I Energy flux direction in a conducting wire?

[coquelicot]

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

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

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

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

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

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

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

 

[renormalize]

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

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

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

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

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

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

 

[coquelicot]

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

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

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

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

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

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

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

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

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

 

[fluidistic]

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

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

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

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

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

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

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

 

[vanhees71]

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

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

 

[hutchphd]

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

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

.

 

[coquelicot]

I'll assume this post is addressed to me.

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

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

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

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

 

[vanhees71]

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

 

[coquelicot]

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

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

 

[vanhees71]

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

 

[fluidistic]

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

.

Sorry hutch for being ignorant, what you have in mind is likely over my head. I was making allusion to Allan's mention of "the real energy flow" when he mentions Poynting vector. I understood it as "the total energy flux". I am just saying that Poynting's vector is not the whole energy flux inside the wire.

 

[coquelicot]

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

So simple when you have grasped the point, isn't it? and that may well open many doors for quantum mechanics etc.

 

[hutchphd]

Sorry hutch for being ignorant, what you have in mind is likely over my head.

Perhaps, but I doubt it. I was trying to point out that there are a panoply of electrectromagnetic interactions happening in the solid and proper treatment thereof would involve very detailed considerations on an atomic scale including the very local and complicated Poynting vector inside. Not useful to calculate. When you treat the wire as "a conductor" you have in some sense decided to only do thermodynamics inside. It is not surprising that the result looks like thermodynamics.

 

[vanhees71]

Of course, the first step is to consider thermal equilibrium and linear-response theory. With this you get the standard constitutive equations explained in all electrodynamics textbooks.

 

[Philip Koeck]

I have a conceptual problem with energy flowing in the wire.
If energy flows from an object A to an object B then I would expect the energy stored in object A to decrease and the energy in B to increase. A good example is a warm object connected to a cold object by a thermal conductor. In that case the energy flow is simply the heat current.
Now if A is the negatively charged plate of a capacitor and B is the positively charged plate, in what sense does the energy stored in A decrease and the energy in B increase when the two plates are connected by a wire? Why is it not the other way round. (Or how can it be either of the two ways?)

 

[renormalize]

Now, let me propose to you a completely observable quantity S′ (according to this definition), different from the Poynting vector and still leading to exactly the same energy transfer results:
A(M1,t)=14πϵ0c2∫j(M2,t′)||M1−M2||dV,
with t′ the retarded time, and
Φ(M1,t)=14πϵ0c2∫ρ(M2,t′)||M1−M2||dV
with the following bound conditions for the integrals: A(M1,0)=0 and Φ(M1,0)=0.
Then define S′=(Φ∇−∇∂A∂t)×B and you are done.

So you have introduced explicit expressions for the scalar potential $\Phi$ and the magnetic vector potential $\boldsymbol{A}$ (which together makeup the relativistic 4-vector potential $A_{\mu}$) and from them defined a modified Poynting vector $$\boldsymbol{S'=}\left(\Phi\nabla-\nabla\boldsymbol{\dot{A}}\right)\times\boldsymbol{B=}\Phi\nabla\times\boldsymbol{B}-\nabla\boldsymbol{\dot{A}}\times\boldsymbol{B}$$ But your operator $\nabla\boldsymbol{\dot{A}}$ as written has two 3-vector indices and it's unclear how you mean to contract and/or cross it with $\boldsymbol{B}$. And does $\nabla$ differentiate both $\boldsymbol{\dot{A}}$ and $\boldsymbol{B}$ or just one of them? Could you please clarify?

This is again exactly analogous to the fact that the energy flux in electromagnetics cannot depend on the four-vector potential, but only on the electric and magnetic fields derived from that potential.

The magnetic and electric fields depend themselves from the potentials, so your last sentence is formally a nonsense.

OK, its my turn to clarify my "nonsense"! The energy flux in electromagnetics depends only on the differentiated 4-vector potential, and only in the specific combinations $\boldsymbol{E}=-\nabla\Phi-\frac{1}{c}\boldsymbol{\dot{A}}$ and $\boldsymbol{B}=\nabla\times\boldsymbol{A}$. What is nonsense is to claim that the undifferentiated 4-potential can appear in the energy flux.

...in the old-new theory "I" propose, the notion of EM flux through an open surface is not intrinsically defined, and need not actually.

I assume you mean to say "...need not actually exist". Given that statement, if EM flux through an open surface might not actually exist in your proposed theory, can you please state clearly what you believe the (open) front face of an illuminated solar panel is receiving from the sun and converting to usable electrical power? After all, the utility of solar panels is an empirical fact independent of any proposed theory.

 

[coquelicot]

So you have introduced explicit expressions for the scalar potential $\Phi$ and the magnetic vector potential $\boldsymbol{A}$ (which together makeup the relativistic 4-vector potential $A_{\mu}$) and from them defined a modified Poynting vector $$\boldsymbol{S'=}\left(\Phi\nabla-\nabla\boldsymbol{\dot{A}}\right)\times\boldsymbol{B=}\Phi\nabla\times\boldsymbol{B}-\nabla\boldsymbol{\dot{A}}\times\boldsymbol{B}$$ But your operator $\nabla\boldsymbol{\dot{A}}$ as written has two 3-vector indices and it's unclear how you mean to contract and/or cross it with $\boldsymbol{B}$. And does $\nabla$ differentiate both $\boldsymbol{\dot{A}}$ and $\boldsymbol{B}$ or just one of them? Could you please clarify?

My bad! the formula was correctly written in the last version of the article I posted. It is
$$\boldsymbol{S'=}\epsilon_0 c^2 \left(\Phi\nabla-\boldsymbol{\dot{A}}\right)\times\boldsymbol{B=}\epsilon_0 c^2(\Phi\nabla\times\boldsymbol{B}-\boldsymbol{\dot{A}}\times\boldsymbol{B}).$$

OK, its my turn to clarify my "nonsense"! The energy flux in electromagnetics depends only on the differentiated 4-vector potential, and only in the specific combinations $\boldsymbol{E}=-\nabla\Phi-\frac{1}{c}\boldsymbol{\dot{A}}$ and $\boldsymbol{B}=\nabla\times\boldsymbol{A}$. What is nonsense is to claim that the undifferentiated 4-potential can appear in the energy flux.

Well, that's the same discussion again and again. I think I have explained the point as far as I could in post #207. Please, take the time to read it carefully; if after that you disagree with my proposition, I think it's a matter of choice and I can do nothing more.

I assume you mean to say "...need not actually exist". Given that statement, if EM flux through an open surface might not actually exist in your proposed theory, can you please state clearly what you believe the (open) front face of an illuminated solar panel is receiving from the sun and converting to usable electrical power? After all, the utility of solar panels is an empirical fact independent of any proposed theory.

My bad again! I meant "the notion of EM flux through an open surface is gauge dependent", which is allowed by "my" theory. In my view, that's the whole body that receives the energy from the sun. But choosing the gauge $\Phi = 0$, you have a useful theorem that says "the energy received by the body is equal to the integral of the flux (equal to the Poynting vector in this gauge) on the exposed surface". That's just because the Poynting vector is directed along the propagation direction of the light.

 

[coquelicot]

I have a conceptual problem with energy flowing in the wire.
If energy flows from an object A to an object B then I would expect the energy stored in object A to decrease and the energy in B to increase. A good example is a warm object connected to a cold object by a thermal conductor. In that case the energy flow is simply the heat current.
Now if A is the negatively charged plate of a capacitor and B is the positively charged plate, in what sense does the energy stored in A decrease and the energy in B increase when the two plates are connected by a wire? Why is it not the other way round. (Or how can it be either of the two ways?)

So, you have also a problem with the density of current $\bf j$. Why is the current flowing in one direction and not the other?
Or, trying to understand the question more in depth, you have also a problem with the circulation of water inside a circular pipe, with a small fan inside the pipe to maintain a constant water stream?

 

[Philip Koeck]

So, you have also a problem with the density of current $\bf j$. Why is the current flowing in one direction and not the other?
Or, trying to understand the question more in depth, you have also a problem with the circulation of water inside a circular pipe, with a small fan inside the pipe to maintain a constant water stream?

A current of charges is no problem. One plate is positively charged to start with and one is negative. The current reduces this unbalance.
I just don't see that the same is true for an energy flow. I can't see that one plate has more energy than the other to start with.

 

[coquelicot]

A current of charges is no problem. One plate is positively charged to start with and one is negative. The current reduces this unbalance.
I just don't see that the same is true for an energy flow. I can't see that one plate has more energy than the other to start with.

Let take a perfect analogy with water. Take a circular pipe full of water. At some point of the pipe, there is a small piston tied to a spring. The spring is tied to the pipe wall. By some mean, you make the piston move from its natural rest position, bending the spring. Then you relax the piston. The piston push the water in the pipe, and you have a current of water, associated with an energy flow. If you have a doubt about that, here is something from Feynman lectures, that will show you that an energy flow is usually associated with the current of particle:

There is an important theorem in mechanics which is this: whenever there is a flow of energy in any circumstance at all (field energy or any other kind of energy), the energy flowing through a unit area per unit time, when multiplied by $1/c^2$, is equal to the momentum per unit volume in the space. In the special case of electrodynamics, this theorem gives the result that g is $1/c^2$ times the Poynting vector:
$$g={1\over c^2}S.$$
So the Poynting vector gives not only energy flow but, if you divide by $c^2$, also the momentum density. The same result would come out of the other analysis we suggested, but it is more interesting to notice this more general result. We will now give a number of interesting examples and arguments to convince you that the general theorem is true.
First example: Suppose that we have a lot of particles in a box—let’s say $N$ per cubic meter—and that they are moving along with some velocity $v$. Now let’s consider an imaginary plane surface perpendicular to $v$. The energy flow through a unit area of this surface per second is equal to Nv, the number which flow through the surface per second, times the energy carried by each one. The energy in each particle is $$m_0c^2\over \sqrt{1−v^2/c^2}.$$ So the energy flow per second is
$$Nvm_0c^2\over \sqrt{1−v^2/c^2}.$$
But the momentum of each particle is $$m_0v\over \sqrt{1−v^2/c^2},$$ so the density of momentum is
$$Nm_0v \over \sqrt{1−v^2/c^2},$$
which is just $1/c^2$ times the energy flow—as the theorem says. So the theorem is true for a bunch of particles.

 

[Philip Koeck]

Let take a perfect analogy with water. Take a circular pipe full of water. At some point of the pipe, there is a small piston tied to a spring. The spring is tied to the pipe wall. By some mean, you make the piston move from its natural rest position, bending the spring. Then you relax the piston. The piston push the water in the pipe, and you have a current of water, associated with an energy flow. If you have a doubt about that, here is something from Feynman lectures, that will show you that an energy flow is usually associated with the current of particle:

I agree that there is an energy flow associated with a particle flow simply because every particle has a momentum and kinetic energy, but I'm not sure that this is what we are discussing.
You get an energy flow with magnitude V I in your theory (using one of the gauges).
I wonder if the flow of kinetic energy of the electrons can be that large.

I also wonder about the direction. If you replace the wire by some device with mobile positive charges and stationary negative charges then the electric current still goes in the same direction, but the particle flow goes in the other direction.

Your result that the power equals V I is obviously right. Maybe the problem just lies in the interpretation. Couldn't it just be that this simply cannot be regarded as an energy flow,
whereas in the other gage (with Φ = 0) you do get an energy flow in form of the Poynting vector?

 

[coquelicot]

I agree that there is an energy flow associated with a particle flow simply because every particle has a momentum and kinetic energy, but I'm not sure that this is what we are discussing.
You get an energy flow with magnitude V I in your theory (using one of the gauges).
I wonder if the flow of kinetic energy of the electrons can be that large.

That's not the flow of kinetic energy (which is negligible), but the flow of electric potential energy, carried by the charges, and equal to $\Phi j$.

I also wonder about the direction. If you replace the wire by some device with mobile positive charges and stationary negative charges then the electric current still goes in the same direction, but the particle flow goes in the other direction.

You are right to say that that's not really the flow of particle (I was aware of that from the beginning, but I didn't want to introduce another inessential problem). That's the algebraic flow of charges that import here. OK, not really a particle flow, but still, sufficiently strongly related. In the same way, the current density is not really the density associated to the current of positive charges, but that's the algebraic flow of the charges (I mean, $J = J_+ - J_-$).

Your result that the power equals V I is obviously right. Maybe the problem just lies in the interpretation. Couldn't it just be that this simply cannot be regarded as an energy flow,
whereas in the other gage (with Φ = 0) you do get an energy flow in form of the Poynting vector?

If you take the example of Feynman in my previous post, you'll see that in addition to the kinetic energy of the particles, Feynman includes their rest energy $m_0c^2$, which is a "kind of" potential energy as far as I understand. So, I see no reason to ban the flow electric potential energy of the charges as well.

 

[Philip Koeck]

That's not the flow of kinetic energy (which is negligible), but the flow of electric potential energy, carried by the charges, and equal to $\Phi j$.
You are right to say that that's not really the flow of particle (I was aware of that from the beginning, but I didn't want to introduce another inessential problem). That's the algebraic flow of charges that import here. OK, not really a particle flow, but still, sufficiently strongly related. In the same way, the current density is not really the density associated to the current of positive charges, but that's the algebraic flow of the charges (I mean, $J = J_+ - J_-$).
If you take the example of Feynman in my previous post, you'll see that in addition to the kinetic energy of the particles, Feynman includes their rest energy $m_0c^2$, which is a "kind of" potential energy as far as I understand. So, I see no reason to ban the flow electric potential energy of the charges as well.

Then I still have the same problems.
I can't see how potential energy can flow to start with.
The other problem is that any flow of energy would mean that there is more energy at one end than the other to start with and the flow evens out this imbalance. I can't see that either.

 

[coquelicot]

Then I still have the same problems.

Ask Feynman

I can't see how potential energy can flow to start with.

Take again the example of the water circuit above, and assume that the pipe is full with a solution containing some chemical energy. Can't you imagine the flow of chemical energy within the pipe? that's just what is said in chemical thermodynamics, or even in the formula that was the subject of this thread.

The other problem is that any flow of energy would mean that there is more energy at one end than the other to start with and the flow evens out this imbalance. I can't see that either.

No, again, you have a circuital flow. In your cap, the potential energy is not stored at the plates, but inside the E-field of the cap (mostly located inside the plates of the cap, but also, in a less extent, outside). To prevent a further question, observe that the discharge of a cap does not produce a steady current, hence truly steady potentials are impossible here. Near the cap, there is an energy flow outside the wires of course. Far from the cap, the potentials are almost steady (with any non foolish gauge), and the energy flow described with respect to this gauge is observed inside the wires.
Of course, you can choose the foolish (in this circumstance) gauge $\Phi = 0$, which leads to a vector potential depending upon the time; then the generalized Poynting vector now reduces to the usual Poynting vector with respect to this gauge, and you are happy. I guess most electrical engineers and thermodynamists will be happy with the steady potentials and the energy power flowing inside the wires.

 

[Philip Koeck]

Ask Feynman Take again the example of the water circuit above, and assume that the pipe is full with a solution containing some chemical energy. Can't you imagine the flow of chemical energy within the pipe? that's just what is said in chemical thermodynamics, or even in the formula that was the subject of this thread.

No, again, you have a circuital flow. In your cap, the potential energy is not stored at the plates, but inside the E-field of the cap (mostly located inside the plates of the cap, but also, in a less extent, outside). To prevent a further question, observe that the discharge of a cap does not produce a steady current, hence truly steady potentials are impossible here. Near the cap, there is an energy flow outside the wires of course. Far from the cap, the potentials are almost steady (with any non foolish gauge), and the energy flow described with respect to this gauge is observed inside the wires.
Of course, you can choose the foolish (in this circumstance) gauge $\Phi = 0$, which leads to a vector potential depending upon the time; then the generalized Poynting vector now reduces to the usual Poynting vector with respect to this gauge, and you are happy. I guess most electrical engineers and thermodynamists will be happy with the steady potentials and the energy power flowing inside the wires.

Energy that flows in a circle is even worse to my way of thinking. Why would it do that?

I'd like to hear what others have to say, though.

The picture that makes sense to me is that heat leaves the wire and this energy loss is balanced by energy that flows from the battery or capacitor via the surrounding space into the wire.
Obviously the total power of this process is V I, but that doesn't indicate an alternative path for the energy flow as far as I'm concerned.

 

[coquelicot]

Energy that flows in a circle is even worse to my way of thinking. Why would it do that?

I have to clarify myself: indeed the energy is flowing in a circle, but it is not constant along the circle. Actually, if the wire as some resistance per unit of length, the energy flow, at the minus of the battery, is null. It is maximal at the + terminal of the battery, and dissipates into heat all along the wire (hence its decreasing). Notice that the situation is worse and much less intuitive with the Poynting vector flow: there the energy flows symmetrically from the two terminals of the battery "near" the wire, decreasing more and more until it becomes null near the middle of the wire (see the article of Harbola I posted somewhere in this thread). That's a bit weird isn't it?

The picture that makes sense to me is that heat leaves the wire and this energy loss is balanced by energy that flows from the battery or capacitor via the surrounding space into the wire.
Obviously the total power of this process is V I, but that doesn't indicate an alternative path for the energy flow as far as I'm concerned.

That's your choice and I respect it, as I use to say.

EDIT: By choosing the gauge $\Phi := -\Phi$, the energy will flow in the opposite direction. So, you have perhaps an answer to your intuitive problem.

 

[Philip Koeck]

Notice that the situation is worse and much less intuitive with the Poynting vector flow: there the energy flows symmetrically from the two terminals of the battery "near" the wire, decreasing more and more until it becomes null near the middle of the wire (see the article of Harbola I posted somewhere in this thread). That's a bit weird isn't it?

That does sound strange, I agree.
My feeling would be that the EM energy flow should balance the heat flow everywhere in steady state.
So if the wire is the same everywhere I would expect the same T, the same heat flow out from the wire and the same energy flow into the wire everywhere along the length of the wire.

 

[coquelicot]

That does sound strange, I agree.
My feeling would be that the EM energy flow should balance the heat flow everywhere in steady state.
So if the wire is the same everywhere I would expect the same T, the same heat flow out from the wire and the same energy flow into the wire everywhere along the length of the wire.

Regarding "the same T", I believe this is almost the case because we have to take into account the heat conduction in the wire. But that's a question of thermodynamics with heat fluxes etc. and I think Fluidistic is better than me to answer it (in fact, I think he has already answered to it, and there are also articles about that). The point is that thermodynamists use the formula that is the subject of the question of the OP, that cannot be justified with the Poynting vector, as far as I can conclude from this thread. This was the starting point of my thoughts, as you know.

 

[vanhees71]

A current of charges is no problem. One plate is positively charged to start with and one is negative. The current reduces this unbalance.
I just don't see that the same is true for an energy flow. I can't see that one plate has more energy than the other to start with.

Of course not, and for a statically charged capacitor the energy flow is 0.

A bit more puzzling is the explanation of the situation, where you have an electrostatic field within a capacitor superimposed by a magnetostatic field. Then the energy flow (Poynting vector), $\vec{E} \times \vec{B} \neq 0$. How to explain this flow in a purely static situation. Hint: Google for "hidden momentum". It's nicely treated in Griffiths's E&M textbook. A very nice collection about these apparent "paradoxes" in E&M is by McDonald:

https://physics.princeton.edu/~mcdonald/examples/

For the question here:

https://physics.princeton.edu/~mcdonald/examples/current.pdf

The answer is that E&M is a relativistic theory and also energy-momentum balance has to be treated relativistically. In fact there is no "hidden momentum" but just the correct definition of the energy-momentum-stress tensor of the em. field + the charged particles.

 

[coquelicot]

A bit more puzzling is the explanation of the situation, where you have an electrostatic field within a capacitor superimposed by a magnetostatic field. Then the energy flow (Poynting vector), $\vec{E} \times \vec{B} \neq 0$. How to explain this flow in a purely static situation. Hint: Google for "hidden momentum". It's nicely treated in Griffiths's E&M textbook. A very nice collection about these apparent "paradoxes" in E&M is by McDonald:

Note: with the theory of relativity of gauges I propose, this is no more a problem. Under any steady gauge, there is no flow of energy in a static state. Another a weird thing that disappears!

 

[vanhees71]

This doesn't make sense. The Poynting vector is the electromagnetic energy-flow density. It's compensated by the corresponding "hidden momentum" of the charges making the current to produce the magnetic field moving in the electric field of the capacitor. All this is of course entirely gauge-independent as it must be for observable phenomena.

 

[coquelicot]

This doesn't make sense. The Poynting vector is the electromagnetic energy-flow density. It's compensated by the corresponding "hidden momentum" of the charges making the current to produce the magnetic field moving in the electric field of the capacitor. All this is of course entirely gauge-independent as it must be for observable phenomena.

In a purely static situation (which I thought was the context), where the magnetic field is created by a magnet and the charge is static, my assertion makes sense because there is no hidden momentum (no charges movement).
In a situation where the magnetic field is created by charges in a solenoid, then we are in the context of magnetostatic, and there is, of course, a field momentum (in my proposed theory). So, no nonsense as well. I will drop your assertion "The Poynting vector is the electromagnetic energy-flow density", since this is a loop inside the same debate, of which I have provided licit arguments.

 

[vanhees71]

There are moving charges (currents). Otherwise there'd be no magnetic field. Of course, to understand permanent magnets you need quantum mechanics to correctly describe it, which is outside of the realm of classical electromagnetism, but also there you have a "current" and "hidden momentum".

 

[coquelicot]

There are moving charges (currents). Otherwise there'd be no magnetic field. Of course, to understand permanent magnets you need quantum mechanics to correctly describe it, which is outside of the realm of classical electromagnetism, but also there you have a "current" and "hidden momentum".

Do you have a reference for you last assertion? Also, does not this hidden momentum statistically cancel?

 

[fluidistic]

In steady state, it's the divergence of the energy flux that must vanish in any volume considered, not the flux itself. Imposing this condition yields the steady state heat equation in the material.

 

[renormalize]

My bad! the formula was correctly written in the last version of the article I posted. It is $$\boldsymbol{S'=}\epsilon_0 c^2 \left(\Phi\nabla-\boldsymbol{\dot{A}}\right)\times\boldsymbol{B=}\epsilon_0 c^2(\Phi\nabla\times\boldsymbol{B}-\boldsymbol{\dot{A}}\times\boldsymbol{B}).$$

Thanks for posting this corrected equation from your article of April 28 in post #176. I have read it and now better understand your motivation for introducing your gauge-dependent energy flux $\boldsymbol{S'}$.
Another form you write for this flux is $$\boldsymbol{S}^{'}=\boldsymbol{S}+\mathrm{\frac{1}{\mu_{0}}}\nabla\times\left(\Phi\boldsymbol{B}\right)$$ where $\boldsymbol{S}=\mathrm{\mathrm{\frac{1}{\mu_{0}}}}\boldsymbol{E}\times\boldsymbol{B}$ is the usual gauge-invariant Poynting vector. This makes it clear that both $\boldsymbol{S}$ and $\boldsymbol{S'}$ have the same divergence and that they are identical whenever the scalar potential $\Phi$ vanishes. Indeed, you deem $\Phi=0$ to be the appropriate gauge for considering a plane wave since the energy flux of EM radiation is well described by the usual Poynting vector. So far so good.
You then introduce another equivalent expression for your energy flux $$\boldsymbol{S}^{'}=\Phi\boldsymbol{J}+\varepsilon_{0}\Phi\boldsymbol{\dot{E}}-\mathrm{\mathrm{\frac{1}{\mu_{0}}}}\boldsymbol{\dot{A}}\times\boldsymbol{B}$$ and consider the steady-state case (like a DC current flowing in a wire) by dropping the time derivatives, resulting in $$\boldsymbol{S}^{'}=\Phi\boldsymbol{J}$$ I note that this flux does indeed have (by construction) the expected divergence $$\nabla\cdot\boldsymbol{S}^{'}=\left(\nabla\Phi\right)\cdot\boldsymbol{J}+\Phi\nabla\cdot\boldsymbol{J}=-\boldsymbol{E}\cdot\boldsymbol{J}$$ (because $\nabla\Phi=-E-\boldsymbol{\dot{A}}$ and both $\boldsymbol{\dot{A}}$ and $\nabla\cdot\boldsymbol{J}$ vanish for a steady state), i.e., this divergence is the negative of the flux of work done by the EM field on the current.
Even so, I remain (as yet) unconvinced when you conclude regarding $\boldsymbol{S}^{'}=\Phi\boldsymbol{J}$ that
The meaning of this equation is that for steady currents, there is no flow of power wherever the space is free of charges: the power is carried by the charges only.
To make this convincing, you need to explicitly solve the relevant Maxwell equations for the scalar potential $\Phi$ and then demonstrate that $\Phi\boldsymbol{J}$ gives a physically reasonable energy flux in the wire.
For example, consider the simple case of a wire of circular cross-section carrying a steady uniform current-density and choose cylindrical-coordinates {$r,\phi,z$} that align the z-axis with the axial-direction of the wire. Since your flux contains just $\Phi$, the only relevant Maxwell equation is Gauss's Law, $\frac{\rho}{\varepsilon_{0}}=\nabla\cdot\boldsymbol{E}=-\nabla^{2}\Phi$. So with no $\phi$-dependence, you have to solve Poisson's equation $$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial\Phi}{\partial r}\right)+\frac{\partial^{2}\Phi}{\partial z^{2}}=-\frac{\rho}{\varepsilon_{0}}$$ Because the electric field external to a current-carrying wire is strictly radial as $r\rightarrow\infty$ and since there is nothing in this simple wire problem that appears to depend on $z$, it's tempting to assume that $\Phi$ depends exclusively on $r$. But of course that can't be right for your theory since it yields an electric field $\boldsymbol{E}=-\frac{\partial\Phi\left(r\right)}{\partial r}$ that's strictly radial even inside the conductor, and hence the field can do no work on a current flowing in the axial direction. (In this scenario, some non-electromotive force, like gravity, must exist to propel the current flow against the resistance.)
Instead, I think you're going to have to find a solution $\Phi(r,z)$ of Poisson's equation that yields an electric field $\boldsymbol{E}(r,z)$ which points radially at infinity. As it gets closer to the wire, the field must bend in the axial-direction such that the component $E_{z}$ reaches the just the right value inside the wire to yield the proper energy flux. I can't claim that this is impossible, but I do wish you luck finding such a scalar potential.

 

[Dale]

Temporarily closed for moderation

Edit: after some internal discussion this thread will remain closed. Participants are reminded that personal speculation is prohibited at PF and all posts must be consistent with the professional scientific literature, not merely PF posts that have been put into a .pdf file