# 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?

• Possibly related:physics.stackexchange.com/q/136431/25301 – Kyle Kanos Dec 10 '18 at 20:36
• It depends on what we call "Kirchhoff's law". What do you call Kirchhoff's law? – hyportnex Dec 10 '18 at 20:38
• – The Photon Dec 10 '18 at 20:57
• I would say a complete statement of KVL should start with the phrase "In a lumped circuit...". In which case KVL applies in a universe that contains a non-lumped circuit. It just doesn't tell you anything useful about the non-lumped circuits. Part of the definition of a lumped circuit is that there isn't any significant changing magnetic flux in the loops formed by the wires interconnecting the circuit. – The Photon Dec 10 '18 at 21:15

...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 voltage 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 Coulomb 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 voltage 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 voltage drop across ideal inductor is unique: $$V.D.(ideal~inductor) = L\frac{dI}{dt}.$$ This voltage 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.