[Kirchhoff's Voltages Law (KVL) failure] would be sorta like doing a momentum analysis between two colliding steel balls without figuring the effect of a magnet under the table.

That's exactly our kind of problem.

Of course, we were trained to identify and deal with it. But it was not so obvious that, in fact, we are "patching" the KVL by adding imaginary batteries to our circuit loop, isn't it? If it were so obvious, we wouldn't bother to talk about it. We are doing that in order to include the external influences of a variable flux. KVL was not meant to include the external influences. Kirchhoff derived the KVL for circuits with batteries and no external fields.

The other Kirchhoff's law, Kirchhoff's Currents Law (KCL) still works just fine no matter the externally magnetic fields. Only KVL doesn't hold, and only in a

*variable* magnetic flux. KVL in a constant flux, again no problem.

Now, is the professor correct, or not? Does Kirchhoff derived his KVL before, or after adding the virtual sources representing induced voltages caused by a variable flux? All the clues indicates that Kirchhoff was not concerned about induced voltages from external fields. He was not even looking for KVL. Kirchhoff was trying to find a way to calculate all the currents in a mesh of linear wires, so he was looking for KCL, not KVL. If I understood it correctly, Kirchhoff was thinking about telegraph wires when he derived KCL and, unintendly, KVL too. (see

https://www.jstor.org/stable/20021539 starting from the last paragraph)

Before taking his degree, Kirchhoff had begun his work in original

research, and published a remarkable paper on electrical conduction in

a thin plate, especially a circular one. His problem was to find the

current in any branch of a network of linear conductors. Starting

from Ohm's familiar law, he derived two results long recognized in

electrical science as Kirchhoff 's laws.

I couldn't find the original papers with the KCL and KVL published by Kirchhoff. All I could find is a followup of the KCL and KVL paper, (which, by the way, seems to be the first analysis of a transmission line:

https://www.ifi.unicamp.br/~assis/Apeiron-V19-p19-25(1994).pdf ). In this followup paper, Kirchhoff started from a real problem of those times: What happens in underwater telegraph wires. Again, the influence caused by an external variable magnetic flux was not a concern for the problem of submarine telegraph cables. All clues indicates that KCL and KVL were originally meant to be used for normal circuits, with batteries, and without considering external induced voltages.

I'll say the professor was correct when he said we can not always apply KVL. Of course, if we first transform the real circuit into a lumped circuit, where we add the externally induced voltages as voltage sources internal to our circuit, then we obtain a new circuit that obeys KVL.