Energy flux direction in a conducting wire?

[Philip Koeck]

1) Right. But you need not to restrain yourself to very small elements. For example, you can compute the total flux passing through a given cross section, or surface. I have done that for the whole wire (open cylinder without caps), and the 2 caps of the wire (it's not in my document). Those quantities make sense. I am not sure about what you are asking, i.e. which fundamental equation you mention. I only used thermodynamics relations.

2) Look up my document. The thermal gradient is radial (has the direction of $-\hat r$). The temperature profile inside the wire is a parabola, whose maximum is reached right at the center of the wire. If you change the boundary conditions, this parabola will be shifted up/down, but it won't change any more than that.

3) Almost. The thermal gradient is not along the wire, it is radial. If you modify your sentence to "Are you saying that there is no current in the wire if there is no temperature gradient in the wire?" then the answer is yes. I was saying it the other way around, but yes, that's a consequence of the math. If there is a non zero current, it is impossible for the wire to be at uniform temperature if the resistivity is not 0, and the thermal conductivity is not infinite. You can see it in my doc.

Okay. I'm fine with 2) and 3) now.

To answer your question about 1):
The fundamental equation of thermodynamics is the one you start with (without P dV):
dU = T dS + P dV + μ dN
It's important that U, S , V and N are state functions of a system.
Obviously you have to clearly define what the system is if you are going to use this equation.
When you introduce dQ this is heat being transferred between the system and the surroundings so you also need to consider where the system ends and where the surroundings start.
Otherwise it's not clear what you are discussing.
If the system is a small subvolume of the wire, for example, that would make sense I believe.

 

[fluidistic]

Assuming you refer to Callen Chap 14 eq. 14.33 and 14.34: there is absolutely no derivation for this formula, and even not a hint about what "energy" he speaks about. This formula is simply parachuted here. Not to speak about (14.33) he says "can be derived like..." which is not obvious at all. If you refer to another section or equation of this chapter, please let me know.
Again and again and again, I have no problem with heat and entropy fluxes, nor with heat equations. These are well defined notions. I have a problem with your alleged "internal energy flux" which "takes into account all possible forms of energies". I don't see that it is defined, and it is even not obvious that the "local energy flux" of Callen is the same as your energy flux.
I don't have access to the article of Domenicali, so, if you can post a snapshot of p 72 to show how he defines your "internal energy flux" (unless that's not that but licit heat equations, involving entropy and heat fluxes, whose I have no problem, again and again and again).

I can do no better than providing the full article, if there is something that should open your eyes, it has to be found there.
It is not "my" internal energy flux, as I said, there is nothing new in what I've done.

Regarding the Callen's objection, I do not see the "can be derived like..." part. Regardless of this, I am now seriously trying to understand what is troubling you. If you say you have no problem regarding the derivation of the heat equation(s), then you have an idea about what $u$ the energy is, right? Callen's nomenclature is fully defined, as far as I know. He also uses and define $\vec {J_U}$ by the way.

Just to be clear, you have a hard time to swallow eqs. 14.33 and 14.34 from Callen's textbook?

Edit: Too bad, the article is too big (>3MB). It's worth checking out though.

 

[fluidistic]

Okay. I'm fine with 2) and 3) now.

To answer your question about 1):
The fundamental equation of thermodynamics is the one you start with (without P dV):
dU = T dS + P dV + μ dN
It's important that U, S , V and N are state functions of a system.
Obviously you have to clearly define what the system is if you are going to use this equation.
When you introduce dQ this is heat being transferred between the system and the surroundings so you also need to consider where the system ends and where the surroundings start.
Otherwise it's not clear what you are discussing.
If the system is a small subvolume of the wire, for example, that would make sense I believe.

My system can be considered as a subvolume of the wire, if you like. It's a portion of the wire between which there is a voltage applied. It needs not be small though.

 

[coquelicot]

Edit: Too bad, the article is too big (>3MB). It's worth checking out though.

If the article is in pdf form, almost every pdf reader contain an option named "snapshot", that allows you to take a snapshot of a page (or a part of a page) and to save it as a picture.

If you say you have no problem regarding the derivation of the heat equation(s), then you have an idea about what $u$ the energy is, right?

Yes whenever $u$ is the heat, a well defined form of energy, and whenever its flux is defined via the Fourier relation with the temperature.
But as I said, I also accept the internal energy of a system as a truth. So, no problem with $u$ again. I have some problem with the flux of $u$, because in order to define a flux, you need to have a flow (that's the words of Callen), that is, a field of velocities. Now, I don't see how a flow could be associated a priori to the internal energy of a system. I repeat again what I said above: I can accept your relation as a definition of the flux of $U$, where $U$ is some energy. But we have to understand what this energy is: you cannot claim a priori (in my opinion that could change) that that's the internal energy of the system. The problem is a problem of interpretation. You have essentially set a definition of $\vec J_U$, but you may be "lying" about what is this $\vec J_U$.

Just to be clear, you have a hard time to swallow eqs. 14.33 and 14.34 from Callen's textbook?

Actually yes. I have not read the whole book, only Chap. 14, so, he may refer (without reference) to some other parts of the book. But as I see it, rel. 14:33 is just parachuted here. I see no justification for this formula at this place.

 

[Philip Koeck]

My system can be considered as a subvolume of the wire, if you like. It's a portion of the wire between which there is a voltage applied. It needs not be small though.

Are you saying that the system is a cylindrical section of the wire?
Then my next question would be: Through which surface does the heat current dQ/dt go, through the caps of the cylinder or the mantle or both?

 

[fluidistic]

If the article is in pdf form, almost every pdf reader contain an option named "snapshot", that allows you to take a snapshot of a page (or a part of a page) and to save it as a picture.Yes whenever $u$ is the heat, a well defined form of energy, and whenever its flux is defined via the Fourier relation with the temperature.
But as I said, I also accept the internal energy of a system as a truth. So, no problem with $u$ again. I have some problem with the flux of $u$, because in order to define a flux, you need to have a flow (that's the words of Callen), that is, a field of velocities. Now, I don't see how a flow could be associated a priori to the internal energy of a system. I repeat again what I said above: I can accept your relation as a definition of the flux of $U$, where $U$ is some energy. But we have to understand what this energy is: you cannot claim a priori (in my opinion that could change) that that's the internal energy of the system. The problem is a problem of interpretation. You have essentially set a definition of $\vec J_U$, but you may be "lying" about what is this $\vec J_U$.Actually yes. I have not read the whole book, only Chap. 14, so, he may refer (without reference) to some other parts of the book. But as I see it, rel. 14:33 is just parachuted here. I see no justification for this formula at this place.

Ok, my PDF reader (zathura) doesn't have this option, I believe. I took 3 screenshots.
Right, it is possible to assign an energy density and a velocity to the energy flux.
Let's consider a simpler case, a wire with a thermal gradient along its length, say. (no electric current). In that case, the energy flux is equal to the thermal flux, in my notation $\vec{J_U}=\vec{J_Q}=-\kappa \nabla T$. In that particular case, we can assign a heat density $q$ and a velocity $\vec{v_\text{heat}}$ such that their product yields the heat flux $\vec{J_Q}$. In this case the velocity is probably the speed of sound (phonons, even though the electrons also carry heat and make up for a good portion of $\kappa$, but they scatter too much to transmit heat faster than sound, I think).

Back to our example, we have exactly the same thing, but the addition of a term coming from the particle's motion along the wire. The total energy flux will therefore have several contributors, and won't be pinpointed to phonons/electrons alone, but also with whatever particle is flowing along the wire.

Since the direction of $\vec{J_U}$ changes along the wire, so does the velocity of every (quasi)particles that makes up for it. At least, that's how I see it.

 

[fluidistic]

Are you saying that the system is a cylindrical section of the wire?
Then my next question would be: Through which surface does the heat current dQ/dt go, through the caps of the cylinder or the mantle or both?

Yes to the 1st question.
The heat is evacuated from the wire through the mantle only. Heat flux/flow is radial, in the system I consider.

 

[coquelicot]

Ok, my PDF reader (zathura) doesn't have this option, I believe. I took 3 screenshots.

You have probably saved the pictures in lossy compression mode, they are blured and almost not readable. What a pity! as the few I have been able to grasp from this article seems to be exactly the answers to my questions.

 

[fluidistic]

You have probably saved the pictures in lossy compression mode, they are blured and almost not readable. What a pity! as the few I have been able to grasp from this article seems to be exactly the answers to my questions.

I could save the pages individually as PDFs. Attached docs.

 

[Philip Koeck]

Yes to the 1st question.
The heat is evacuated from the wire through the mantle only. Heat flux/flow is radial, in the system I consider.

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)?

 

[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.