :(.

 

[universal_101]

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.

I don't know how I missed this post, but this is exactly my line of thinking. But this, inevitably changes the classical electrodynamics Laws or atleast render them as incomplete.

 

[Meir Achuz]

I don't know how I missed this post, but this is exactly my line of thinking. But this, inevitably changes the classical electrodynamics Laws or atleast render them as incompete.

Shouldn't that give you some second thoughts?

 

[Jonathan Scott]

I don't know how I missed this post, but this is exactly my line of thinking. But this, inevitably changes the classical electrodynamics Laws or atleast render them as incompete.

My feeling is that terms like "current loop" and "magnetic flux" are macroscopic simplifications of what is happening at a microscopic level, in a somewhat similar way to concepts such as "dielectrics", so sometimes one must look at a more detailed level to understand exactly what is happening.

 

[universal_101]

Shouldn't that give you some second thoughts?

What about, missing something obvious?

 

[universal_101]

My feeling is that terms like "current loop" and "magnetic flux" are macroscopic simplifications of what is happening at a microscopic level, in a somewhat similar way to concepts such as "dielectrics", so sometimes one must look at a more detailed level to understand exactly what is happening.

Well, the problem is not with the understanding of the current example, it is the incompleteness of the Maxwell's equations, implied by the incompatibility of theoretical classical electrodynamics with classical mechanics. That is, it is the Maxwell's Equations which are the source of the problem.

 

[Jonathan Scott]

I'm still not entirely satisfied with my own idea about the Faraday EMF from the changing flux being effectively caused by apparent charge density separation due to the time lag during the change, mainly because it doesn't take into account the reverse effect.

I've Googled Shockley James paradox and I see there is a pair of PDF files with "Theoretical question 1" in the title, one "question" and the other "solution", which between them explain the hidden momentum solution.

I don't fully understand it yet, but it seems to suggest that the momentum of the charge carriers in the loop is modified by the presence of the nearby charge, and that if the current dies down, this momentum is transferred to the loop, giving it an impulse in the opposite direction to the impulse on the charge. If this is a correct interpretation, I'm not at all sure I buy it, in that it implies a non-zero net momentum in the initial system with charge at rest and current flowing round the loop.

 

[vanhees71]

I'm not sure in which sense you come to the conclusion the Maxwell equations are incomplete or inconsistent. As far as phenomena are concerned that are describable in the classical approximation (i.e., neglecting quantum effects) the Maxwell equations are a complete description. The here discussed issues with "hidden momentum" are somewhat misleading, and I don't like this notion at all. There is no "hidden momentum". At the moment I'm writing on a little manuscript about this issue, where I try to avoid this idea of "hidden momentum". What's called "hidden momentum" is nothing else than a correct consideration of all sources of energies and momenta, including mechanical (kinetic), the electromagnetic, and the intrinsic stress of macroscopic bodies as an effective classical description of quantum phenomena (which on a microscopic level are also mostly electromagnetic in origin but go partially beyond the classical approximation, like the permanent magnetism caused by spins and many-body effects like exchange forces).

The apparent problems are from several sources of wrongly applied (mostly non-relativistic) approximations. One of the most simple cases is explained nicely in Griffiths's book on electrodynamics and illuminating in understanding the issue of so-called "hidden momentum" clearly. Unfortunately he expresses this issue in a somewhat oldfashioned way, because he uses the traditional way to present electromagnetism first in a kind of non-relativistic approximation as far as the mechanical part is concerned. This is almost always justified for everyday household currents but not always, and the example for the socalled "hidden momentum" he discusses seems to be pretty mysterious, but if you reformulate it only a bit using the exact relativistic expressions everywhere, all the mystery vanishes and it occurs that no part of the momentum was ever hiding somewhere, except in the sloppy mind of the physicist treating the charge carrier's mechanical momentum in the non-relativistic approximation, forgetting that the neglected terms are precisely the total momentum of the electromagnetic field which is precisely the opposite of the piece neglected in the non-relativistic approximation of the charge carriers' mechanical momentum. So let me reformulate the problem in the strictly relativistic form (although I'm using the non-covariant 3D treatment, which is more intuitive than the manifestly covariant 4D tensor formalism but nevertheless fully exact concerning relativistic effects).

He consideres a rectangular loop at rest carrying a steady current as a model for a magnetic dipole in an additional electrostatic field (which take as homogeneous across the loop) parallel to the vertical segments of the loop. The momenta of the charge carriers that make up the current in the two vertical pieces cancel but the momenta in the upper segment are different, because there is a change in energy due to the electrostatic potential $\Phi=-E y$. Let $N_{<}$ be the (constant) number of charge carriers in the lower horizontal segment at $y=0$ and $N_{>}$ the one at the upper segment at $y=h$. The total mechanical momentum of the charge carriers in the loop thus is (I set $c=1$ in this posting for simplicity)
$$\vec{p}_{\text{mech}}=(N_{>} E_{>} v_{>}-N_{<} E_{<} v_{<}) \vec{e}_x.$$
Now due to the stationary continuity equation $\vec{\nabla} \cdot \vec{j}=0$ the current is the same everywhere in the loop and thus
$$I=\frac{N_>}{l} Q v_>=\frac{N_<}{l} Q v_{<}$$
and thus
$$N_> v_>=N_< v_<=\frac{I l}{Q}.$$
Plugging this into the formula for the mechanical momentum
$$\vec{p}_{\text{mech}}=\frac{I l}{Q} (E_>-E_<) \vec{e}_x=I l E h \vec{e}_x,$$
because we have
$$E_>-E_<=(m+Q E h)-m=Q E h.$$
Now we need to evaluate the total field momentum. The momentum density is given by Poynting's vector, and thus the total momentum by
$$\vec{p}_{\text{em}}=\int \mathrm{d}^3 \vec{x} \vec{E} \times \vec{B}.$$
In our case, it's most easy to get this, if we could find an expression only involving the electric potential and the current density. We find such an expression by writing
$$(\vec{E} \times \vec{B})_j=-\epsilon_{jkl} (\partial_k \Phi) B_l = -\epsilon_{jkl} [\partial_k (\Phi B_l)-\Phi \partial_k B_l].$$
Integrating over the entire space gives 0 for the first term due to Stokes's integral theorem and the vanishing of the magnetic field at infinity and the second piece is
$$\vec{p}_{\text{em}}=\int \mathrm{d}^3 \vec{x} \Phi (\vec{\nabla} \times \vec{B}) = \int \mathrm{d}^3 \vec{x} \Phi \vec{j}.$$
The contributions from the vertical pieces of the loop cancel obviously. The constributions from the horizontal parts give
$$\vec{p}_{\text{em}}=-I E l h \vec{e}_x.$$
As we see, the total momentum is
$$\vec{p}_{\text{tot}}=\vec{p}_{\text{mech}}+\vec{p}_{\text{em}}=0,$$
as it must be due to the general theorem that in relativistic(!) physics any closed(!) system with a center of energy at rest must have total 0 momentum. Our closed system consists of the moving particles and the electromagnetic field and fufills the general theorem. An apparent paradox only occurs when one treats the momenta non-relativistically, which is wrong in this case no matter how slow the charge carriers might be when it comes to the balance between the mechanical and field momentum. The example also clearly shows that there is no mysterious "hidden momentum". It's only the wrong assumption we could use the non-relativistic approximation for the momentum of the charge carriers.

It's also very illuminating to think about the current as produced in an ideal-fluid picture. There it turns out that the "hidden momentum" occurs from the fact that the pressure has to be appropriately taken into account of the momentum in the upper and lower segment of the loop. Again, there's nothing mysterious or hidden about any part of the momentum, it's just the proper fully relativistic treatment of all parts of the setup.

There are a lot of similar examples. The historically most famous problem of this kind is the classical model for charged particles. To keep the charged particle stable one has to take into account the mechanical stresses holding the charges in place, because otherwise the like-sign charges would repel each other and the construct would simply blow appart (although even then there is no paradox if one can treat everything fully relativistically). The apparent paradox in this case was that the energy-momentum relation $E^2-\vec{p}^2=m^2$ for the model for a charged "particle" seemed to be violated, because one took the integral of the electromagnetic field energy and its momentum although for this tensor the equation of continuity doesn't hold and the fields alone do not form a closed system, but one has to take into account the charges and the mechanical stresses of their binding on a body to a static charge distribution. You find a very clear and very general treatment in Jackson, Classical Electrodynamics, 3rd edition, referring to a paper by Julian Schwinger, who wasn't only a master of quantum but also classical electrodynamics.

http://link.springer.com/article/10.1007/BF01906185

As in relativistic electrodynamics the mass of a particle is an empirical/phenomenological parameter which has to be adapted by tuning other parameters in the theory in the sense of "renormalization". Schwinger clearly shows that this is due to the ambiguity in defining the mechanical stresses needed to stabilize the particle (Poincare stresses) from considerations within electromagnetics alone.

 

[Jonathan Scott]

I'm not sure in which sense you come to the conclusion the Maxwell equations are incomplete or inconsistent. As far as phenomena are concerned that are describable in the classical approximation (i.e., neglecting quantum effects) the Maxwell equations are a complete description. The here discussed issues with "hidden momentum" are somewhat misleading, and I don't like this notion at all. There is no "hidden momentum". At the moment I'm writing on a little manuscript about this issue, where I try to avoid this idea of "hidden momentum". What's called "hidden momentum" is nothing else than a correct consideration of all sources of energies and momenta, including mechanical (kinetic), the electromagnetic, and the intrinsic stress of macroscopic bodies as an effective classical description of quantum phenomena (which on a microscopic level are also mostly electromagnetic in origin but go partially beyond the classical approximation, like the permanent magnetism caused by spins and many-body effects like exchange forces).

Thanks for a very clear description of the "hidden momentum" concept, which now makes sense to me at least in this static context.