But since Stokes’ is just vector math, and our concept of vector fields is a simplification of the real situation of trillions of atomic orbiting electrons in one region of space influencing trillions of conduction electrons in another region of space, is it not possible that Stokes’ would not apply in certain cases?
A vector field is first and foremost a model. In the case of the magnetic field, the B field allows us to characterise space locally rather than using the billions of billions of equations describing each interaction at a distance with a delay, of each charge with each other.
If anyone has a better model, don't hesitate to provide it.
You can always imagine that since it's a model, and since it would be a simplification, phenomena could deviate from it. Even if we don't assert that something is true because it hasn't been shown to be false, we assume it, which is simply the argument from ignorance, and therefore an irrelevant argument.
Even if the experiment we're talking about produces a current, we don't even need to assume that the field model or Stockes' theorem wouldn't apply, as long as we don't have experimental confirmation that not only is there a real current produced but that no field model would explain it, for example by adding the field of a source we hadn't thought of.
On the other hand, since no experiment has yet shown a deviation from field theory, we can say that it is highly improbable that this would be the case here, and that it is up to those who are convinced of the contrary to demonstrate it.
...When you examine that and use E=-dA/dt instead of flux cutting E=vXB you find that both conductors endure the same E field so the closed loop induction is zero. My take is that the flux cutting rule cannot be used under all situations but E=-vXB is more fundametal and can be used.
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The flux cutting rule is generally false, except when applied in the near-field approximation, i.e. when field propagation times can be neglected. Here, it is correct. Sometimes it's also difficult to identify, as in the case of the Faraday disk.
As for using A instead of B, we've already done quite a bit of work on this, remember, and it turns out that the electromagnetic results agree, since what links them is not a physical theory but a purely logical mathematical relationship.
The only problem we have is when we limit ourselves to 3D Euclidean space when we live in a 4D universe.
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We even see, which surprised me at first, that a scalar potential φ can appear in the charge referential when it moves in a place where there is only the vector potential. And vice versa”.
https://www.overunityresearch.com/index.php?topic=4389.msg103488#msg103488When we use the 4-vector, many surprises disappear!
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