The Romer-Lewin ring: an unlumpable circuit

This is a homemade demo of the Romer-Lewin ring in action.
A toroidal transformer creates an EMF of 380 mV (rms). A Ring with two resistors, 2.2 ohms and 15 ohms goes around the core. Probes are placed across the very same two points midway between the resistors. The measured voltages are clearly different: they are in opposition of phase and their amplitudes are in the same ratio as the resistors.
Computed voltages (excluding the mV drops in the highly conductive copper arcs) with nominal resistor values are 48.6 mV and 331.5 mV; measured values with a true RMS voltmeter are 48 mV and 332 mV (with less than a mV dropped along any copper arc); values shown on oscilloscope are 332 - 333 mV and 49.8 - 50.4 mV.
The position of the voltmeter or oscilloscope probes (including the loop formed by the ground clip) - as long as they do not go around the core - is irrelevant.
This is NOT a probing issue.
When we compute the total electric field Etot = Ec + Eind inside and around the ring, we see that the voltmeters are actually measuring the voltages along the branch of the circuit 'near to them'. (With that I mean the branch of the circuit that along with the probes and the voltmeter does not encloses the variable magnetic flux region.)
Voltages (i.e. path integrals of the total electric field Etot) computed along the paths that follow the left and the right branch of the circuit have to be different.
This being a video shot in potatorama, it can be reduced to three stills that happen by chance to be in focus:
1:43 the note on paper
1:52 the actual ring around the core
1:34. the scope screen
A detailed analysis of the circuit can be found at the following links:
electronics.st...
electronics.st...
An extensive list of references can be found in the description of my other videos.
#faradayslaw #kirchhoff #lewin

[@woodcoast5026]

For the Lewin Circuit. Here is a concise explanation for why the voltmeter readings are in accordance with the resistor values.
Voltmeters display the value of the electric field that lies along the path of their internal resistance. For the Lewin circuit, Faraday's law dictates that the current always flows in one direction around the flux and so dictates which resistor defines the value of the electric field lying along the path of the resistance of the voltmeter.

[@woodcoast5026]

Hi Copernico Felinis,
Faraday's law mandates that a current that is the product of induction can only flow around a closed loop that is spacially defined such that it encircles the flux that generated it, and in one direction only. In accordance with that law the voltmeters and their wires are parts of different current loops, depending on how they are physically arranged and so different currents flow through the meters. Do you concur.

[@copernicofelinis]

I would say that each voltmeter and its probes are part of multiple loops. The current in the voltmeter is the result of the composition of the effects (not the currents) of all loops.
For example, in Lewin's ring, voltmeter A and its probes form three loops: one with R1, one with R2, and one with the other voltmeter. If you consider these loops alone, *deleting the other components* , you get three different currents (namely 0, emf/(R2+Rvm1), emf/(Rvm1+Rvm2) ) which are different from the actual current flowing in the voltmeter when all elements are present (it's a rather complicated function, but we can approximate it with emf/Rvm1 R1/(R1+R2) ).
Of course we can look at the measurement loops *with the other elements present* . What we can say in that case is that in the loop with R1 - which does not contain variable flux - we can apply KVL, so the voltmeter will measure the voltage along the branch with R1 correctly (voltage is the same for the path going along the branch and for all paths in the space across the terminals, as long as I stay in the magnetic-free region), while the other loops will produce measurements of the other branches that are altered by the linked emf. The nice part is that these measurements will be altered so as to produce the same voltage associated with the first loop.
I've read many times the objection "but the other loops go around the changing flux, so you will induce a current in the probes!!!"
The point is that the voltmeter (as well as its probes) will see almost none of that current because it is shunted by the nearest resistor.

[@woodcoast5026]

@@copernicofelinis Hi Copernico. I analysed the circuit again. I worked it out and posted my new comment on the subject in the comment section for this video.

[@yelloweater5506]

Unfortunately I don’t have an oscilloscope and also I think I can’t afford it

[@copernicofelinis]

You can use a voltmeter to measure the AC voltages, like I showed in my other video.

[@ferferfferf6644]

Hello, I come from electroboom and Dr Lewin videos. I saw you had a very informed opinion about the topic and I would like to read in depth what you have to say. Sadly, it's very uncomfortable (basically impossible) to read all the comments you replied to in their videos so I would appreciate it if you could synthesize it here for me and those who might come to your channel under similar circumstances.

[@copernicofelinis]

Yeah, I could write a book by putting all those comments together. :-]
What I know comes from electromagnetism books. Three books in particular have this problem clearly explained. Here are the relevant quotes (from one of my comments in another channel's video, slightly edited to remove non-relevant parts):
The first book is *Purcell and Morin* , "Berkeley Physics vol 2: Electricity and Magnetism"
Section 7.5
"A consequence of Faraday's law of induction is that Kirchhoff's loop rule (...) is no longer valid in situations where there is a changing magnetic field. Faraday's law has taken us beyond the comfortable realm of conservative electric fields. The voltage difference between two points now depends on the path between them."
And then it adds:
"problem 7.4 provides an instructive example of this fact."
And what is problem 7.4? Lewin's ring, around a toroidal core like the one depicted in this video. The problem is solved in the end of the book. And concurs with Lewin, of course.
Let's move to MIT with *Haus and Melcher* , "Electromagnetic Field and Energy".
This book is entirely online at web DOT mit DOT edu SLASH 6.013_book SLASH www SLASH (I hate you, youtube!)
The chapter on magnetoquasistatics opens with Lewin's ring around a toroidal core. There is even a video demonstration (Demo 10.0.1: Nonuniqueness of Voltage in an MQS System). Again, in line with what Lewin says.
The third book has a more applicative slant: *Ramo, Whinnery and vanDuzer* , "Fields and Waves in Communication Electronics".
This book explain clearly why voltage has to be multivalued and when and how we can treat it as single valued to simplify our treatment of circuits. It lliterally takes you by hand expanding the integrals and showing you precisely where the L di/dt formula come from. In particular at page 174 of the third edition it says:
"let us take a closed integral of electric field along the conductor of the coil, returning by the path across the terminals. Since the contribution along the part of the path which follows the conductor is zero, all the voltage appears ACROSS the terminals."
Moreover, on page 179, it explains how in lumped circuit theory we pretend to treat voltage as if it were a potential difference:
"In the above we seem to be treating voltage as potential difference when we take voltage of a node with respect to the chosen reference, but note that this is only after the circuit [path] is defined and we are only breaking up the integral of E.dl into its contributions over the various branches. As illustrated in the preceding section, we do have to define the path carefully whenever there are inductances or other elements with contribution to voltage from Faraday's law."
Finally, to better understand the difference between voltage and potential difference and how the two concepts are linked in the presence of a variable magnetic field, I would suggest *Popovic and Popovic* , "Introductory Electromagnetics", sec. 14.4 "Potential difference and voltage in a time-varying electric and magnetic field". You can find an extended quote searching "Assigning a notion of voltage even when there is a changing magnetic field" on Electrical Engineering Stack Exchange.