That's a lot of stuff to digest. But one aspect seems obvious, no matter which "cult" you adhere to: In order to actually observe the path dependence of the voltage between A and D, you need to probe in a way that allows "selecting" the path you want to observe.
It's very simple. If the measurement loop formed by voltmeter, probes, and that branch DOES NOT enclose (cut) any (appreciable) variable magnetic flux then the electric fields obeys E = -grad V without the additional term dB/dt and can be seen as conservative. If we limit ourselves to planar circuits to make it easier to see, all paths in the area delimited by that measurement loop give the same value for the path integral of E, so voltage between any two points inside that area is the same no matter how you choose to join them (it only depends on the endpoints): along the measured branch, in the space between them, or along the probes through the voltmeter. I.E. the value shown by the voltmeter is also the voltage computed on the path along the branch, and even across it if you stay in the space that is magnetic-free.
Watching the MIT video that was linked here by @rfeecs, it is obvious that this selection is done only by the layout of the probe wires. If I understood the reasoning correctly, the selector is not the magnetic flux, because the paths that lead to the instrument are outside of it.
The selector is the presence or better the absence of magnetic flux inside the measurement loop.
That means the chosen path must never encompass any area that is inside the magnetic flux.
Yes, but you need to think in terms of closed paths. When you put one voltmeter on the outside of the ring, it will form two loops: one with the nearest resistor - that does not enclose the dB/dt region, the other with the farthest resistor - that does enclose the dB/dt region.
The voltages between two points in the region of space delimited by the first loop are all equal, no matter the path - hence the reading on the voltmeter corresponds to the correct voltage of the nearest branch.
The other measurement loop is not reading the correct voltage of the farthest branch, though, because it is affected by the presence of the variable magnetic field linked.
The beauty of it is that [correct voltage of far branch] + [emf with correct sign] = [voltage read by multimeter]
What I still have a hard time coming to terms with is that there is no voltage contribution by the other possible path through the second resistor
Oh, but there is. The current flowing in the near resistor - the current that gives rise to the voltage you read on the the voltmeter - would not be there if not for the second resistor.
Now, why is there no voltage contribution by the wire itself, this is the most counter-intuitive aspect and the core of the claim that KVL doesn't hold. "Silicon Soup" explains that the rotational electric field caused by the changing magnetic flux and the electric field inside the wire caused by charge separation cancel each other out and the net electric field "in the wire" is 0. D'oh! So it's there but you cannot observe it because there's another electric field of same "strength" just in the "opposite direction". I put that in quotes not to express doubt but to denote that I understand that it's a simplification and we're talking about vectors here. But it is still hard to stomach because you can observe the effect of the charge separation through the voltage measurable at the terminals.
It is exactly that charge separation that produces the coloumbian field that obliterates the induced field in the wire. If the wire is a perfect conductor, then the total electric field inside is zero. If the wire is a real conductor with high conductivity such as copper then there is a tiny electric field of the order of a handful of microvolts per meter that is compatible with the local form of Ohm's law.
So, is it really "not there" or is it just difficult to find a path along which it would be observable?
It's not there because the electric field has been cancelled (exactly in a perfect conductor, almost entirely in a good conductor.)
If there is no field, then what did Mabilde measure in his setup? Because his setup modeled after the McDonald paper shows 0.25V across a quarter segment of the loop, and that is obviously way to much for just Ohms law in action.
Mabilde has put its probes INSIDE the variable magnetic field region. He is cutting flux on purpose to induce a voltage in his measurement loop so that it can cancel the contribute of the induced electric field. This leaves only the contribute of the coloumbian field that is a conservative field. It's a nice technique, but he does not understand that he is measuring a partial contribute only. The field the electrons in the wire and the space experience is the total field
Etot = Eind + Ecoul
He is measuring the effects of Ecoul alone which admit a scalar potential phi. Which is fine if you realize that phi alone is not sufficient to completely describe the system. You also need the vector potential A, as McDonald shows.
What McDonald did was to apply the Helmoltz decomposition of fields to the total electric field and then associate the scalar potential phi to Ecoul and the vector magnetic potential A to Eind.
What the KVLers understood is only half of it. They stopped at phi and thought: "see? the potential is uniquely defined" without realizing that such potential is referred to a part and not all of the electric field.
In one of my answers on EESE I quote a paragraph of Popovic & Popovic where the actual expression of voltage in the presence of a variable magnetic field is given.
I'm still not done thinking this through but at this point I have an inkling that when I'm done, I will have to apologize to @bsfeechannel ![Face Palm :palm:](https://www.eevblog.com/forum/Smileys/default/facepalm.gif)
From time to time you read, on Lewin's YT channel, the posts of someone apologizing to him after realizing he has been right the whole time.
an electric field in the wire if it's not stationary,
Get hold of Purcell. It has pictures for that.
And for the variac... we know the field is in the space between turns, and that is what you measure with a voltmeter on the outside. But if you have trouble computing that, why don't you use the other measuring loop, the one following the conductor around the core? You just have to apply Faraday and the concept that there is no appreciable voltage drop in the conductor: [voltage measured outside] = [negligible voltage drop in the turns tapped] + [emf with correct sign multiplied by number of turns]
That also explains why you cannot measure partial turn voltage, but only integer multiples.
(Edit: corrected a sentence to make it clear which area, which loop, which points)