The Romer-Lewin ring with resistors: two voltmeters read different voltages
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- Опубліковано 22 сер 2024
- This is an example of UNLUMPABLE circuit - a Romer-Lewin ring linking the variable magnetic flux of a toroidal transformer. Now with voltmeters!
The measured emf is a little more than 370 mV, slightly less than in the previous video. The discrepancy can be attributed to momentary variation in mains voltage or in the different kind of probes used.
One of the voltmeters, the UT139C is not the ideal choice for measuring such low voltages at low currents, and just getting near its probes would make the reading fluctuate wildly. As a matter of fact, it could not manage measuring the voltage when both voltmeters ran around the core (with roughly 20 megs of total resistance, the current was probably too small to give an accurate measure.)
Theoretical values with a 370 mV emf would be:
Iring = 370 mV / 17.2 ohm = 21.5 mA
Vrl = 2.2 ohm * 21.5 mA = 47.3 mV
Vrh = 15 ohm * 21.5 mA = 322.5 mV
measured values with low-cost equipment were:
Iring = 20 mA
Vrl = 47.8 mV
Vrh = 314 - 320 mV
A more detailed description, with timestamps, has been added as a pinned comment.
What follows is a list of references that explicitly deal with multivaluedness of voltage and even with the Romer-Lewin ring itself.
Textbooks
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Purcell & Morin
Electricity and Magnetism
Berkeley Physics volume II, 3rd edition
2013, Cambridge University Press
For certifying the death of KVL and the nonuniqueness of voltage (p.359)
it also features the ring with two resistors as a solved exercise (p. 710).
Zoya Popovic, Branko D. Popovic
Introductory Electromagnetics
1999, Prentice Hall
a gentle introduction to undergrad EM; for the decomposition of voltage, see sec. 14.4 "Potential difference and voltage in a time-varying electric and magnetic field" (yes, he uses the same symbol V for the scalar potential, but he makes the decomposition of the actual voltage explicit.)
J. A. Brandão Faria
Electromagnetic Foundations of Electrical Engineering
2008, Wiley
lots of interesting examples, but mentioned here because it's another textbook showing the decomposition of voltage and the nonuniqueness of voltage. It uses the 'German' notation for the arrows that represent voltage across a component - some might prefer it.
Markus Zahn
Electromagnetic Field Theory: A Problem Solving Approach
1979 Wiley - 2003 Krieger Publishing Company
a good intro textbook with many examples, including the ring with two resistors - predates Romer
The full textbook is freely downloadable from the MIT OCW website
Herman A. Haus, James R. Melcher
Electromagnetic Fields and Energy
1989, Prentice Hall
for nonuniqueness of voltage and the role of surface charge in multitap coils (how multitap transformers and autotransformers work).
This text, too, is freely available on the MIT OCW website
Ramo, Whinnery, VanDuzer
Fields and Waves in Communication Electronics
3rd ed. 1994, Wiley
an engineer's point of view on the multivaluedness of voltage and when it is still possible to pretend it's single-valued, like a potential difference. The authors stress the fact that voltage ALONG the coil is zero, while voltage ACROSS the coil is not.
José Roberto Cardoso
Electromagnetism through the Finite Element Method
A Simplified Approach Using Maxwell's Equations
2018, CRC Press
You can read about the multivaluedness of voltage - even applied to the Romer-Lewin ring on p. 54
Papers
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D. R. Moorcroft
Faraday's Law, potential and voltage: discussion of a teaser
American Journal of Physics 38, 376 - March 1970
References a previous 1968 note of the same author proposing the (Moorcroft-Purcell-Zahn-whoknowshowmanyothers)-Romer-Lewin ring as a teaser. This paper also highlights the need to distinguish between scalar potential and path integral of the total electric field, suggesting to call the latter 'voltage'.
Robert H. Romer
What do voltmeters measure? Faraday's law in multiply connected regions
American Journal of Physics vol 50, no 12, December 1982
It shows the ring with two resistors and what a voltmeter actually measures,
a very often cited paper
J Roche
Explaining electromagnetic induction: a critical re-examination. The clinical value of history in physics.
Physics Education, Volume 22, Number 2 - IOP Publishing Ltd
This paper considers all components of voltage: from mutual or self induction, from resistive losses and, obviously, from the coulombian field of the charges (what he calls the 'masking electrostatic potential difference')
John Belcher
Massachussets Institute of Technology lecture supplement
(updated by Walter Lewin, April 2002)
Considers - among other things - the case of a uniform resistive ring in a changing magnetic field, and a ring made of two half-circles of different resistive materials.
Sorry I have run out of space...
00:00 *Introducing the ring*
I built it by soldering two naked strands of copper to a 2.2 ohm resistor on one side, and a 15 ohm resistor on the other side. I added a few soldering blobs here and there to prevent the strand from undoing itself. I left one arc longer than the other, then I cut it so that I could open the ring and then tie it around a transformer core.
00:28 *The toroidal transformer*
Here I am measuring the emf due to the magnetic flux linked by one turn around the core of a toroidal transformer.
It came out as 372 mV (all measurements are RMS, and I am ignoring the error in the voltmeter reading, for simplicity)
Note that the shape of the secondary coil - realized with the probes and closed on the internal resistance of the voltmeter - is irrelevant because all of the flux is confined inside the core.
The induced electric field, which runs in circle around the core, behaves as a conservative field in all regions of space that do not contain the core. But does not behave as a conservative field when we go around the core.
00:53 *The open ring around the core*
Here I placed the ring around the core, but I left it open and placed the voltmeter across the gap. We are basically closing the loop with the internal resistance of the voltmeter. Let's say its 10 megaohm (it's actually higher when the dial in on the millivolt range for this particular multimeter): the full resistance of the loop becomes a little more than 10 megaohm. The current flowing in the ring with the added voltmeter is
272 mV / 10 megaohm = 27.2 nA
What are the voltage drops on the resistors, in this setting?
2.2 ohm * 27.2 nA = 59.84 nV
15 ohm * 27.2 nA = 408 nV
Since less than half a millivolt drops along the resistors, the entire emf is basically appearing in the form of voltage drop across the internal resistance of the voltmeter.
01:16 *Closing the ring around the core and using a voltmeter to probe an arc of conductor (nearly 2 inches, or about 5 cm.)*
When we close the ring around the core, a much bigger current will flow in the ring than when the 10 meg internal voltmeter impedance was part of it.
Iring = 372 mV / (2.2 + 15) ohm = 21.6 mA
The computed value agrees with what I measured with a clamp on amperometer at 01:46 . The UT136 gives a reading of 20 mA - close enough to the theoretical value, considering the accuracy of the instrument.
This current flowing in 5 cm of copper will drop a voltage of the order of a microvolt, and in fact the voltmeter reads nearly 0V.
Now, this is a crucial point in understanding how to correctly apply KVL or Faraday in this circuit: the loop formed by the voltmeter, its two probes and the 5cm arc of conductour is sitting in a region of space that does not link any variable magnetic flux from the transfomer (remember, all the flux is contained in the core and this measurement loop does not run around the core). So *we can apply KVL to THIS loop* : we can consider voltage as path independent in this loop and avoid to refer it to a particular path, by specifying the endpoints only. The voltage along the copper arc between the tips is equal to the voltage along the path going through the probes and the internal resistance of the voltmeter and shown by the instrument.
Compare this to the case of open ring: now that the gap has been replaced by 5cm of copper, the voltmeter is no longer able to read the open circuit voltage.
Path independance *does not apply to the ring itself* , though, because the ring forms a loop that does run around the magnetic core.
02:05 *And now... the full fledged ring with both voltmeters*
Here I placed the probes of both multimeters at the same points on the ring (more or less). Now we can identify six loops of four kinds: the ring itself (running around the core), the external loop formed by both voltmeters and their probes (running around the core), the two measurement loops formed by each voltmeter with the 'distant' resistor (still running around the core), and finally the two measurement loops formed by each voltmeter with the 'near' resistor. Only these last two loops do not run around the core, therefore they do not link any (appreciable) variable magnetic flux - and certainly the do not link the variable magnetic flux that is confined in the toroidal core.
It is these two measurement loops (voltmeter UT61E and 2.2 ohm resistor on one side, and voltmeter UT139C and 15 ohm resistor on the other) that we can apply KVL to. This means that the voltage across the internal resistance of each voltmeter is equal to the voltage across (and along) the 'nearest' resistor.
And, lo and behold, the UT61E reads 48 mV - what we expect by applying Ohm's law to a 2.2 ohm resistor with a 21.6 mA flowing through it. The other voltmeter, the UT139C reads something between 314 and 329 mV - accuracy is significant less in this instrument - still in line we what we expect when 21.6 mA flow through a 15 ohms resistor.
KVL works fine in the outer loops, the ones that are not linking the magnetic flux, but *it is no longer valid in the ring itself* .
The ring runs around a variable magnetic flux and therefore the circulation of the (total) electric field is no longer zero. This means that voltage (which is defined as minus the path integral of the total electric field) in the ring depends on the particular path between two points. And sure enough, voltage along the path that goes through the 2.2 ohm resistor (and also along many other paths in the nearby region) is 48 mV, while voltage along a path joining the very same endpoints but going through the 15 ohms resistor is 320some mV. (There is also an inversion of phase, as can be seen if we use an oscilloscope instead of voltmeters like I did in my other potatorama video.)
As Lewins says: KVL is for the birds. And it is for the birds in the ring itself, and in all measurement loops that happen to run around the core.
We can still apply Faraday, tho.
Hello C F, you stated incorrectly that Kirchhoff loop rule can be applied to the meter and adjacent resistor. It cannot , because two conductors in parallel do not form a loop. The rule for them is not Kirchhoff it is the Ohms law for conductors in parallel .
@@woodcoast5026 two resistors in parallel form a loop. The rule for parallel resistors comes from applying KVL to such a loop. The voltage across either resistor is the same exactly because of KVL.
@@copernicofelinis You just stated "two resistors in parallel form a loop" that is an absurd statement. When two resistors are in parallel the current runs from one end of the split path to the other end of the split path. KVL applies to conductors in SERIES. The meter and resistor are not in series they are in PARALLEL.
@@woodcoast5026 a loop is any closed path in a circuit. See here: www.electrical4u.com/nodes-branches-and-loops-of-a-circuit/
@@copernicofelinis It is not a loop, it is not a closed path. It goes through the meter and resistor in parallel , and then continues around the ring.
Some people mistakenly think there is a circulating current going around and around a loop of the internal resistance of the voltmeter and the resistance of the adjacent resistor.
You could DISPROVE this by a demonstration, replacing R1 with two 4.4 Ohm resistors side by side and replacing R2 with two 30 Ohm resistors side by side. Now with that arrangement with 6 resistances the circuit behaves as before and on one side of the circuit there are three resistances, and the concept of the current circulating around the two resistors and then going on through the internal resistance of the voltmeter is impossible.
At 1:40, the voltage across the wire segment is 0V, same as my measurement.
Yes, in fact the multimeter shows practically zero. The 0.0004 volt is nearly zero considering the error of measurement and the fact that a couple of inches of copper wire with 20 mA in it will drop a handful of microvolts. I will add some numbers in the description.
I wanted to add titles and drawings, but I was not able to install OpenShot (cx_freeze error) so I just stitched the videos together.
For the Romer 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 Romer Lewin circuit, Faraday's law dictates that the current always flows in one direction around the flux and so therefore there is a meter/resistor pair in series with another meter/resistor pair, and as the resistor of a pair is in parallel with the meter in that pair the resistor defines the value of the electric field lying along the path of the resistance of the voltmeter in that pair.
@Biskwit2
Current going clockwise.
R Right, the current goes into the top of R Right and out of the bottom of R Right. E field high at top of R Right.
R Left , the current goes into the bottom of R Left and out of the top of R Left. E field High at bottom of R Left.
@Biskwit2 Yes, current flows along an E field from High to Low. There are two fields , one for R Left, and one for R Right.
@Biskwit2 With that arrangement there is no E field along the copper. There is only an E field along R Right.
You need to consider the total electric field, which is a superposition of the induced electric field (going around the loop) and the coloumbian electric field, which has sources and sinks in corresponendence of the boundaries between resistors and conductor.
This total field is nearly zero in the conductor, then becomes suddenly strong inside the small resistor, returns to nearly zero inside the conductor and then suddenly much much stronger in the bigger resistor. Always going around the ring in the same direction.
I wanted to add a picture but I couldn't install the video editing program. I might add another video with just the pictures.
Biskwit2,
Copernico Felinis comment directly above is useful.
But more importantly It is not clear what you mean by " voltage without induced EMF" that is impossible. The current is produced by the EMF, the voltage for the resistor is produced by the the current and the resistance. Ohms Law V=I*R. The Electric field from the solenoid exists in free space, not in the wire.
Hi Copernico
Is the field in the vicinity of your transformer mainly electric or is it a mix of magnetic and electric.
Ideally all the magnetic field is inside the toroidal core. The physical circuit (resistors and copper wire) can occupy a region of space where there is no magnetic field at all.
(In practice if a current is flowing due to the induced electric field, this current will generate a magnetic field around it, but in this case it is so small that we can neglect it - and in doing so we also neglect the self inductance of the ring)
Maybe a correction at 45 seconds. The internal resistance of the meter is not low. It is very high.
That's my broken English. I tried to say: "...and the internal resistance of the meter is the load". It came out as unintelligible maybe, but iirc the automatic translation got it right, despite my pronunciation. (Not so much "portion", which apparently I pronounced "Persian")
@@copernicofelinis yes I hear it . Load