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.

 

[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.