I had this answer written and put on hold while I was looking for a quote in a reference, and then I forgot to post it. Might as well do it now, even if I don't find who I was responding to. I guess this is the internet equivalent of being in a room and asking oneself: "what am I doing here?" But it was about how to add and remove energy from the field without radiation.Ok, 'quasi-statics' is a misleading term, in this context. In my language constant current flow system are classified as steady-state electrodynamics. They are not electrostatics, they are not magnetostatics. In these system we neglect the energy we put both into the magnetic field around the current, and into the electric field between conductors.
I'll get back to the battery-wires-resistor circuit later, but for now, let's consider the system where a charge is forced to move at constant velocity towards a part of the circuit that is charged with the same sign. We can even simplify this example further by considering just two same sign charges. When I push one against the other I need to do work. The energy I lost is put into the system (I would say 'in the field'). This is an electrostatic system when we start, and is an electrostatic system when we finish. If we are careful and do not accelerate the charge much (I was thinking of a very slow transformation, what in thermodynamic is termed 'reversible', see note 1) we lose almost nothing to radiation.
And yet energy is put into the system even when the charge is moving at a steady constant velocity v.
Where is the changing field in this picture? The electric field of the whole system is a function of the position of both charges. So, even when velocity is constant in magnitude and direction, the field changes. This is where we can see the Poynting vector appear: the moving charge is basically a current, a magnetic field ensues, and time after time the product E x H is nonzero and shows energy flowing into the field.
When I stop the charge (again, making sure not to radiate any appreciable amount of energy) I end up with a static system whose energy has increased.
If I wiggle the charge very slowly, so that I do not experience any appreciable acceleration, I can add and subtract energy to the system in an almost ("quasi") reversible way - see note 1. So, where does energy flow in this case? I would say it flows into the field, in the space between (and around) the charges.
Let's get back to the resistor circuit, again with the charge in the space between wires. Let's say the top of the resistor is positively charged (I'm talking about surface/interface charges, the same charges responsible for the strong electric field inside the resistor's body). If I try to push my positive charge against the top of the resistor, I have to do work. The field of the system circuit+charge will increase at my expenses. Where did the energy go? Into the field: the physical system in this new configuration has acquired the capacity to do more work. If I attach my charge to a little spring, it will be repelled by the top of the resistor. The spring will compress and mechanical energy will be put into the mechanical system. Where did that energy come from? I say it comes from the EM field. My tiny charge was in the space between wires, and has received energy while it was in the space between wires. Did this energy flow out of the wires? Or is it just the result of the interaction with the field - generated by surface charge - that already was outside the wires?
Regarding the resistor itself, here we reach the limits of classical electrodynamics since Ohm's law cannot be quantitatively explained classically. In any case, if we choose to go along with the classical model of resistivity, the constant velocity of the charges inside the conductor is the result of an equilibrium between acceleration and deceleration due to the constant electric field in the material and the collisions with lattice atoms. Constant velocity is a macro-generalization.
The simple battery-wires-resistor circuit is much more complicated that it appears at first sight, and in my opinion the model where a uniform and constant current density is the result of a magically established electric field E=j/sigma inside the good and bad conducting parts of the circuit is a gross oversimplification. It makes people think that the electric field is only inside the material. It does not take into account at all the role of surface charge: the surface charge that is responsible for the electric field around the battery, the wires, the resistor. The surface charge that shapes the EM field in the space inside AND outside the circuit itself. The surface charge that is responsible for the energy in the EM field in the whole of space.
If I have to choose between a description (phi J) that only works in the material (where the E field is constant and directed along the conductor just 'because' it has to follow Ohm's law) and a description (E x H) that works in the material and in the space around it (explaining how the surface and interface charge interact to produce that curiously shaped E field inside the conductor), I choose the latter. YMMV.
I add here an extract from
Edward G. Jordan, Keith J. Balmain
Electromagnetic Waves and Radiating Systems 2e
1968, Prentice Hall
(which I should have edited in in the philosophical appendix of my previous post)
p. 169: 6.02 A note on the interpretation of ExH"Most engineers find acceptable the concept of energy transmission through space, either with or without guiding conductors, when wave motion is present. However, for many engineers this picture becomes disturbing for transmission line propagation in the DC case.
When E and H are static fields produced by unrelated sources, the picture becomes even less credible. The classic illustration of a bar magnet on which is placed an electric charge is one which is often cited. In this example a static electric field is crossed with a steady magnetic field and a strict interpretation of Poynting's theorem seems to require a continuous circulation of energy around the magnet. This is a picture that the engineer generally is not willing to accept (although he usually does not question the theory of permanent magnetism which requires a continuous circulation of electric currents within the magnet)."
Incidentally, this example is also discussed in
Pollack and Stump's "
Electromagnetism", on p. 421. Somewhat unsatisfactorily, they 'solve' the paradox by cutting the Gordian knot with twenty-two (s)words:
"The energy flow associated with S in this case is merely formal; it has no physical significance because it cannot be detected."
Back to Jordan and Balmain, they go on explaining how integrating over a closed surface will make the apparent paradox disappear and how this implies that we cannot say where the energy is because if what matters is the integral, then we can add a zero-divergence vector to ExH and still get the same result. And then they draw a parallel with the gravitational potential energy case: where is the energy of a rock raised at an height h over the ground?
Following
Ramo Whinnery VanDuzer (and Stratton), I like to think that the energy is in the space where the fields are. RWvD have this to say:
p 140 3eAlthough it is known from the proof only that total energy flow out of a region per unit time is given by the total surface integral
fig poynting theorem
it is often convenient to think of the [Poynting] vector P defined by P = E x H as the vector giving direction and magnitude of energy flow density at any point in space. Though this step does not follow strictly, it is a most useful interpretation and one which is justified for the majority of applications.
and then they point to the example depicting a resistor where the Poynting vector is directed radially inward and say
p 142 3e (italics theirs, bold mine)"We know that this result does represent the correct power flow into the conductor, being dissipated in heat. If we accept the Poynting vector as giving the correct density of power flow at each point, we must then picture the battery or other source of energy as setting up the electric and magnetic fields, so that the energy flows through the field and into the wire through its surface.
The Poynting theorem cannot be considered a proof of the connectedness of this interpretation, for it says only that the total power balance for a given region will be computed correctly in this manner, but the interpretation is nevertheless a useful one."
In the third edition they also add this:
p 141 3e"It should be noted that there are cases for which there will be no power flow through the electromagnetic field. Accepting the foregoing interpretation of the Poynting vector, we see that it will be zero when either E or H is zero or when the two vectors are mutually parallel.
• Thus, for example, there is no power flow in the vicinity of a system of static charges that has electric field but no magnetic field.
• Another very important case is that of a perfect conductor, which by definition must have a zero tangential component of electric field at its surface. Then P can have no component normal to the conductor and there can be no power flow into the perfect conductor."
Note 1: Regarding the transformations so slow as to be considered almost static so they would not lose energy to radiation, I have found comfort in
Stratton,
Electromagnetic Theory 1941:
p. 131, italics mine"Poynting’s Theorem: In the preceding sections of this chapter it has been shown how the work done in bringing about small variations in the intensity or distribution of charge and current sources may be expressed in terms of integrals of the field vectors extended over all space.
The form of these integrals suggests, but does not prove, the hypothesis that electric and magnetic energies are distributed throughout the field with volume densities respectively
fig densities formulae
"The derivation of these results was based on the assumption of reversible changes; the building up of the field was assumed to take place so slowly that it might be represented by a succession of stationary states".
"It is essential that we determine now whether or not such expressions for the energy density remain valid when the fields are varying at an arbitrary rate. It is apparent, furthermore, that if our hypothesis of an energy distribution throughout the field is at all tenable, a change of field intensity and energy density must be associated with a flow of energy from or toward the source."
On the arbitrariness of the assumptions relating Poynting's theorem Stratton has this to say:
p. 133"As a general integral of the field equations, the validity of Poynting's theorem is unimpeachable. Its physical interpretation, however, is open to some criticism. The remark has already been made that from a volume integral representing the total energy of a field no rigorous conclusion can be drawn with regard to its distribution. The energy of the electrostatic field was first expressed as the sum of two volume integrals.
Of these one was transformed by the divergence theorem into a surface integral which was made to vanish by allowing the surface to recede to the farther limits of the field. Inversely, the divergence of any vector function vanishing properly at infinity may be added to the conventional expression u = 1/2 E.D for the density of electrostatic energy without affecting its total value. A similar indefiniteness appears in the magnetostatic case."
But all in all
pp. 134-135, bold mine"The classical interpretation of Poynting’s theorem appears to rest to a considerable degree on hypothesis. Various alternative forms of the theorem have been offered from time to time,’ but none of these has the advantage of greater plausibility or greater simplicity to recommend it, and it is significant that thus far no other interpretation has contributed anything of value to the theory.
The hypothesis of an energy density in the electromagnetic field and a flow of intensity S = E x H has, on the other hand, proved extraordinarily fruitful. A theory is not an absolute truth but a self-consistent analytical formulation of the relations governing a group of natural phenomena. By this standard there is every reason to retain the Poynting-Heaviside viewpoint until a clash with new experimental evidence shall call for its revision."
Now, what
experimental evidence have we?
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