I Energy flux direction in a conducting wire?

[coquelicot]

There were several notions I didn't master or even know. Thanks to the article of Domenicali posted by Fluidistic, I think I have finally understood the main point. And that's rather simple actually, once you know the definition of the electrochemical potential $\mu$ (I thought it was something else).
I allow myself to explain in simple words what I understood, in order for other persons not to be mystified by the formulae.

So, the electrochemical potential is the potential energy (density) of a kind of particles, which includes both the chemical potential energy and the usual electrical potential of the particle. The chemical potential energy of the particle stem from its natural tendency to move toward (or from) some another chemical compound. Now, we can neglect the potential chemical energy of the electrons in the wire, at least here for the sake of simplicity.
So, $\mu$ is the electrical potential of the electrons with respect to the electrodes. I mean, if $\varphi$ is the electrical potential (which decrease linearly in the wire from the + electrode to the - electrode), an electron at position $z$ in the wire as a potential $e\phi(z)$.

Now, the thermodynamic energy is equal to HEAT + ELECTRICAL POTENTIAL ENERGY (EPE) of the electrons (if we assume only electrons are relevant here).
Fluidistic has in fact just written that the heat flux, + the flux of the EPE is equal to the flux of the thermodynamic energy, which stem directly from this truth. The heat flux can be shown to be radial and the flux of the EPE axial. There is nothing new regarding the heat flux, so let me focus on the flux of the EPE; that's after all very natural: all what is said here is that the electrons are moving from the + electrode to the - one because they want to reduce their potential electrical energy, and thermodynamists delight at defining fluxes, so they define a flux of electrical potential energy (more generally a flux of electrochemical energy) just to say that such or such kind of particles are moving in order to decrease their potential energy, which is transformed into heat by some process as they move. That's just that! Of course, the flux follows the direction of the movement of the electrons etc.

Now the interesting point: this idea is very natural after all, even without involving thermodynamics. Why should we say that the electrons move in the wire because of the EM flux materialized by the Poynting vector, and not just because of the decreasing electrical potential from the + to the -. There is no problem after all to define a EPE energy flux, just as thermodynamists do. But then, how to conciliate the EM flux with this flux?

That's annoying and I have no real answer, but perhaps an analogy: Assume we have a vertical pipe. At the top of the pipe, some apparatus is continuously relaxing dust at a fix rate. Due to the gravity and the friction with air, the dust falls inside the pipe at constant speed. At the bottom of the pipe, the apparatus pumps the dust that has gathered here to the top of the pipe, generating a constant current of dust inside the pipe.
Notice that during its falling, the dust reduces its potential energy of gravity which is converted into heat by friction with air, and evacuated radially from the pipe.

Now, my question is: what has actually created the current of dust inside the pipe? is it the apparatus that is pumping the dust?, or is it the potential energy of gravity of the dust?

 

[fluidistic]

Then I wonder why you describe this heat current using the expression for heat conduction.
Shouldn't it be radiation (assuming the wire is in vacuum to keep things simple)?

My system is the inside of the wire. The wire's surface are the boundaries of my system, radiation effects, if they are to be dealt with, should appear as boundary conditions to the heat equation. Inside the material, the temperature obeys a Fourier conduction term + heat source heat equation.

Again, radiation effects, if you want to tackle them, will only have a uniform shift in temperature everywhere in the system, leaving the temperature gradient intact, the whole analysis intact.

 

[fluidistic]

There were several notions I didn't master or even know. Thanks to the article of Domenicali posted by Fluidistic, I think I have finally understood the main point. And that's rather simple actually, once you know the definition of the electrochemical potential $\mu$ (I thought it was something else).
I allow myself to explain in simple words what I understood, in order for other persons not to be mystified by the formulae.

So, the electrochemical potential is the potential energy (density) of a kind of particles, which includes both the chemical potential energy and the usual electrical potential of the particle. The chemical potential energy of the particle stem from its natural tendency to move toward (or from) some another chemical compound. Now, we can neglect the potential chemical energy of the electrons in the wire, at least here for the sake of simplicity.
So, $\mu$ is the electrical potential of the electrons with respect to the electrodes. I mean, if $\varphi$ is the electrical potential (which decrease linearly in the wire from the + electrode to the - electrode), an electron at position $z$ in the wire as a potential $e\phi(z)$.

Now, the thermodynamic energy is equal to HEAT + ELECTRICAL POTENTIAL ENERGY (EPE) of the electrons (if we assume only electrons are relevant here).
Fluidistic has in fact just written that the heat flux, + the flux of the EPE is equal to the flux of the thermodynamic energy, which stem directly from this truth. The heat flux can be shown to be radial and the flux of the EPE axial. There is nothing new regarding the heat flux, so let me focus on the flux of the EPE; that's after all very natural: all what is said here is that the electrons are moving from the + electrode to the - one because they want to reduce their potential electrical energy, and thermodynamists delight at defining fluxes, so they define a flux of electrical potential energy (more generally a flux of electrochemical energy) just to say that such or such kind of particles are moving in order to decrease their potential energy, which is transformed into heat by some process as they move. That's just that! Of course, the flux follows the direction of the movement of the electrons etc.

Now the interesting point: this idea is very natural after all, even without involving thermodynamics. Why should we say that the electrons move in the wire because of the EM flux materialized by the Poynting vector, and not just because of the decreasing electrical potential from the + to the -. There is no problem after all to define a EPE energy flux, just as thermodynamists do. But then, how to conciliate the EM flux with this flux?

That's annoying and I have no real answer, but perhaps an analogy: Assume we have a vertical pipe. At the top of the pipe, some apparatus is continuously relaxing dust at a fix rate. Due to the gravity and the friction with air, the dust falls inside the pipe at constant speed. At the bottom of the pipe, the apparatus pumps the dust that has gathered here to the top of the pipe, generating a constant current of dust inside the pipe.
Notice that during its falling, the dust reduces its potential energy of gravity which is converted into heat by friction with air, and evacuated radially from the pipe.

Now, my question is: what has actually created the current of dust inside the pipe? is it the apparatus that is pumping the dust?, or is it the potential energy of gravity of the dust?

The electrochemical potential is not a potential, it's really an energy (per particle, or mole, depending on the def. but here it's per particle). There are some worked out examples related to it in the appendix of the paper. The paper is worth it.

 

[coquelicot]

The electrochemical potential is not a potential, it's really an energy (per particle, or mole, depending on the def. but here it's per particle). There are some worked out examples related to it in the appendix of the paper. The paper is worth it.

I have not said that it is a potential, but that it is a potential energy. By the way, that's also the way Domenicali call it in his article.

 

[Philip Koeck]

My system is the inside of the wire. The wire's surface are the boundaries of my system, radiation effects, if they are to be dealt with, should appear as boundary conditions to the heat equation. Inside the material, the temperature obeys a Fourier conduction term + heat source heat equation.

Again, radiation effects, if you want to tackle them, will only have a uniform shift in temperature everywhere in the system, leaving the temperature gradient intact, the whole analysis intact.

Okay. So the system you are considering is a volume that is completely inside the wire and has no contact with the surface. The effect of the surface is introduced later as a boundary condition.

 

[Philip Koeck]

I have not said that it is a potential, but that it is a potential energy. By the way, that's also the way Domenicali call it in his article.

This might be worth looking into:

The fundamental equation of thermodynamics:
dU = T dS + P dV + μ dN

You can also write:
dU = ∂U/∂S dS + ∂U/∂V dV +∂U/∂N dN

since
U = U(S, V, N)

In words this means that the inner energy U of the system changes when different state variables change.
For example μ is the change of U when the number of particles in the system increases by 1 and the two other variables are kept constant.

There might be an important thing to consider here, but I'm not sure:
dS is a change of the state variable S in the system.
dQ is a small amount of heat transferred between system and surroundings and there is no state varible (or function) that dQ would be the change of.

For reversible processes T dS = dQ.
Per time-unit this gives: T dS/dt = dQ/dt
In the above equation dS/dt is a rate of change, whereas dQ/dt is a current, since it has a direction (into or out of the system).

I'm wondering whether there might be a problem with introducing energy and entropy flows as vectors.
dU/dt, dS/dt etc. are just rates of change of state-variables of the system, whereas dQ/dt is actually a current.
Not sure whether this is a problem for the derivation, though.

Addition: As an example of the possible problem I see imagine a smallish sub-volume of the wire. The inner energy of this sub-volume, which I regard as the system, can change without an inner energy current. There can, for example, be a heat current into or out of the system, which leads to a change of inner energy.
So a change of inner energy doesn't necessarily imply an inner energy current.

 

[coquelicot]

I'm wondering whether there might be a problem with introducing energy and entropy flows as vectors.
dU/dt, dS/dt etc. are just rates of change of state-variables of the system, whereas dQ/dt is actually a current.
Not sure whether this is a problem for the derivation, though.

This was actually my problem during most of the posts above, that is, to understand if these fluxes are licit, in particular the flux of "internal energy" of Fluidistic. But in fact, this internal energy flux is not so important, only the flux of some well defined energy need be considered here, namely the electrochemical potential energy of the electrons + heat. Now, it is a fact that electrodynamists define and use the heat and electrochemical energy fluxes.
Regarding the flux of entropy, since entropy is already defined as $TdS = dQ$, and since the heat flux is already defined, there is no reason not to define the "entropy flux" by $\vec TJ_S = \vec J_Q.$
On the other hand, the the electrons are moving, and they carry with them a potential electrical energy (as well as a negligible kinetic energy), and possibly some chemical potential energy which is probably nonexistent or negligible. So, there is no apparent reason not to define the flux of potential electrochemical energy as the transfer of this energy through a surface by the electrons. This is even rather natural. In fact, neglecting the chemical energy of the electrons, if any, this almost too simple view could have been formulated even if the context of electrodynamics.

The main problem we have all not been able to understand till now is how to conciliate the EM view with the thermodynamic view. Thermodynamics shows that that's the potential electrical energy flux of the electrons that causes the heating. EM shows that that's the EM energy flux that carries the energy to the wire. What is going on here?

Addition: As an example of the possible problem I see imagine a smallish sub-volume of the wire. The inner energy of this sub-volume, which I regard as the system, can change without an inner energy current. There can, for example, be a heat current into or out of the system, which leads to a change of inner energy. So a change of inner energy doesn't necessarily imply an inner energy current.

Isn't heat a form of energy too? If you have a heat flow, you have a flow of energy as well. Again, I think the main problem that caused most of my confusion with fluidistic is that we are not defining precisely the energies we are speaking about. Energy is a term designing a class of physical notions; it's a way to say:
1. "Work" belong to the class "energy"
2. if something can be transformed totally or partially into an element of the class "energy", then it belongs to this class.
But in fact, it suffices to consider only the relevant energies and the problem vanishes. This is common in thermodynamics after all.

 

[Philip Koeck]

This was actually my problem during most of the posts above, that is, to understand if these fluxes are licit, in particular the flux of "internal energy" of Fluidistic. But in fact, this internal energy flux is not so important, only the flux of some well defined energy need be considered here, namely the electrochemical potential energy of the electrons + heat. Now, it is a fact that electrodynamists define and use the heat and electrochemical energy fluxes.
Regarding the flux of entropy, since entropy is already defined as $TdS = dQ$, and since the heat flux is already defined, there is no reason not to define the "entropy flux" by $\vec TJ_S = \vec J_Q.$
On the other hand, the the electrons are moving, and they carry with them a potential electrical energy (as well as a negligible kinetic energy), and possibly some chemical potential energy which is probably nonexistent or negligible. So, there is no apparent reason not to define the flux of potential electrochemical energy as the transfer of this energy through a surface by the electrons. This is even rather natural. In fact, neglecting the chemical energy of the electrons, if any, this almost too simple view could have been formulated even if the context of electrodynamics.

The main problem we have all not been able to understand till now is how to conciliate the EM view with the thermodynamic view. Thermodynamics shows that that's the potential electrical energy flux of the electrons that causes the heating. EM shows that that's the EM energy flux that carries the energy to the wire. What is going on here?
Isn't heat a form of energy too? If you have a heat flow, you have a flow of energy as well. Again, I think the main problem that caused most of my confusion with fluidistic is that we are not defining precisely the energies we are speaking about. Energy is a term designing a class of physical notions; it's a way to say:
1. "Work" belong to the class "energy"
2. if something can be transformed totally or partially into an element of the class "energy", then it belongs to this class.
But in fact, it suffices to consider only the relevant energies and the problem vanishes. This is common in thermodynamics after all.

I think one has to be careful. For a particular system it's quite possible that dU = dQ.
That means that heat entering the system increases the inner energy of the system and nothing else changes.
It also means that a heat current entering the system (dQ/dt) leads to a rate of increase in inner energy of the system dU/dt.
It doesn't necessarily mean there is a current of inner energy.
So dU/dt is just the change of a state function of the system. It's sort of localized.
dQ/dt is a heat current, which of course is an energy current.

At least that's how I see it.

However, I'm not saying there can't be any energy currents, entropy currents etc.
I'm just saying there don't have to be any just because there is a heat current.

Part of the difficulty might come from the system-surroundings-thinking in thermodynamics, which is not really used in other fields, I believe.

I also find it hard to picture what this current of inner energy along the wire would be.
It's not heat, since heat only flows radially.
There's no temperature gradient along the wire.
The only thing that flows are the electrons. Are we discussing the kinetic energy of the electrons?
About the potential energy: I don't think that flows with the electrons. It's more like the electrons use it up while they fall through the potential. Not sure about that.

 

[coquelicot]

I also find it hard to picture what this current of inner energy along the wire would be.

That's the electrical potential energy of the electrons, that's just that.

About the potential energy: I don't think that flows with the electrons. It's more like the electrons use it up while they fall through the potential. Not sure about that.

A flow of potential energy can be defined without doubt, and is apparently used successfully by thermodynamists.

Let me make things even more simple, without thermodynamics.
Let $\vec J$ be the current density inside the wire, $\vec J = \rho \vec v$.
Let $\phi(z)$ be the electrical potential at position $z$ in the wire. The potential decreases from the + electrode to the - electrodes (linearly if the resistance per unit length is constant).
Define a priori the electrical potential energy flux by
$$J_\phi = \rho \phi \vec v = \phi \vec J.$$
So, the potential energy flowing through a cross section of the wire per unit time is
$$\int _{cross\ section} J_\phi dA = \phi I,$$ where $I$ is the intensity of the current.
Thus, the integral of the potential energy flow on the surface of the cylindrical volume made by a length L of wire between $z_1$ and $z_2$ is equal to
$$\phi_{z_2}I - \phi_{z_1}I = (\phi_{z_2}-\phi_{z_1})I = V I,$$
where $V$ is the potential difference stemming from the resistance of the wire between points $z_1$ and $z_2$.
One recognize the well know law for the power dissipated by a resistor: $P = VI$. That makes sense!

 

[bob012345]

Could someone please succinctly summarize what the main diverging points of this discussion are? I'm totally lost. Thanks.

 

[Philip Koeck]

Could someone please succinctly summarize what the main diverging points of this discussion are? I'm totally lost. Thanks.

When the 2 poles of a battery are connected by a wire so that a current flows is there some sort of energy flux going through the wire or is all the energy transported from the battery to the wire by the Poynting vector, so to say.

 

[hutchphd]

I suggest the first chapter of Wald's new book in addition to Feynman lecture 27. Poynting Vector.
No need for a voodoo resister chemical potential, but to each his own. Energy is conserved

/

 

[coquelicot]

Could someone please succinctly summarize what the main diverging points of this discussion are? I'm totally lost. Thanks.

Philip Koeck said that right.
Note: I will be out this weekend, and come back tomorrow evening, just to let persons know.

 

[bob012345]

I suggest the first chapter of Wald's new book in addition to Feynman lecture 27. Poynting Vector.
No need for a voodoo resistor chemical potential, but to each his own. Energy is conserved

/

It might be interesting and instructive to also read what Poynting himself says;

https://royalsocietypublishing.org/doi/epdf/10.1098/rstl.1884.0016

Interesting that this was a couple of years before the work of Heinrich Hertz on Maxwellian waves.

 

[hutchphd]

It might be interesting and instructive to also read what Poynting himself says;

That's a very nice paper. Thanks.

 

[coquelicot]

I suggest the first chapter of Wald's new book in addition to Feynman lecture 27. Poynting Vector.
No need for a voodoo resister chemical potential, but to each his own. Energy is conserved

/

Unfortunately, I don't have this book. If you can post a snapshot of the pages you think are relevant, this may help.

 

[hutchphd]

Here is the intro chapter to chew on:

https://press.princeton.edu/books/h...0/advanced-classical-electromagnetism#preview

 

[Philip Koeck]

That's the electrical potential energy of the electrons, that's just that.
A flow of potential energy can be defined without doubt, and is apparently used successfully by thermodynamists.

Let me make things even more simple, without thermodynamics.
Let $\vec J$ be the current density inside the wire, $\vec J = \rho \vec v$.
Let $\phi(z)$ be the electrical potential at position $z$ in the wire. The potential decreases from the + electrode to the - electrodes (linearly if the resistance per unit length is constant).
Define a priori the electrical potential energy flux by
$$J_\phi = \rho \phi \vec v = \phi \vec J.$$
So, the potential energy flowing through a cross section of the wire per unit time is
$$\int _{cross\ section} J_\phi dA = \phi I,$$ where $I$ is the intensity of the current.
Thus, the integral of the potential energy flow on the surface of the cylindrical volume made by a length L of wire between $z_1$ and $z_2$ is equal to
$$\phi_{z_2}I - \phi_{z_1}I = (\phi_{z_2}-\phi_{z_1})I = V I,$$
where $V$ is the potential difference stemming from the resistance of the wire between points $z_1$ and $z_2$.
One recognize the well know law for the power dissipated by a resistor: $P = VI$. That makes sense!

I like your result, but I'm uncertain about the interpretation.

First I'd like to point out that your result accounts for the total heat production by the current in the wire. So it's not just an additional energy flux on top of the Poynting vector, it's the whole thing.
It's more like an alternative description for how the energy, that is then radiated off as heat, is delivered to the wire.

I'll try to describe in words what I think is going on in terms of thermodynamics, also as an alternative to the Poynting vector.

The chemical potential μ is the energy required to add 1 electron to a system (with constant V and S). Now we can picture the wire as series of connected systems starting at the minus pole all the way to the plus pole. Let's call these systems sections, since they really are just sections of the wire.
At the minus pole μ must be largest and then it decreases as you go closer to the plus pole.
Every time an electron is removed from one of the sections down to the next there is a small amount of excess energy that is given off as heat.
I haven't done the maths, but I'm quite sure that the total heat given off per second due to this process is exactly what you get, U I.

About the interpretation:
I believe that this flow of electrons with the associated conduction of heat from the center of the wire to the surface and then radiation from the surface is the only thing that happens thermodynamically in a steady state situation.
Steady state means that the dU/dt and dT/dt is zero everywhere in the wire. There's a radial temperature gradient that is constant in time.
The only current is the current of electrons along the wire and the heat current radially away from the wire. The inner energy of the battery decreases with time and at the same rate heat is given off by the wire.
In this picture there's no balancing of heat currents, which is quite typical for thermodynamics I would say. If a hot object cools due to radiation the heat current is also only balanced by the decrease of inner energy and not by an incoming energy current.

So, I think, this thermodynamic picture is really just an alternative description of the EM picture with the Poynting vector.
Importantly, no additional energy current is needed.

 

[coquelicot]

I like your result, but I'm uncertain about the interpretation.

First I'd like to point out that your result accounts for the total heat production by the current in the wire. So it's not just an additional energy flux on top of the Poynting vector, it's the whole thing.
It's more like an alternative description for how the energy, that is then radiated off as heat, is delivered to the wire.

I'll try to describe in words what I think is going on in terms of thermodynamics, also as an alternative to the Poynting vector.

The chemical potential μ is the energy required to add 1 electron to a system (with constant V and S). Now we can picture the wire as series of connected systems starting at the minus pole all the way to the plus pole. Let's call these systems sections, since they really are just sections of the wire.
At the minus pole μ must be largest and then it decreases as you go closer to the plus pole.
Every time an electron is removed from one of the sections down to the next there is a small amount of excess energy that is given off as heat.
I haven't done the maths, but I'm quite sure that the total heat given off per second due to this process is exactly what you get, U I.

About the interpretation:
I believe that this flow of electrons with the associated conduction of heat from the center of the wire to the surface and then radiation from the surface is the only thing that happens thermodynamically in a steady state situation.
Steady state means that the dU/dt and dT/dt is zero everywhere in the wire. There's a radial temperature gradient that is constant in time.
The only current is the current of electrons along the wire and the heat current radially away from the wire. The inner energy of the battery decreases with time and at the same rate heat is given off by the wire.
In this picture there's no balancing of heat currents, which is quite typical for thermodynamics I would say. If a hot object cools due to radiation the heat current is also only balanced by the decrease of inner energy and not by an incoming energy current.

So, I think, this thermodynamic picture is really just an alternative description of the EM picture with the Poynting vector.
Importantly, no additional energy current is needed.

I appreciate this interpretation, but I think I have completely solved the paradox in the mean time. I have almost finished to write an article on this subject, and I will post a first draft here in one hour or so. Be patient, you may like what you'll see.

 

[coquelicot]

Here is the article I wrote, that completely solves the paradox in my opinion. This is only a first version, and I have to add the bibliography and few other things. Also, there probably remains many English mistakes, but I think it is quite understandable for now. You are of course invited to warn about mistakes, errors and comments.
I will probably throw this article somewhere, say in Arxiv. So, if someone here thinks he should be cited, acknowledged etc. , please, let me known.

 

[hutchphd]

Very nicely written and clear. I am a little bit uncertain as to what happens within this framework for AC power. It seems to me not generalizeable in any simple way.

 

[coquelicot]

Very nicely written and clear. I am a little bit uncertain as to what happens within this framework for AC power. It seems to me not generalizeable in any simple way.

I think the flux has been shown to be equivalent to the Poynting vector in full generality regarding energy transfer. See the various expressions of the power flow in the last section.

 

[Philip Koeck]

I think the flux has been shown to be equivalent to the Poynting vector in full generality regarding energy transfer. See the various expressions of the power flow in the last section.

I have difficulties with the concept of a potential energy flow.
We can look at a mechanical example: Let's say we have a stone on a shelf inside a room.
The room is filled with honey all the way up to the shelf.
Now this stone falls from the shelf and slowly glides through the honey until it hits the floor.
The potential energy of the stone is converted to heat during the fall apart from a very small amount of kinetic energy that the stone still has when it reaches the floor.
Where is the flow of potential energy?

 

[coquelicot]

I have difficulties with the concept of a potential energy flow.
We can look at a mechanical example: Let's say we have a stone on a shelf inside a room.
The room is filled with honey all the way up to the shelf.
Now this stone falls from the shelf and slowly glides through the honey until it hits the floor.
The potential energy of the stone is converted to heat during the fall apart from a very small amount of kinetic energy that the stone still has when it reaches the floor.
Where is the flow of potential energy?

Basically, in your example, you cannot speak about "flow" because there is only a single stone: the electrical equivalent would be a single point charge moving in the electrical wire. A better image would be a bag of sand on the shelf, which would pour slowly and uniformly inside the honey. Then, yes, this would make sense.
(that's not to say that my alternative density would not work for a single point charge, but that a single point charge is not a "steady regime").

Notice also that the $\rho \vec j$ can be interpreted as a potential energy flow in my paper, but that's not necessary. You could just see it as a term. Then the definition of $S'$ in my paper, which reduces to $\rho \vec j$ for steady regimes, shows the energy flows only where there are charges, in the direction of the wire (for steady regime again).

There are much more problematic things than that, to say the full truth: the Poynting vector is more than just used to compute the energy flow: it is also used for the linear and angular momentum conservation, related to Maxwell's stress tensor. To argue that the Poynting vector could be replaced by my alternative definition, I'll have to show that an alternative Maxwell stress tensor and angular momentum can be defined up to a divergence (of tensors). I think this is the case, but that will demand much more work to add to this article. Hope I will be able to do it.

 

[fluidistic]

Wow, I had been busy and wasn't warned by PF that there were replies to this thread. What a nice surprise! Especially the paper of coquelicot.

I will rewrite a bit my PDF and publish it in a github page (aka a website). I don't think my PDF is serious enough even for Arxiv.

 

[coquelicot]

Wow, I had been busy and wasn't warned by PF that there were replies to this thread. What a nice surprise! Especially the paper of coquelicot.

I will rewrite a bit my PDF and publish it in a github page (aka a website). I don't think my PDF is serious enough even for Arxiv.

I'm happy for that.
Unfortunately, there is a computation mistake at the last line of my paper, that produced a wrong formula. I have corrected this error and the basic idea remains the same. But my ideas have very progressed from the time I posted this paper. I am now aware that the formula I proposed in not an "alternative form" of the energy flux, but the "general form" of the energy flux, that includes the poynting vector as a particular case: the key idea to understand what is boiling down is the notion of "gauge". For example, if the chosen gauge fulfills the condition $\Phi = 0$ everywhere (it is always possible to use this gauge), then my general formula simply becomes the pointing vector. In contrast, if we are in the case of steady currents, a gauge can be chosen that fulfills ${\partial A\over \partial t} = 0$. Then my formula becomes $\Phi \bf J$, that is, the formula needed in your thermodynamics. It is impressive that the theory of energy flux has been so badly shaped. They have simply arbitrarily fixed a particular form of the energy flux (which amounts to a particular gauge), and destroyed its inner structure with several degrees of freedom.
My next version of my paper will be much involved and deep, and will also involve the field momentum. I am entirely rewriting it, but this may take some time (say 1-2 weeks). For the moment, in order to let you see the correct formulae, I join a draft here.

Regarding your paper, if you wish to include some of my ideas, could you please wait 1-2 weeks until I finish mine? In this way, you could conveniently cite the suitable formulae.

 

[hutchphd]

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

 

[coquelicot]

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

Thank you so many for this paper! I'm not sure the formula I've provided in my paper appears there (I have to check that carefully). But without doubt, I will include this paper in the bibliography. In any case, it appears I have deeper insight than many previous persons that worked on this subject. Notably, the fact that a change of gauge does provide various useful concepts of energy flow, and that's the key point. There is not a single "energy flux vector", but infinitely possible fluxes that derive one from the other by a change of gauge.

 

[renormalize]

There is not a single "energy flux vector", but infinitely possible fluxes that derive one from the other by a change of gauge.

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

 

[bob012345]

Thank you so many for this paper! I'm not sure the formula I've provided in my paper appears there (I have to check that carefully). But without doubt, I will include this paper in the bibliography. In any case, it appears I have deeper insight than many previous persons that worked on this subject. Notably, the fact that a change of gauge does provide various useful concepts of energy flow, and that's the key point. There is not a single "energy flux vector", but infinitely possible fluxes that derive one from the other by a change of gauge.

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.