Does Kirchhoff's Law always hold? There's a bit of furore from this question on Youtube involving Dr. Walter Lewin and another Youtuber. With Dr. Lewin claiming Kirchhoff's Law doesn't always hold when magnetic fields are involved, and that two voltmeters attached to identical places in a circuit can give different readings. Is this the case?
Original Video :
https://www.youtube.com/watch?v=0TTEFF0D8SA
Dr. Lewin's response :
https://www.youtube.com/watch?v=AQqYs6O2MPw
 A: 
...claiming Kirchhoff's Law doesn't always hold when magnetic fields are involved, and that two voltmeters attached to identical places in a circuit can give different readings. Is this the case?

It is true that two voltmeters connected to same pair of points in a circuit can show different values. The reason is the voltmeters can be affected differently by a solenoidal electric field which is present since there is changing magnetic flux.
What is the point of contention here between prof. Lewin and his oponents is terminology(semantics) of the words like potential and voltage.
As others on this site have observed, whether KVL holds depends on:

*

*which version of KVL we use: there is the historical version about sum of electromotive forces in a circuit, and there is the modern textbook form about potential drops along a circuit;


*what we mean by voltage (or electromotive force, if we use the original version of KVL).
Prof. Lewin seems to understand voltage as either 1) integral of total electric  field along some path connecting two points 2) whatever voltmeter connected to two points shows. In both cases, his voltage is ambiguous, as it depends either on the chosen paths in space, or on how the probes of the voltmeter are arranged in space. With voltages defined as integrals of total electric field, the modern KVL indeed does not hold, because their sum equals minus net electromotive force for the circuit and this emf is arbitrary, depending on how we arrange the integration paths (or wires).
But the textbook authors writing KVL applies to such situations with changing magnetic flux are not wrong, because their concept of voltage between two points is that it is difference of electric potentials of those points, irrespective of whether a voltmeter or other device can measure it. This is a prevailing convention because:

*

*electric potential can always be naturally and uniquely defined as the potential of the conservative part of the electric field, i.e. the Coulomb-integral potential of all charges around, even in high frequency AC circuits with solenoidal fields everywhere (such potential is less useful in such high frequency cases but there is no problem in its definition)


*this definition is very useful in lumped elements model of circuit, because then each pair of terminals in a circuit has unique potential drop, equal to difference of Coulomb potentials on those terminals.
With prof. Lewin's implied definition (integral of total electric field), we would encounter various problems. For example, line integral of electric field taken along path connecting two terminals of an ideal solenoid in which electric current is changing would be arbitrary, dependent on the chosen integration path. It could always be made zero if the path is chosen entirely inside the perfect conductor of the solenoid. Thus the inductor would have no single voltage and we could not use the usual tools of analysis for lumped element model.
Of course, the standard theory asserts that potential drop across ideal inductor is unique:
$$
p.d.(ideal~inductor) = L\frac{dI}{dt}.
$$
This potential drop cannot be understood as integral of total electric field (which would depend on the choice of path), but it is integral of electrostatic part of it (which does not depend on that choice), or equivalently, minus electromotive force inside the solenoid - the "solenoidal component of electric field" or the "induced field" integrated along the path that is entirely in the wire of the solenoid.
