It looks like you are trying to say you can define emf only if you can identify an area to associate to it.

Well, that's progress. That's what Faraday has tried to tell you since the nineteenth century.

**Alright i have to apologies because i did get one detail about voltage in an open loop wrong**. Straight wires require a special case to produce no EMF in the field (I will go back and add a note about this in my previous posts)

I finally found an article that explains how voltages in stationary open loops work when exposed to a varying uniform field:

Induced voltage in an open wire by K. Morawetz, M. Gilbert, A. Trupp

www.eevblog.com/forum/chat/does-kirchhoffs-law-hold-disagreeing-with-a-master/?action=dlattach;attach=581270It is indeed solved by closing the loop using a wire flowing a path that generates no EMF. However such a wire does not directly connect the ends of the open loop wire segment. What is required instead is two straight wires that travel to the geometric origin of the uniform magnetic field. Such straight wires that pass trough or touch the geometric origin point of the field generate no EMF. This was causing problems for me because i did not realize that even ideal uniform fields have a center point. However in the case of a close loop coil the center point becomes irrelevant because it affects the entire loop and averages out by the time you get all the way around. So technically the beloved closed loop is just a special case of a open loop that closes in on itself(Or the open loop being a special case of a closed loop, whatever way around you want to think about it).

So there you do get voltage induced in a open loop of wire (Proven using Maxwell even).

Correct, but not as weird as you think.

I suggest to brush up your physics on some good book, like Purcell (Electricity and Magnetism, second volume of the Berkeley physics series), and then look up the practical applications of the basic concepts in books like Ramo, Whinnery, VanDuzer (Fields and Waves in Communication Electronics).

I do have at least some understanding on all of those areas, just that some i don't know well in to the detail and especially not down into the deep math behind it. Its just not something i deal with on a regular basis. Electronics engineering has so many abstractions in place that there is no need to delve this deep into the fundamental math under it all. Hence why most electronics engineers know about Maxwell and what he did, but they never used his equations on the job. Its just easier and faster to use the derived "easy bake" equations for calculating everything you would need, but if you dig down and dissect a lot of those equations you tend to find some Maxwells equation somewhere in there.

Contrary to popular belief engineers are mostly not math geniuses, they are just really good at looking up the right equation and quickly punching it into a calculator. Its simply the fastest way to get work done on a deadline.

As for the further goalpost shifting at the end of your post, please... Leave caps out - we are trying to keep things simple here. If we are having trouble understanding each other with a simple circuit like that, what do you think would happen if you introduce another paradox generating element, like the two caps back to back?

The capacitor in an inductive loop is just my proposed experiment to show that an open loop can generate a voltage. Circuit analysis is well understood for RLC circuits so we can easily use it to predict the behavior, then actually do the experiment and see if the results match. Capacitors don't create any more of a paradox than inductors. Can you propose a simpler experiment that shows or disproves the presence of voltage in open loops of wire (aka fractional turns)? If the experiment can be done with equipment and materials found in a reasonable electronics lab i will recreate it.

The purpose of this experiment was to show that fractional turns can indeed take part in a circuit and pick up EMF just like complete loops can.

And no, KVL did not work with Lewin's circuit. You had to introduce that magical emf term to make your numbers check. That's Faraday at work. In fact, you cannot locate that voltage anywhere with a voltmeter, can you? I am talking about that circuit, do not try to modify it. Let down those scissors, I tell you!!!

Its not introducing a magical emf out of nowhere.

The cirucit mesh model simply needs to know about the properties of a wire. You do agree that a coil of wire with 100 turns placed across two points in a circuit is modeled using an inductor symbol right? Well these inductors are not closed loops as they have two terminals that connect to other components of a circuit (just like a straight wire).

A straight piece of wire is basically the same thing except with much less inductance since magnetic flux is not being reused multiple times on the same wire by coiling. This website provides a helpful calculator for this:

https://www.eeweb.com/tools/wire-inductance . Among other things it also provides a calculator for loop inductance that comes useful later(mutual inductance)

If you place two such 100 turn coils in close proximity you can get some of the same magnetic field passing trough both coils. This turns them into coupled inductors where they not only have self inductance, but also something called a mutual inductance (This is essentially a transformer). The value of self and mutual inductance for each is all that is needed to describe the magnetic properties of them. Any number of coils can be added to this magnetically coupled inductor, not just two. The coupled inductor model is the "mathematical adapter" that brings Maxwells equations into a form that fits into circuit analysis theory. Once it fits inside the circuit analysis abstraction all other circuit analysis tools can be applied(Kirchhoff being only one of them). Its sort of like a software API, but with math rather than code. By putting an inductor into the mesh we simply create an instance of the inductor model that deals with magnetic effects for us.

So since a straight piece of wire is simply a inductor with less turns than a coil of wire we can model a piece of wire in the exact same way. Tho due to it having essentially zero turns means the self inductance is pretty low(but NOT zero) while the mutual inductance is likely significantly larger as soon as it forms a larger loop with other components of the circuit that it connects to. This mutual inductance is where the loop inductance is if you connect multiple segments of wire together into a complete loop.

We are still using Maxwells equations and Faradays law deep inside the equation that calculated the inductance value in Henrys. So why is modeling a length of wire as an inductor incorrect? Is using Maxwells equations as part of another equation forbidden?

EDIT: Repetitia juvant.

What happens if we pull the resistors out of the loop and make sure we cannot interfere with the flux that is generating the emf? That we have a series of two resistors and a black box with two terminals. Now you can call that the secondary of a hidden transformer. Now you can located the voltage it 'generates' with a voltmeter. It's right there, at its two terminals! Now you can delude yourself KVL works, and call it, instead of Faraday's Law, "extended KVL" or "modified KVL" or "modern KVL". Lumped circuit theory works, all voltages we can measure are uniquely defined. Now the quarrel "KVL vs Faraday" is just a language barrier.

But when the resistors are inside the loop, say goodbye to lumped circuit theory and uniquely defined voltages. You have to take paths into account.

Yes that is correct, see its not that hard to think in terms of circuit mesh models. You can indeed fix things by adding a black box transformer into the circuit. However Dr. Lewins experiment is about the voltage on points A and B. Once we lump all of the loop inductance into one black box we loose points A and B.

But wait! We can fix that. Instead of lumping all of it into one black box we can just lump each wire segment into its own black box. This way we get 4 such black boxes that are located between the resistors terminals and the points of interest.

This way points A and B are maintained while each black box is now one of the 4 secondary coils to the solenoid coil creating the original field. They all know about each other trough mutual inductance. If we suddenly also want a point C that's halfway between A and the left resistor we simply cut the black box into two blackboxes with half the inductance value. Point C now pops out as the midpoint between the two new blackboxes. This is why i was trying to explain above that open wire segments can interact with magnetic fields just as much as wire loops can, the black boxes are models of a open wire segment.

As i said KVL is not just a different form of Faradays law. Its is like comparing an apple to a car, they are two completely different things. Faradays law calculates the relationship between voltage and magnetic flux change in loops. Kirchhoffs voltage law just calculates voltage relationships in abstract electrical circuit meshes. It has nothing to do with magnetic or electric fields. Its just a law that is part of circuit analysis methods, those will then call upon Faradays law whenever circuits have to deal with inductance. Circuit analysis wraps Faradays law into the form of an inductor. The process of mesh analysis eventually marries together the equation for an inductor model (That contains Faradays law deep inside it) with Kirchhoffs voltage and current laws to produce a mathematical model of the circuit. Both KVL and Faradays law exists together in that resulting circuit equation, you can't see it in the form that Faradays law is written but if you expand the equations backwards far enough you could eventually get it to pop out Faradays law in textbook form. KVL can't deal with anything other than voltages so it relies on other laws to do it instead. It can't even understand resistance, it needs help from Ohms law to tell it the voltage on a resistor. Because of that we don't say KVL is for the birds because Ohms law handles it better.

So the only thing that Faradays law and KVL have in common is that during circuit analysis KVL makes use of Faradays law to be able to understand magnetic fields. They are great friends even if circuit analysis sometimes demands it to do unusual things such as fractional loop segments(That still work fine, see above). Is it forbidden to plug the results from one law into another law?

Circuit models are not meant to describe the underlying physics, they are about as real as the imaginary part of complex numbers is real. But circuit models use just enough physics to accurately describe the behavior of the circuit on a macroscopic level.