IT DEPENDS ON THE PATH
Of course it does.
But then why do you ask, and i quote, "
Is there potential across the ends or not?" without specifying the path?
Anyway, my yesterday answer was focused on the the long straight conductor through the center of the torus. For some reason (well, I had your sphere machine in mind) I thought you were talking about that all along and that the long U shaped path was just another question at the end (I even skipped it as non related). So, there has been a misunderstanding but what I wrote in my previous answer still applies. It is nevertheless clear that with that U shape the "potential across" meant the voltage on a path joining them in the proximity of the terminals. Sincere apologies for this misunderstanding on my part.
I nevertheless confirm that you should not use the word potential and in general - and even in this case because there are paths that can be run around the core.
The pictures I had drawn yesterday are mainly centered on the partial turn that according to some represent the 'elementary battery', but at this point I might as well post them. The second one was about the nearly complete but still partial turn and that is basically the same as your U shaped turn.
What I wrote in the previous answer still apply - but I would have worded it in a different manner. Can I blame the tequila?
Ok, here is the partial turn and some paths
https://i.postimg.cc/jqJ7FwKh/screenshot.pngand here is still another partial turn that runs almost - but not - completely around the core. You can consider this a representation of your long U coil.
https://i.postimg.cc/QN5Fq2k4/screenshot-2.pngAs you can see it all depends on the path. And the need for a closed (mathematical, non necessarily material) path is that it's the only kind of path that allows me to apply Faraday's law to get the answer. We already discussed that the local form of Maxwell equations doesn't really change this, because they are partial differential equations that requiire boundary conditions. (Also look up at the definition of curl and divergence and you will see that surfaces and boundary paths, volumes and boundary surfaces are built in into the equations.)
Now, let me answer your question about the U coil. You ask about the electrometers at the end and I have seen the experiment performed with FET electrometers. They put them near an outlet and they sense the (alternating) accumulation of charge at the terminals. The core is in the transformer on a pole near the house in the US, or even a mile away in a big transformer that delivers power to whole city blocks in some parts of the rest of the world. So, I'd say that with a linearly increasing excitation you will see the leaf open - for as long as you can ramp up the current in the primary.
What I am perplexed about is your fixation with 'where there are no fields'. If the charges are there, there is at least their field, so you can't do away with it. If there are no fields to push and keep them there, then the charges are no reason to accumulate there.
To put this in context, I was comparing three phenomena- a conductor in a static, irrotational conservative field, a conductor in a rotational non-conservative field derived from dB/dt and a conductor moving across magnetic lines as in a generator, where the charges experience the Lorentz force. So in the first case, I think we all agree with your assertion. Apparently, and don't attribute this assertion to me, the third case does result in a potential gradient over the length of the conductor. Is that right or wrong? And if it is right, then how do you differentiate the effect of the Lorentz force from the local Eind force in the second example?
Well, yesterday I also did a couple of pictures for the straight rod near a variable magnetic flux region. They are no different from the ones above, I only separated the paths that give zero voltage, like these
https://i.postimg.cc/cJ2JRhKG/screenshot-4.pngfrom the ones that, by linking the emf, give a voltage different from zero (and a multiple of the EMF)
https://i.postimg.cc/1zk9nzT9/screenshot-5.pngThe second picture is the reason I wrote - adding in the edit: IN GENERAL
"and this is again (EDIT: in general) wrong for the reason above."
The paths above show that
in general VBA can be nonzero.
Immediately after I added (emphasy added now):
"But, if you consider a region of space that contains all of your piece of conductor and none of the magnetic flux, like a cylinder around the rod in the middle of the torus, in that region of space all paths you can imagine will never enclose the magnetic flux region. in this sense you can consider it as equipotential."
And this is basically the situation depicted in the former picture. If I consider paths that do not run around the variable magnetic region. I can consider - AND PLEASE NOTE THAT I AM PUTTING IT IN QUOTES - the conductor as 'equipotential', meaning that the voltage (difference) between any two points of the conductor (and in particular on the surface) is zero. This is apparently indistinguishable from the conductor polarized by immersion in the electrostatic field generated by two charged plates.
I believe that there could be a distinction, tho. It's in the nature - I would say 'the shape' - of the field. In both cases, if you stay in your 'safe space' or 'magnetic free bubble' you can consider the conductor 'equipotential' (THE QUOTES! THE QUOTES!!!), but I seriously doubt that you could manage to make the field lines (and presumably the distribution of charge on the surface) the same in both cases (maybe a few lines will be reasonable close but then the fields will differ). What I mean is that the shape of the field lines coming out of charged plates will be straight if the plates are parallel, with the field magnitude constant, or hyperbolic shaped if the plates are at an angle, but you probably won't be able to replicate both the shape of the lines and the way the field increase or decrease on large portions of space. (I am not 100% positive.)
This in turn means that even the distorted field will be different.
But from the point of view of an observer of a polarized body that is in a room next to the room where a giant solenoid is being powered, the voltage along any path he can devise inside the room will be zero.
Does this answer your question?
I am not going to touch the motional case even with a ten foot pole. We are already wasting too much space and time to address the Lewin ring and we would fall down a relativistic rabbit hole with no end. I suggest you look up Purcell for an introduction to that kind of stuff.