I Energy flux direction in a conducting wire?

[coquelicot]

But how many of those possible fluxes remain if one restricts attention to only gauge-invariant expressions?

If you demand gauge invariance, I believe only the Poynting vector. But that's the fundamental mistake. It is now well acknowledged that the scalar and vector potentials are real, and perhaps more real than the E and M fields. These potential fields can be made felt more concrete by choosing a gauge, but choosing arbitrarily one special gauge hides the reality. This is exactly what they do when they choose the Poynting vector as the unique possible representation of the energy flux. There are several implication that I will explain in a future version of my paper. First, at a practical viewpoint, this stupid constraint complicates the computations of energy transfer (and momentum): by choosing adequately a gauge corresponding to a given context, the computations can be made quite simple. Second, this leads to counter intuitive (albeit not false) results: for example, to the fact that the energy is flowing outside the electrical wires and enter normally into the wires. On the contrary, by choosing the suitable gauge which is also the most natural one in context, it appears that the energy "flows" inside the wires, and this provides the needed basis for some equations of thermodynamics that were discussed hard in this thread. Of course, this is only one way to see the energy flow, but it corresponds to our intuition, and that's usually desirable for the development of physics. Last but not least, the theoretical viewpoint: CONCRETE vs REAL; the energy flow should not be regarded as a "concrete" flow, where one can identify the particles in a fluid. In this case, the flow would follow the particles movement and would be uniquely determined. But that's not the case: the energy flow can be expressed differently by a great variety of gauges, nonetheless, it is real.

 

[coquelicot]

Could you please give a very short and simple explanation of what constitutes a different gauge for the less initiated like myself. Is it like a different reference frame? Thanks.

Not exactly. Here is the point. I assume you know what is the scalar potential $\Phi$ (the usual V for electrical engineers) and the vector potential A. There holds $E = -\nabla \Phi - {\partial A\over \partial t}$ and $B = \nabla \times A$ (you can see these relations as a definition of $\Phi$ and $A$, because a theorem of math says that such fields exist). Thanks to these relations, Maxwell's equations and all the electrodynamics can be described in term of $\Phi$ and $A$ only. Notice that this description is often simpler, and is in fact used everywhere. Moreover, according to Feynman, that's the right way to do electrodynamics and has deep implication in quantum fields theory. Now comes the surprising fact: there are in fact infinitely many pairs of fields $(\Phi, A)$ that satisfy the above relations. More precisely, it is found that if $(\Phi', A')$ is another pair, then there is a scalar function $\Lambda$ such that
$$\Phi - \Phi' = -{\partial \Lambda \over \partial t}\ and \ A - A' = \nabla \Lambda.$$
Conversely, given any scalar function $\Lambda$ (possibly depending on the time), you can transform this way a pair $(\Phi, A)$ into another pair $(\Phi', A')$ , and the result will fulfill the relations above, provided that they are fulfilled by $(\Phi, A)$. Thus you have some freedom in the choice of $(\Phi, A)$, and you can demand it to fulfill additional relations, as far you can show it is possible to find some $\Lambda$ that leads to a pair $(\Phi', A')$ satisfying those relations. For these reasons, any partial or total specification of $(\Phi, A)$ (the relations) of this kind is called a gauge.

Example: Let me show that there is a gauge for which $\Phi = 0$ identically: Start with some licit $(\Phi, A)$ (which always exists). Let $\Lambda$ be the integral of $\Phi$ along $t$.
So, $$\Phi - {\partial \Lambda \over \partial t} = 0.$$
Define $A' = A + \nabla \Lambda$. Then the pair $(\Phi'=0, A')$ is valid, since $\Lambda$ yields it according to the transformation above.

Famous gauges are the Coulomb gauge: condition $\nabla\cdot A = 0$, and the Lorenz gauge: condition $\nabla \cdot A - {\partial \Phi\over \partial t} = 0$. For these gauges, it can be shown that a $\Lambda$ exists.

 

[coquelicot]

I found this paper amusing at first read. Might be useful:
https://physics.princeton.edu/~mcdonald/examples/poynting_alt.pdf

OK, so it is now clear that the form I proposed is not new: that's what was proposed by Sepian (eq 18 in the article you posted). I think I have more to explain, but the literature on this topic is huge. There are several articles cited by the author that I should read before writing my article. Unfortunately, I'm not affiliated to any institution, and I cannot afford these articles. What a pity!

 

[hutchphd]

What a pity!

Depending upon where you live, you may have more access than you know. Because I pay state taxes, I have library privileges through the local state college branch. Also Phys. Rev. offers free copies to public (and public school) libraries. check it out

 

[Philip Koeck]

OK, so it is now clear that the form I proposed is not new: that's what was proposed by Sepian (eq 18 in the article you posted). I think I have more to explain, but the literature on this topic is huge. There are several articles cited by the author that I should read before writing my article. Unfortunately, I'm not affiliated to any institution, and I cannot afford these articles. What a pity!

You might find preprints on Research Gate or similar.

 

[coquelicot]

You might find preprints on Research Gate or similar.

That's usually the case for relatively recent articles, but not for older ones like those cited in this paper.

 

[coquelicot]

Could you please give a very short and simple explanation of what constitutes a different gauge for the less initiated like myself. Is it like a different reference frame? Thanks.

I recommend googling at "Aharonson-Bohm" effect, and Wikipedia.
Also, you may want to read the beginning of the paper I join here, where the reality of gauges is discussed.

 

[coquelicot]

Here is another article full of historical details that shows that our discussion is not only old, but has involved the top geniuses. The debate is in fact still opened (I think). Worthy to read it.
Where_is_electromagnetic_energy_located

 

[Philip Koeck]

If you demand gauge invariance, I believe only the Poynting vector. But that's the fundamental mistake. It is now well acknowledged that the scalar and vector potentials are real, and perhaps more real than the E and M fields. These potential fields can be made felt more concrete by choosing a gauge, but choosing arbitrarily one special gauge hides the reality. This is exactly what they do when they choose the Poynting vector as the unique possible representation of the energy flux. There are several implication that I will explain in a future version of my paper. First, at a practical viewpoint, this stupid constraint complicates the computations of energy transfer (and momentum): by choosing adequately a gauge corresponding to a given context, the computations can be made quite simple. Second, this leads to counter intuitive (albeit not false) results: for example, to the fact that the energy is flowing outside the electrical wires and enter normally into the wires. On the contrary, by choosing the suitable gauge which is also the most natural one in context, it appears that the energy "flows" inside the wires, and this provides the needed basis for some equations of thermodynamics that were discussed hard in this thread. Of course, this is only one way to see the energy flow, but it corresponds to our intuition, and that's usually desirable for the development of physics. Last but not least, the theoretical viewpoint: CONCRETE vs REAL; the energy flow should not be regarded as a "concrete" flow, where one can identify the particles in a fluid. In this case, the flow would follow the particles movement and would be uniquely determined. But that's not the case: the energy flow can be expressed differently by a great variety of gauges, nonetheless, it is real.

The way I learned about gauge fields is that they leave E and B and thus the "actual physics" unchanged.
(I'll take the risk that Feynman is watching this thread and is now displeased.)
To me that would mean that the real flow of energy should be unaffected by the gauge one chooses for the potentials.

 

[vanhees71]

Of course, gauge-dependent quantities are not physical observables. The electromagnetic potentials are not observable but only gauge-independent quantities expressed by them. Often it is claimed otherwise, because in quantum mechanics the Aharonov-Bohm effect seems to indicate a dependence of observables (interference effects) on the potentials, but the observable phase factor is not gauge-dependent but can be expressed through the magnetic flux, which is a gauge-independent quantity.

From the point of view of Noether's theorem the densities of conserved quantities are only defined by the symmetry up to a socalled pseudo-gauge transformation (which has nothing to do with gauge transformations of the electromagnetic potentials), and thus one has to be careful also here how to interpret these local densities and current densities. Also here, what's observable must be independent of the choice of "pseudo-gauge". Indeed what's observable are the total quantities like the total energy, momentung, and angular momentum of a closed system, and these are pseudo-gauge independent.

For more details on this, see

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

For the energy-momentum tensor, within general relativity, you have to additional constraint that it must be symmetric and locally conserved. Using the Hilbert action to derive Einstein's field equation from the action principle it is defined as the variational derivative of the "matter action" wrt. the metric and coincides usually with the Belinfante energy-momentum tensor, which is in addition also gauge invariant for the em. field.

 

[coquelicot]

The way I learned about gauge fields is that they leave E and B and thus the "actual physics" unchanged.
(I'll take the risk that Feynman is watching this thread and is now displeased.)
To me that would mean that the real flow of energy should be unaffected by the gauge one chooses for the potentials.

Well, that's also the way I learned about gauge fields. I am now aware that that's a choice of the physicist, that may well not the best choice (actually, I am convinced that it's not). I project to explain this point very thoroughly in the next version of my article (that will have few in common with the present version), and I believe my arguments will be sufficiently strong to convince a lot of people. I will post here the article when it is ready.

 

[coquelicot]

Of course, gauge-dependent quantities are not physical observables. The electromagnetic potentials are not observable but only gauge-independent quantities expressed by them. Often it is claimed otherwise, because in quantum mechanics the Aharonov-Bohm effect seems to indicate a dependence of observables (interference effects) on the potentials, but the observable phase factor is not gauge-dependent but can be expressed through the magnetic flux, which is a gauge-independent quantity.

How "observable" is defined here? the electric field is not "observable" as well, only its action on a point charge is. Similarly, a kinematic movement is not observable without referencing it to a coordinate system. Why not admit that the scalar and vector potentials are also observable (by their effects), but need to be described with reference to a gauge? It suffices to admit that a change of gauge is homologue to a change of coordinate system. So, the energy flow become observable with reference to a gauge. By changing the gauge, you change the description of the energy flow, but not the energy flow itself. That's completely natural after all, and the advantages of this view are numerous.

 

[bob012345]

How "observable" is defined here? the electric field is not "observable" as well, only its action on a point charge is. Similarly, a kinematic movement is not observable without referencing it to a coordinate system. Why not admit that the scalar and vector potentials are also observable (by their effects), but need to be described with reference to a gauge? It suffices to admit that a change of gauge is homologue to a change of coordinate system. So, the energy flow become observable with reference to a gauge. By changing the gauge, you change the description of the energy flow, but not the energy flow itself. That's completely natural after all, and the advantages of this view are numerous.

But the very definition of an electric field is the force per unit charge which certainly is observable in principle at every point.

 

[coquelicot]

But the very definition of an electric field is the force per unit charge which certainly is observable in principle at every point.

I meant, you cannot observe the field at every every point as a whole. You observe it as some points, and then you imagine that there is a field by completing mentally the lines. So, the decision to consider the field as a real thing is yours (and also mine), but it was considered as an artificial mathematical construction by some physicists of the 19th century who preferred the "more" physical action at a distance. I think it is important to grasp this point.

I would add the following: assuming the E-field and B-field are observable, is the $S = E\times B$ field observable? the cross product is a mental construction, and no experiment can show it directly. So, the argument of "observable" seems to me rather vague, unless you can provide me a rigorous definition of what is "observable". If I'm not wrong, the argument of "observable" is not defensible because it cannot be given a clear definition; there remains eventually only this "credo" according to which "physical notions should be gauge invariant". But that's just a belief unduly erected as an axiom. It is licit and does not lead to a contradictory theory, but if we just forget it, everything become clear and simpler. Scalar and vector potentials are now believed to be real, and are described with respect to a gauge, exactly like movements need reference systems of axes to be described. In this view, there are some notions that are gauge invariant, like there are some notions like "distance between points" that are invariant under a change of coordinate systems. And there are some notions that are not gauge invariant, like the power flux, which need to be described with respect to a gauge. There are even more than that: I hope to show that in the same way there are "privileged systems" of axes to describing some kinematical situations, there are also "priviledged gauges" for describing a given EM situation. In the case of steady currents in electrical wires for example, the privileged gauge is a gauge where $\Phi$ and $A$ are independent of $t$ (the Coulomb gauge is just fine). In the case of the propagation of a plane wave, the privileged gauge is the zero potential gauge $\Phi = 0$. That has probably something to do with the symmetries of the EM configuration, Noether thms etc, and I'm not sure I'm sufficiently skilled to materialize this idea, but that would be nice.

 

[renormalize]

I would add the following: assuming the E-field and B-field are observable, is the S=E×B field observable? the cross product is a mental construction, and no experiment can show it directly.

By this logic, the angular momentum $\boldsymbol{L}=\boldsymbol{r}\times\boldsymbol{p}$ is not observable and no experiment can show it directly. Nonsense!

And there are some notions that are not gauge invariant, like the power flux, which need to be described with respect to a gauge.

Can you offer an E&M textbook citation supporting your claim that the power flux depends on a choice of gauge? Or are you speculating based on a personal theory?

 

[alan123hk]

Now, from a thermodynamics point of view, there exist a relation between the internal energy U and the electrochemical potential μ―. This relation implies that the internal energy flux J→Q=μ―J→ where J→ is the current density (I am ignoring thermoelectric effects for simplicity here). However this means that the energy flux's direction is along the wire, not perpendicular to it, i.e. the direction is perpendicular to that of S→.

This discussion thread has a very in-depth discussion of energy flow in wires and extends to other rich content. Please excuse me for not having time to read every reply post.

Below I try to analyze the simplest conductor segment using Poynting vector. The results show that the real energy flow enters from the outer space vertically through the conductor surface and remains in the same direction until the the conductor center. The energy flux on the conductor surface is exactly equal to voltage multiplied by current (ohmic losses). But I really don't understand and suspect that there is an internal energy flow inside the conductor in a different direction than the Poynting vector?

Note that this is not a waveguide that assumes no ohmic losses inside the conductor​

 

[alan123hk]

A true overall picture of the flow of energy flux in a circuit (including batteries, waveguides, and ohmic resistors) should look like the following.

Energy Transfer in Electrical Circuits: Poynting Vector, Surface Charges, and All That
https://www.physics.udel.edu/~bnikolic/teaching/phys208/lectures/poynting_vector.pdf

 

[coquelicot]

By this logic, the angular momentum $\boldsymbol{L}=\boldsymbol{r}\times\boldsymbol{p}$ is not observable and no experiment can show it directly. Nonsense!

You have completely missed the point. I haven't said that something is not (directly, physically) observable because it is the cross product of something; I said that if two vectors $E$ and $B$ are believed to be observable, this does not imply a priori that their cross product is physically observable. And actually, that's the main concern of the Poynting vector. Nobody has found a way to directly observe the energy flux (please, read the historical deep debate of the greatest geniuses about that, in the article I inserted in post #188). The only thing that we can observe is the integral of the flux on a closed surface. So the only observable here is defined up to the divergence of a field. Again, that's not a scoop but a debate that is very old and seems to have been never solved in a satisfying manner.

Can you offer an E&M textbook citation supporting your claim that the power flux depends on a choice of gauge? Or are you speculating based on a personal theory?

I don't know if there are textbooks that support this claim, but there are a bunch of articles that define gauge dependent power fluxes (which are equivalent in fact). I don't even know if the "claim" above is new (I believe it is not, but I'll have to check at least the articles of Sepian and a few other cited in the article of post #177). You are probably right to say that this is not main stream though. I've forgotten that this site is not really a free discussion site, so, I will take that into account, and be more quiet from now.

 

[coquelicot]

This discussion thread has a very in-depth discussion of energy flow in wires and extends to other rich content. Please excuse me for not having time to read every reply post.

Below I try to analyze the simplest conductor segment using Poynting vector. The results show that the real energy flow enters from the outer space vertically through the conductor surface and remains in the same direction until the the conductor center. The energy flux on the conductor surface is exactly equal to voltage multiplied by current (ohmic losses). But I really don't understand and suspect that there is an internal energy flow inside the conductor in a different direction than the Poynting vector?

View attachment 300869
Note that this is not a waveguide that assumes no ohmic losses inside the conductor​

Hello Alan123hk. Yes, you have apparently missed some important posts. I tried to give the idea in the doc I posted in post #176 (that I somewhat abusively called "article"). Please, have a look there. Also, you may want to have a look at the historical account about this old debate in articles posted in posts #188 (and also #177).

 

[renormalize]

I haven't said that something is not (directly, physically) observable because it is the cross product of something; I said that if two vectors E and B are believed to be observable, this does not imply a priori that their cross product is directly observable.

Your statement is simply untrue. If you can simultaneously observe (measure) the 3 components each of the electric field $\boldsymbol{E}$ and the magnetic field $\boldsymbol{B}$, you can always compute their cross-product (a mathematical operation) and thereby arrive at the observed value of $\boldsymbol{E}\times\boldsymbol{B}$. This procedure is exactly analogous to measuring the electrical power $P$ dissipated in a resistor. You use a voltmeter to measure the voltage $V$ across the resistor while simultaneously monitoring with an ammeter the current $I$ flowing through the resistor. By forming the ordinary product $VI$ (a mathematical operation!) we have thereby measured $P$. What could be more "directly observable" than that?

Nobody has find a way to directly observe the energy flux ... . The only thing that we can observe is the integral of the flux on a closed surface.

More nonsense. Do you really think that a huge 1 AU radius spherical closed surface is required to determine the solar irradiance arriving at the earth? According to https://www.nasa.gov/mission_pages/sdo/science/Solar Irradiance.html the light energy flux from the sun at the top of the atmosphere (integrated over the visible & IR wavelengths) is $1.366~kW/m^{2}$. This is measured by satellite-borne light power meters fabricated from small open surfaces. Indeed, every home with solar panels on its roof is in effect measuring the Poynting vector, integrated over the open panel surfaces, whenever the panels deliver power to the home.

I've forgotten that this site is not really a free discussion site, so, I will take that into account, and be more quiet from now.

Thanks for acknowledging this.

 

[coquelicot]

Your statement is simply untrue. If you can simultaneously observe (measure) the 3 components each of the electric field $\boldsymbol{E}$ and the magnetic field $\boldsymbol{B}$, you can always compute their cross-product (a mathematical operation) and thereby arrive at the observed value of $\boldsymbol{E}\times\boldsymbol{B}$.

On one hand, you say that it is a "computed value", and on the other hand, that it is observable. Please, before calling something a nonsense, read the historical debates in the aforementioned articles. I don't think Heaviside, Maxwell, Sepian and many other were totally stupid.

This procedure is exactly analogous to measuring the electrical power $P$ dissipated in a resistor. You use a voltmeter to measure the voltage $V$ across the resistor while simultaneously monitoring with an ammeter the current $I$ flowing through the resistor. By forming the ordinary product $VI$ (a mathematical operation!) we have thereby measured $P$. What could be more "directly observable" than that?

Yes, but are you aware that V is exactly the $\Phi$ potential you have (implicitly) supposed to be a mathematical artifice? are you aware that V is defined up to a constant, that is, needs a gauge to be described ? Of course, regarding the energy transfer, only the difference of potential is used in your resistor, but you are still considering the V potential as a real thing.

EDIT: When you use VI, you just use directly my (actually the Slepian) formula for the energy flow. You can arrive to the same power transfer result with the Poynting vector, but in a much more intricate way.

More nonsense. Do you really think that a huge 1 AU radius spherical closed surface is required to determine the solar irradiance arriving at the earth? According to https://www.nasa.gov/mission_pages/sdo/science/Solar Irradiance.html the light energy flux from the sun at the top of the atmosphere (integrated over the visible & IR wavelengths) is $1.366~kW/m^{2}$. This is measured by satellite-born light power meters fabricated from small open surfaces. Indeed, every home with solar panels on its roof is in effect measuring the Poynting vector, integrated over the open panel surfaces, whenever the panels deliver power to the home.

"Small" is a relative concept. What is measured here is still the integral of the flux over a closed surface. Again, the theory advocated by Sepian and many many others leads to exactly the same results. No experiment of this kind can prove one form or another.

 

[renormalize]

On one hand, you say that it is a "computed value", and on the other hand, that it is observable. Please, before calling something a nonsense, read the historical debates in the aforementioned articles.

I don't intend to delve willy-nilly into the historical debate until you can provide specific citations to support the assertions on which we disagree. For example, you draw a distinction between "computed value" and "observable". So if I measure the two distinct sides of a rectangle and multiply to get the area of that rectangle, is this area an "observable" or just a "computed value"? By what objective criteria do you distinguish the two possibilities? And how do you apply these criteria to the cross product defining the Poynting vector?

... are you aware that V is exactly the Φ potential you have (implicitly) supposed to be a mathematical artifice? are you aware that V is defined up to a constant, that is, needs a gauge to be described ?

Yes, I am aware of that. Indeed, circuit theory is distilled from Maxwell's equations and is, by construction, gauge invariant. As you point out, only voltage differences are physical, meaning that the computed power cannot depend on the gauge choice (the arbitrary constant). This is again exactly analogous to the fact that the energy flux in electromagnetics cannot depend on the four-vector potential, but only on the electric and magnetic fields derived from that potential.

What is measured here is still the integral of the flux over a closed surface.

What exactly do you mean by a closed surface? To my understanding, a closed surface has no boundary curve separating the two sides of the surface (e.g., the surface of a sphere), whereas an open surface has an boundary curve (e.g., a solar panel with its edges). So to me, all localized readings of an energy flux must rely on open detector surfaces. And in principle, you can just keep making these surfaces smaller and smaller until you are satisfied that you have measured the energy flux at some particular point to your desired accuracy. This is like clocking the speed of a particle by measuring its position x(t) at two different times, then shortening the time interval Δt until the ratio Δx/Δt approximates the instantaneous speed to some desired accuracy.

 

[coquelicot]

I don't intend to delve willy-nilly into the historical debate until you can provide specific citations to support the assertions on which we disagree.

What a pity. The article I provided is full of extremely interesting things, not only historical, but also theoretical, and is very understandable. It is not necessary to read it in full, but I think that reading some parts of would give you a more thorough understanding of what is boiling down.

For example, you draw a distinction between "computed value" and "observable". So if I measure the two distinct sides of a rectangle and multiply to get the area of that rectangle, is this area an "observable" or just a "computed value"? By what objective criteria do you distinguish the two possibilities? And how do you apply these criteria to the cross product defining the Poynting vector?

So, you want to set the following definition: if $O_i$ are observables, and if $O$ is a quantity computed from the $O_i$, then $O$ is an observable. That's a licit choice and I will not contradict it.
Now, let me propose to you a completely observable quantity $S'$ (according to this definition), different from the Poynting vector and still leading to exactly the same energy transfer results:
$$ A(M_1, t) = {1\over 4\pi \epsilon_0 c^2}\int {{\bf j}(M_2, t')\over ||M_1-M_2||} dV,$$
with $t'$ the retarded time, and
$$\Phi(M_1, t) = {1\over 4\pi \epsilon_0 c^2}\int {{\rho}(M_2, t')\over ||M_1-M_2||} dV$$
with the following bound conditions for the integrals: $A(M_1, 0) = 0$ and $\Phi(M_1, 0) = 0$.
Then define $S' = (\Phi \nabla - \nabla {\partial A\over \partial t})\times {\bf B}$ and you are done.

How could you prove that the Poynting vector is the correct power density, while the above "observable" formula is not?

Yes, I am aware of that. Indeed, circuit theory is distilled from Maxwell's equations and is, by construction, gauge invariant. As you point out, only voltage differences are physical, meaning that the computed power cannot depend on the gauge choice (the arbitrary constant). This is again exactly analogous to the fact that the energy flux in electromagnetics cannot depend on the four-vector potential, but only on the electric and magnetic fields derived from that potential.

The magnetic and electric fields depend themselves from the potentials, so your last sentence is formally a nonsense. Regarding your first two sentences: 1) I have not pointed out that only voltage differences are physical, and 2) I have never claimed that the computed power should depend on a gauge choice. On the contrary, I have shown that the computed power is always the same, no matter what gauge is chosen.

What exactly do you mean by a closed surface? To my understanding, a closed surface has no boundary curve separating the two sides of the surface (e.g., the surface of a sphere), whereas an open surface has an boundary curve (e.g., a solar panel with its edges). So to me, all localized readings of an energy flux must rely on open detector surfaces. And in principle, you can just keep making these surfaces smaller and smaller until you are satisfied that you have measured the energy flux at some particular point to your desired accuracy. This is like clocking the speed of a particle by measuring its position x(t) at two different times, then shortening the time interval Δt until the ratio Δx/Δt approximates the instantaneous speed to some desired accuracy.

That's the most interesting part of your objections. Indeed, there is a notion of "flux of light" through an open surface. Pay attention that this notion is mainly used for light in the context of plane waves. Now, in the classical theory of Poynting (which I never said is wrong, but is unsatisfying, and complicates things uselessly), there is no problem of defining this notion, since $S$ is unambiguously provided. But if you are careful, you'll remark that the only way to measure the flux of light through a surface is to measure the energy transferred to the body whose surface absorbs the light. In other words, that's a way to say: put your surface orthogonal to the propagation of light, then the integral of the Poynting vector on the surface is equal to the flux through the whole body (actually a theorem). That's of great practical value, but that's only a definition to be used inside a theorem. In the same way, I can set the following definition in the context of waves propagating through free space: The flux through an open surface is the integral on the surface of the generalized Poynting vector, for which the gauge is set to $\Phi = 0$ identically (so, this is nothing but the usual Poynting vector). I claimed above that there are privileged gauges for describing the energy flux naturally, and that this gauge is just adapted to plane wave, or perhaps more generally, to all waves propagating in free space. Again, this is a definition of practical value, but in the old-new theory "I" propose, the notion of EM flux through an open surface is not intrinsically defined, and need not actually.

 

[fluidistic]

This discussion thread has a very in-depth discussion of energy flow in wires and extends to other rich content. Please excuse me for not having time to read every reply post.

Below I try to analyze the simplest conductor segment using Poynting vector. The results show that the real energy flow enters from the outer space vertically through the conductor surface and remains in the same direction until the the conductor center. The energy flux on the conductor surface is exactly equal to voltage multiplied by current (ohmic losses). But I really don't understand and suspect that there is an internal energy flow inside the conductor in a different direction than the Poynting vector?

View attachment 300869
Note that this is not a waveguide that assumes no ohmic losses inside the conductor​

You should read the thread more in details, my last PDF for example. Otherwise the thread will grow over and over just repeating the same things over and over.

1) Poynting vector is not the whole energy flux. A quick way to see this is to compute $\nabla \cdot \vec S$. In steady state, the divergence of the energy flux must vanish, everywhere in the wire. However, if you do that with the Poynting vector, you will get that it doesn't vanish, in fact it will be equal to the Joule heat. This is a clear indication that the Poynting vector is not the whole energy flux in the wire.

2) I have shown that if the conductor has a non zero resistivity, then it cannot be isothermal when there is a current going through it. This has an implication, that its center is hotter than its surface. The thermal energy flux, which is given by Fourier's law ($\vec J_Q =-\kappa \nabla T$) points radially outwards the wire. This is true for any point inside the wire.

3) Yes. From the thermodynamics relation $dU = TdS+\overline{\mu}dN$, one can see that there is an energy flux component that goes in the direction of the current. In the case of the wire, this means along it, i.e. perpendicular to Poynting's vector.

 

[vanhees71]

We should for sure make clear that indeed only gauge-independent quantities are observable quantities with a physical meaning. There have been several claims to the contrary, but it's obvious that a gauge-dependent quantity cannot be physical, because they are arbitrary, i.e., they are not determined by the fundamental physical laws nor are they in an operational way definable as measurable quantities.

One example above was "voltage" as used in circuit theory. Of course the world "voltage" has to be read with a grain of salt since almost never is it a difference of a scalar potential. As is also clear, the fundamental laws governing electromagnetic phenomena (as far as quantum effects can be neglected) are Maxwell's equations connecting the directly observable electromagnetic field, $(\vec{E},\vec{B})$ with the charge-current distribution $(\rho,\vec{j})$, and Kirchhoff's Laws which make up circuit theory, are derived from the special cases of electrostatics (DC circuits) or the quasistationary approximations of the Maxwell equations (AC circuits). What occurs as "voltage" in this analysis is usually an electromotive force. Take the utmost simple example of a battery connected by a wire with a finite resistance in the DC situation. There you use Faraday's Law (one of Maxwell's equations) and integrate it along the wires and the battery, making a closed loop. Within the wire you have an electric field, and the corresponding line integral along it gives the potential difference of this static field (in the here of course used Coulomb gauge of magnetostatics, where $\vec{E}=-\vec{\nabla} \Phi$) at the ends of the wire. Within the battery you have an electromotive force due to the "chemistry" of the battery, leading to the simple law $\mathcal{E}_{\text{bat}}=U=R i$. As you see from this argument, the entire derivation involves only gauge-independent quantities, although it's at the end expressed by the potential difference with the potential chosen in a specific gauge, but that of course doesn't make the result gauge dependent.

 

[hutchphd]

1) Poynting vector is not the whole energy flux. A quick way to see this is to compute ∇⋅S→. In steady state, the divergence of the energy flux must vanish, everywhere in the wire. However, if you do that with the Poynting vector, you will get that it doesn't vanish, in fact it will be equal to the Joule heat. This is a clear indication that the Poynting vector is not the whole energy flux in the wire.

I do not understand this straw man. One cannot build a wire using classical elctromagnetic theory alone. Of course the energy supplied by the influx indicated by the Poynting vector will end up as heat (there are alternate degrees of freedom afforded by QM). This heat will diffuse essentially isotropically.
A wire is complicated. Why is this interesting?

.

 

[coquelicot]

I'll assume this post is addressed to me.

We should for sure make clear that indeed only gauge-independent quantities are observable quantities with a physical meaning. There have been several claims to the contrary, but it's obvious that a gauge-dependent quantity cannot be physical, because they are arbitrary, i.e., they are not determined by the fundamental physical laws nor are they in an operational way definable as measurable quantities.

That's your choice (and admittedly the choice of most physicists) and I respect it. But that's only a choice. And adding terms like "we should for sure make clear", "it's obvious" etc. adds nothing to the value of this choice. Again, your choice does not leads to a contradictory theory, nor to wrong results. In the same way, scientists before Nicolas Copernic used the geocentric system to describe the movements of the planets; that was licit and they were able to predict accurately eclipses etc. But the heliocentric system of Copernic simplified considerably the analysis, and lead to further progress. Similarly, the ether was a licit hypothesis after Lorentz invented his theory of contraction of length etc, essentially equivalent to RR. But this choice was useless, needed weird ad hoc axioms (like the contraction of length and time) and would have prevented the progress of physics. So, physicists adopted the view of Einstein were all the Galilean referentials are equivalent for the description of the physics.
So, you choose to ban the scalar and vector potential as physical notions BECAUSE they are seemingly ambiguously defined, and it follows immediately (and rightfully) that only gauge independent notions defined by their mean have a physical meaning. Please, pay attention that you have a priori decided that the potentials are mathematical artifices without physical meaning, and from these premises follows that only gauge independent notions are physical. That's a choice that can be understood (and that was also mine before I entered in this thread).
Now I propose you to understand my choice. For me, the potentials are truly physical notions. According to my view, one should not say potentials are defined with respect to a gauge, but that they are described with respect to a gauge. That's fundamentally different. There are very good analogies with the description of movements with respect to a system of axes (please, read my post #194, I will not repeat here). Once you admit potentials are physical, there is no more need to infer that only gauge invariant notions are physical. On the contrary, that would be unnatural. So, other physical notions involving the potentials can now be described with respect to gauges as well. So are the power flux density, the energy momentum, or the power flux through and OPEN surface. Of course, there are still gauge invariant notions, like the EM fields, the energy density, the energy transfer rate etc., exactly like their are invariant notions by the Lorentz transformations, like space-time distance etc.

You may ask: but why doing so, if overall, the results are the same?
Well, there are very good reasons:

• Practically, the computations can be greatly simplified by just choosing the adequate gauge;
• intuition. By choosing an adequate gauge, the energy flux is described in a way that feels intuitive, and avoid the "weird" flows of the Poynting vector, noted my most authors. As an example, for steady state currents, the energy is just flowing inside the wires. That's exactly what is needed for electrical engineering. Intuitive behavior of physical notions is very important for the smooth development of physics.
• Thermodynamics. The formula that was the subject of this thread is now justified.
• Quantum field theory. The Aharonson-Bohm effect is now demystified. There is no more problem with admitting that the potentials have produced the effect.
• Last but not least, the theoretical point of view: the Poynting vector just corresponds to the zero potential gauge $\Phi = 0$. It is very adequate for waves in free space, because it is directed in the direction of the propagation of the wave. But according to my view, choosing it for everything is just like choosing the geocentric system for describing all the movements in the universe.

 

[vanhees71]

It's not a choice. How can a quantity that depends on an arbitrary choice describe an objective property of some phenomenon in Nature? Nature doesn't care about our way to describe her!

 

[coquelicot]

It's not a choice. How can a quantity that depends on an arbitrary choice describe an objective property of some phenomenon in Nature? Nature doesn't care about our way to describe her!

Wow! either you are able to read at the speed of light, or you have only read the first sentence of my answer. Sorry, I think I have explained everything very well there. So, you may want to read my post again, (and post #194 indicated above). You should especially address the analogy with coordinate reference systems that are, well, just an arbitrary choice to describe movements.

 

[vanhees71]

It's a good analogy. Indeed, coordinates have no direct physical meaning either. Particularly in GR, what's observable are not the coordinates but only the (local) coordinate-independent quantities described by tensors.