Recently, Prof. Walter Lewin and YouTuber ElectroBOOM started a discussion about KVL, after Dr. Lewin claimed that KVL did not hold in the presence of an magneto-dynamic field. I would argue that Dr. Lewin is incorrect in his interpretation, and KVL does hold under scrutiny.
Lewin's argument follows from the given definitions. The electric potential, $\phi$, is defined as the line integral of the electric field, $\vec{E}$,
$$\phi = \int_C \vec{E} \cdot d\vec{l}$$
And in the general case, e.g. when a changing magnetic field is present, $\vec{E}$ is non-conservative, so $\phi$ is path-dependent. That is, the potential difference between two points in space is not unique, and therefore "voltage" is not well-defined. This is a direct result from vector calculus.
Now, as an extension, from the Faraday-Maxwell law,
$$ \oint_C \vec{E} \cdot d\vec{l} + \int_S \frac{\partial \vec{B}}{\partial t} \cdot d\vec{s} = 0$$
Where $C = \partial S$ is the boundary of surface $S$. This can be interpreted as an "induced EMF" term, the flux of the time derivative of $\vec{B}$, contributing to the potential around a given loop, $C$. Thus, when the induced EMF is accounted for, the net potential around a given loop is zero. Yet, the first term, the closed line integral of $\vec{E}$, is still ambiguous.
We can provide the circuit-analysis technique with the following definitions. We assume that radiation may be neglected, and that components are lumped. Quasi-static conditions are assumed. We rewrite the Faraday-Maxwell law as a sum of $n+1$ discrete paths which are piece-wise continuous, such that
$$ \sum_{k=0}^n C_k = C$$
Where $C$ is defined as above. From the definition of $\phi$, we substitute the first term in Faraday's law, using
$$ \sum_{k=0}^n \oint_{C_k} \vec{E} \cdot d\vec{l} = \sum_{k=0}^n \phi_k $$
And we replace the symbol $\phi$ by $V$ to put the sum in circuit theory symbols…
$$ \sum_{k=0}^n V_k $$
Faraday's law then becomes
$$ \int_S \frac{\partial \vec{B}}{\partial t} \cdot d\vec{s} + \sum_{k=0}^n V_k = 0 $$
We now make the substitution
$$ V_i = \int_S \frac{\partial \vec{B}}{\partial t} \cdot d\vec{s} $$
Where $V_i$ is called the induced EMF for its ability to induce current in a conductor (hence being an electromotive force). This yields
$$ V_i + \sum_{k=0}^n V_k = 0 $$
Which is Kirchhoff's voltage law as seen in circuit theory. This definition appears to hinge on $V_k$ being well-defined, but in fact it does not require a path-independent $V_k$, simply one that is defined along the circuit in question, as we will see.
To prove his point, Dr. Lewin places a loop of wire with two resistors in series around a solenoid, and switches the solenoid on, while probing the same position with two oscilloscope probes. The two probes detect different voltages when the switch is thrown, and because the same point in a circuit shouldn't measure two different voltages, KVL has failed, and the potential is shown to be path-dependent.
This is, however, an incorrect conclusion. In general, the integration path for the Faraday-Maxwell law is arbitrary, but the circuit path is fixed. The only ambiguity in the potential around the loop is due to the nature of induced EMF, which depends on the area of the loop. If the integration path is perturbed, the potential between two points will change, but the induced EMF term changes as well, directly opposite. By accounting specifically for the path taken by the circuit (from a macroscopic perspective, e.g. with one-dimensional wires and resistors), the potential difference around the loop is still zero.
The probing of the circuit, as ElectroBOOM pointed out, is responsible for the perceived failure of KVL. While Dr. Lewin correctly demonstrated the path-dependence of electric potential, he neglected the straight-forward behavior of this dependence. By careful choice of integration path, including probing loops, there is no discrepancy in the total potential measured around a loop.
While it is possible to consider $\vec{E}$ in terms of a scalar and vector potential,
$$ \vec{E} = -\nabla \Phi – \frac{\partial \vec{A}}{\partial t} $$
And to observe that the conservative and rotational components of $\vec{E}$ allow direct application of KVL to the scalar potential, this is unnecessary, and further, $\Phi$ is only an accurate measure of the electric potential in the electrostatic case. So while using potentials is mathematically convenient for general classical electrodynamics, they are not required to resolve path dependence of $\phi$, and the failure of $\vec{E} = \nabla \phi$.
While Kirchhoff's voltage law is not an electrodynamic law, it is a circuit theory law, and it is upheld in the example which Lewin provides, contrary to his results. The definition of KVL given above is necessary to analyze circuits including such as elements as transformers, below the limit where transmission line analysis techniques are in common use (typically when component lengths are $>>\lambda$ for a signal propagating in the circuit).
Have I missed anything in my derivation, or misinterpreted a result? I've been mulling this over for a few days now, and I originally defended Dr. Lewin's position, however the logic against him appears quite sound.
Best Answer
This is not the usual form of KVL that we use in circuit theory. For example,
These are all essentially just the $\sum_k V_k$ term from your statement of the law.
By including the $V_i$ term to account for changing magnetic flux encircled by the circuit, you have indeed formed a version of the law that accounts for changing magnetic flux encircled by the circuit. But that doesn't mean that the most common forms of KVL account for it.
This isn't really true. Usually we simply model transformers or inductors as devices that directly affect only the voltages between the nodes they're directly connected to. For example we model an ideal inductor connected between nodes a and b with the relation
$$ V_{ab} = L\frac{{\rm d}I}{{\rm d}t}.$$
With this relation the inductor voltage is simply one of the $V_k$ in our KVL equation, and we don't need to introduce $V_i$.
What we can't model without the $V_i$ term, and don't normally model under KVL, is the effect of flux through the loops enclosed by the circuit elements and the wires that connect them.
For example, in this simple circuit
the usual KVL can't account for EMF produced by changing magnetic flux in the loop formed by the switch, cell, and bulb and the wires connecting them.
(image source here)