Is voltage the integral of electric field or energy per unit charge? See this video on the debate about the definition of Kirchoff's voltage law. See around 8:15. The difference in views seems to hinge on the definition of voltage.
In one case voltage is defined as
$$
V^1_{ab} = \int_{\Gamma_{ab}} \boldsymbol{E}\cdot d\boldsymbol{l}
$$
This definition is only independent of that path $\Gamma_{ab}$ if $\boldsymbol{E}$ is conservative implying $d\boldsymbol{B}/dt=0$ everywhere. If $\boldsymbol{E}$ is not conservative then it is generally said that voltage $V^1$ is undefined.
In another case voltages is defined as
$$
V^2_{ab} = \frac{1}{q}\int_{\Gamma_{ab}} \boldsymbol{F}\cdot d\boldsymbol{l}
$$
In this case the force $\boldsymbol{F}$ may be supplied by elements other than the electric field $\boldsymbol{E}$. In this case the voltage may again be path dependent, but for the case of circuits, this doesn't seem to be a big problem since there is only one path that is important, that of the circuit itself.
Some questions about this state of affairs:

*

*Regarding the second definition: Under this definition I believe it is argued that an inductive element applies a force to the charges in a circuit. I suppose I believe this, but it is hard for me to see it from the equations. Faraday's law gives a relationship between electric and magnetic fields. If the magnetic field in an inductor puts a force on the charges in a loop it is going to be complicated mathematically because there can be no net force on charges in a conductor (the charges redistribute in a negative feedback loop to prevent charge build-up and net forces). If this is the case could someone clarify the form of the force on charges $\boldsymbol{F}$ that arises on charges due to the presence of an inductor and Faraday's law?

*Which definition is preferred by which groups? For example do physicists prefer one while audio engineers prefer another and microwave engineers prefer another?

*Could you please provide lists of references which use each of these definitions for comparison?

 A: I only scanned the video to see the background of this debate but I was confronted with this question several times I try to give an answer:
The voltage across a loop is zero. Voltage is something you can measure with a voltage meter. But what is a loop? The word "loop" is just not defined as a physical object.
And that is where the confusion comes from.
Let there be an electrical field: we define: an electrical charge, e.g. an electron placed on the tip of an imaginary rod, is somehow moved along a path in space and for every point along this path we measure the force needed to move the electron, multiply it by the incremental path element and integrate this energy. As the "field" defined this way is conservative the integral amount of energy is zero (or vice versa).
So the loop is just the path followed by the rod.
Now we get material. If we realize the path using an (ideal) conductor every voltage difference along the path will immedeately lead to a current flow of infinite strength (pure electrical case) and so no voltage difference is possible. That said: the voltage along the wire is zero.
Now if you cut the wire in a place there still will be no voltage across this cut. Until you change the magnetic flux (what ever it is) "through" the loop. Then you will be able to measure a voltage.
The point is: if you close the gap and still change flux, where is the voltage?
The answer is quite simple: you just can not change the flux through an loop formed by an ideal conductor! That is the trap you are caught in.
