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The *potentials* are not unique, even for the same gauge. But under certain conditions of regularity at infinity, the Helmholtz decomposition of the *fields* in a simply connected region (like the whole of space) is unique. That is: the decomposition in Eirr and Esol is unique. There is a mathematical theorem that states that. (There are also extensions to finite and even not simply connected regions, but let's keep it simple...)

Thank you for your comment. The 'paper' (it's more like a letter) by Moorcroft I am going to discuss in my next video, and where he proposed to use the term voltage for the line integral of the total electric field and PD for the conservative component, dates back to 1969. If we haven't found common ground now, I doubt we ever will :-) . But in the light of the IEC and ISO definitions (which I discussed in another video) I believe there is some hope. I think the problem here is in the teaching stage, and I will show in another video later on why the definition that uses the total electric field makes more sense and, most importantly, does not lead to contradictions in certain situations. Note that this does not mean that the "PD + induced voltage" approach is inherently wrong. The two approaches are actually equivalent, but one has to acknowledge that, in the presence of variable magnetic fields, PD alone does not suffice to describe the system, and that the determination of how voltage decomposes into PD and induced voltage requires the knowledge of the sources (and I have two interesting examples to make). As a matter of fact, the uniqueness of the Helmholtz decomposition into Eirr and Esol is subordinated not only to a certain regularity of the fields at infinity, but also to the exact knowledge of the sources of the fields (charges and currents). Knowledge of Etot, on the other hand, is local and so you can tell what the voltage is along a path that is accessible to you without knowing the exact position and strength of the sources. I do agree that surface and interface charges are important, and as I anticipated in another comment (maybe to another video) I intend to cover this topic as well. There is a partial discussion in my video "Dispelling the Lewin KVL Paradox: the total electric field", and I also wrote an answer on Stack Exchange about it (I don't remember the title now, something along the lines of "the electric field in a wire is constant").

​@@copernicofelinis When I commented, I had only watched this one video. Now that I have watched your other videos, I see that you address my issues very nicely. This is excellent stuff! Are you thinking of making videos to discuss non-static circuits where the work per charge is integral of vxB instead of integral of E? Do you refer to that as "voltage"?

The next video will be about 'loop voltage', 'potential difference' and how to treat voltage in lumped circuit (what certain authors call 'terminal voltage'). The IEC chose not to define (or rather discourages the use of) the term 'electromotive force', but I still like to use it, with due care. Now, what the IEC call 'loop voltage' is what I would call "EMF of electromagnetic origin" and yes, in general it should include the motional term (and the IEC definition includes that). We can nonetheless see it as a 'specialization' of voltage when applied to closed loops: the conservative component gives zero contribution, and all that's left is the 'EMF'). So, yes I intend to do a video on the EMF and its relation with the voltage at the terminals, but I will still consider the stationary case to avoid confusion. I have then a few more videos on the measurement of voltage and an illustration of the fields that explains why a voltmeter measures what it measures, plus a video on why defining voltage as potential difference can give you problems, and then if I am faster than the grim reaper I intended to do videos on voltage in a motional context, voltage in transmission lines and maybe even one on voltage in radiating systems.

Feline minds think alike. 🐈😀

Did you mean at 9:45? Yeah, I agree they are concomitant in that they come out of the particular decomposition of the electromagnetic field in a given frame of reference. Sometimes concise expressions can be imprecise, as in this case. I also consider voltage and current to be two sides of the same coin but from time to time I find it more concise to say "the current flowing in the resistor R will produce a voltage..." or vice versa.

@@copernicofelinisYes I would say that the change of a magnetic field correlates to an induced voltage in a conductor perpendicular to the field, but I’m not sure if it’s [always] the cause. Say for example we have an ideal toroid transformer of very high permeability. We could expect some leakage flux at the primary winding with little to no leakage flux at the secondary. Yet we still have an induced voltage in the secondary proportional to the rate of change of B, even though no line of magnetic flux touches the secondary coil. Is this action at a distance? Or does it involve the magnetic vector potential, and if so does that correlate to it being a physical reality?

@@cosmicyoke I tend not to read cause and effect in any physical equation unless there is an explicit reference to time in it. I consider them to be just relationships between physical quantities. So I do not consider changing electric fields to cause magnetic fields or changing magnetic fields to cause electric fields (even if sometimes - actually often - I use colloquial imprecise language for brevity). For example, in the case of the toroid: there is essentially zero magnetic field outside if it, but there is an associated curling electric field all around it and it's that induced electric field that is responsible for the current flowing in a loop of copper around the core. We tend to see this as a chain of cause +effects ("the changing flux produces the induced field and this in turn produces a current") when in reality these are concomitant effects and not really one the cause of the other. Regarding the last part of your comment, in classical physics there are three ways to describe EM phenomena: via fields, via potentials and ultimately via the position, velocity and acceleration of charges. I like to see this last one to be more physical; fields are an expression of how charges are disposed and move, while potentials are a mathematical aid that lead to simplified equations. But all three ways are essentially equivalent. In quantum physics it seems that the potentials are the most relevant descriptor of reality, but I'm not getting into that black magic here :-)

Thank you. I remember reading your excellent comments on Electroboom's channel. I believe we shared some fight together, but at the time I used a different handle.

In quasi-static there is no propagation of potentials (or fields). So, there are no waves to which associate a wave vector. Can you be more specific about what you mean?

Fourier transform in space, the wavenumber is just the transform variable of position. I didn't mean to imply a Fourier transform in time.

Well, the last part was a bit too condensed since the full explanation of what is meant by quasi-static - and in particular electro-quasistatic and magneto-quasistatics would have lengthen the video to a total duration of about 45 minutes. And that would have been too much. I plan to make a dedicated video on quasistatics and the orders of approximation in the solution of Maxwell's equations, but that won't be anytime soon. I have three videos almost ready (IEC notation, what a voltmeter measures, how a voltmeter measures what it measures) before even considering that. So, don't be worried by the fact that it wasn't clear, because due to excessive summarization, it ended up not being clear. Maybe if I added Maxwell's equations for each case it would have been better - but I figured "these are just concluding observations..."

@@copernicofelinis Mate, you have done wonders for me. I was wondering about "EMF around a circle", and I fully got that now. I have seen a professor talk nonsense, and you made perfect sense. I really do appreciate your efforts. I trust you. Subscribed. I learn so much. Thank you!

I thought I had already added them in the video description, but I didn't. I did it a few minutes ago. Here's the list: Edward M. Purcell, David J. Morin Electricity and Magnetism 3e 2013, Cambridge University Press Branko D. Popovic Introductory Engineering Electromagnetics 1971, Addison Wesley Zoya Popovic, Branko D. Popovic Introductory Electromagnetics 1999, Prentice Hall Kenneth R. Demarest Engineering Electromagnetics 1998, Prentice Hall Markus Zahn Electromagnetic Field Theory: A Problem Solving Approach 1979 Wiley, 2003 Krieger Publishing Company Herman A. Haus, James R. Melcher Electromagnetic Fields and Energy 1989, Prentice Hall J. A. Brandão Faria Electromagnetic Foundations of Electrical Engineering 2008, Wiley David J. Griffiths, Introduction to Electrodynamics 3e 1999, Prentice Hall Wolfgang K. H. Panofsky, Melba Phillips Classical Electricity and Magnetism 2e 1962, Addison Wesley (reprinted 1983, Dover) Hans C. Ohanian Classical Electrodynamics 2e 2007, Infinity Science Press John D. Jackson Classical Electrodynamics 3e 1999, Wiley Guest Stars: John D. Kraus Electromagnetics 2e 1973, McGraw Hill William H. Hayt, John A. Buck Engineering Electromagnetics 4e 1981, McGraw Hill

@@copernicofelinis great thanks!

Thank you for your feedback. And for your interesting videos.

​@@copernicofelinis I watched all your videos on the torriod puzzle, all the time trying to rationalise what was happening. You seem to have the theory and the maths, but my take on the effect is more pragmatic:. If you omitted the resistors and kept the two digital volt meters, the meters and their leads would make one turn of the torriod. Each meter would comprise half of the turn, and assuming the DVMs had identical resistance there would be identical voltages across each. Introducing a 'shunt' resistor across one meter would cause that meter to read lower than it did originally as current is diverted away from it. The other meter could then be shunted with a different value resistor and its value would also be lower than the original. Perhaps a bit too pragmatic? But I just loved the way you made it a mystery! It really held my attention. Many many thanks to you. Richard.

@@richardlangner it is pragmatic, and it can be used to find the voltages, but there are some subtleties that need to be considered. One: you say "voltage across" the half turns, but since voltage is path dependent you should specify a path. If the path is along the half turn, then voltage along it is nearly zero. In fact, when you place the two voltmeters around the core, you end up with all the voltage to drop on the internal resistors - none has been left in the leads and probes. I found out that the best way to explain voltmeter readings is by introducing the concept of measurement loop, i.e. the loop formed by voltmeter, leads, and the branch of circuit you are interested in (there may be several, connected to the same two endpoints). If the measurement loop for branch X does not include any relevant changing magnetic field, then the voltmeter reads the correct branch voltage. If the measurement loop for branch Y (attached to the same points as branch X!) links some dB/dt, then you need to correct the reading by discounting the linked emf. I have already completed the scripts for two upcoming videos on "What a voltmeter reads" and "How a voltmeter measures what it measures", that will be published after the "IEC voltage" video. (I am temporarily stopped because I destroyed my good phone, so I have no device to record the voiceover with decent quality). The shunting effect you describe is correct, and I will address it in the circuital treatment of the Lewin ring (some of it is in one of my answers on EE Stack Exchange, but I can't give a link because yt will delete it).

Glad it was helpful!

Thanks!

The next video ("The IEC definition of voltage") will still be on the boring side, but the one after that ("What a voltmeter measures") will be really interesting with actual practical implications and little to no math.

Thanks for the feedback. Yes, all of this is pretty basic physics, and that is why I was surprised to witness such a fierce opposition to the idea that voltage (a path integral!) could be a path dependent quantity. I had a look at the English, French, Spanish, German, Italian pages of Wikipedia where voltage is defined, and so far only the German page has the correct, complete definition that agrees with the IEC definition of voltage (next, upcoming video).

The problem with the analogies required to reduce the concept to laymen's terms is that they always leave out some key aspect, or worse incur into contradiction. In my opinion the surefire way to understand voltage is from the definition I gave in my previous video (Voltage for grown-ups): it is work (energy, if you will) per unit charge, and this work is in general path dependent. Here is an analogy: the "gaseolage" is the amount of gasoline per unit passenger that a car requires to go from A to B. As you well know, this amount depends on the road you choose. And even if all roads from A to B had the same length, in a world where there is friction and air resistance, "gaseolage" like voltage will still depend on the particular path. Now, this analogy can be made more stringent by considering spring powered cars, running on smooth rails so that in the absence of air resistance "gaseolage" would reduce to height difference (or rather a difference in gravitational potential energy per unit passenger) - much like voltage would reduce to a difference in electric potential energy per unit charge. The path dependency comes into play when we introduce air vortexes (analogous to the circulating lines of the solenoidal electric field associated with changing magnetic fields) but... It's getting a bit complicated and I am not sure this explanation is simpler than the original one...

The potential difference is defined between any two points, it is simply the difference in the electric field strength at the two points. The voltage between two points, however, is non-sensical. You must define voltage along a path, the journey matters if you will. This is why there can be a voltage across a loop, even though it ends and starts at the same point. This all only comes into effect in the dynamic case (ie. changing electric fields), and in the static case this "correction" vanishes and potential difference equals voltage.

Yes, understood. Unless I continue on from Pre-calculus I may have to use Mathematica or some other math simulator to follow along. Your measurement at two different points on the same conductor was an interesting complement to your articulated points. I also found it helpful to dive into the transcript to tackle the vocabulary I lack practice with.

@@gary.richardson The electric field is a vector field, with an arrow at each point. This field (when no charge is moving) has the remarkable property that it is conservative. This means there is no "curl" in the field (a term perhaps to google as an image speaks a thousand words: vector field curl). This allows us (although not always, we get lucky with the e field) to create a potential function for the e field. We have to do integration and all sorts to find this function but simply put, there exists a function of (x,y,z) position such that it spits out a single number for each position. The gradient of this surface at a point gives us back the vector e field at that point. I'm sure there are very good videos on this, I think 3b1b has done great videos on vector calculus. The short of it is that as we can have this function that is uniquely defined at each point, it doesn't matter how we travel between points, the change in this function will just depend on the start and end point. This difference in potential is named as such: potential difference. Analogously we can think of a mountain or other landscape. When we climb we move to a spot with higher gravitational potential, and when we descend we move to a position with lower gravitation potential. As this potential is a function of the spot only, the change in potential does not matter on the journey, but just the start and end points. Unfortunately, in the case with moving change (hence magnetic fields) these potential analogies break down. This is as we can now get curl in the field (it is this curl that induces current in coils in changing magnetic fields and upon which all AC electronics depends) and it is no longer conservative. This means we cannot make a potential from it, and the whole landscape goes out the window. Instead we must define a voltage, which is path dependent. It is defined with the complex integral in the video, and equates to adding up the small bits of the field along the path. When you take this formula and set the change in magnetic field to zero, you get out a function that is identical in value to the values from the potential function; the voltage with no moving charge is the potential difference. I hope that helps give a broader picture of the maths, if you have any questions do ask. It helps me to better understand it when I explain it!

Actually, AC circuits are covered by the quasi-static approximation of Maxwell's equations (Look at the figure in the third note at the end: they are firmly under the quasi-static column). Translation: this video is about AC circuits.

Thank you for pointing out the new video by Trevor. It seems he, SiliconSoup and I have 'specialized' in covering different angles of the Lewin ring: Trevor is more focused on areas and geometry, SiliconSoup on charge density, while I prefer to look at fields and measurement loops (this last part will be clear when I'll upload the "What a voltmeter measures" video.)

Thank you for your kind words, 'armchair fellow' . (For everybody else: 'armchair fellow' is kind of an insider joke. SiliconSoup and a few others will understand it). 😊

Yeah, I get that a lot. 😭 Out of curiosity, do you think the audio level in my previous video "Dispelling the Lewin KVL paradox" to be acceptable? I was almost shouting, there.

@@copernicofelinis I have some ideas for you if you're looking to improve your audio! I'm guessing there's some stuff with your microphone, recording and finalizing process, etc. that can be changed to help the volume out a fair bit.

​@@no1unorightnowI'm open to suggestion. I have already tried 'fixing my audio' using win10's diagnostic tool but to no avail. I am now thinking that my mic got damaged when I left the headset in a room where I used ozone.

The answer to the question in the title is in the video linked at the end of the description. th-cam.com/video/qTzqIYrc_Hs/w-d-xo.html

I have just finished updating a corrected version of the subtitles. The problem with the voiceover is twofold: first, my abysmal pronunciation (compounded by the fact that the recording was initially meant only to estimate the length of the video); second, the low sensitivity of the mic. I have a Logitech headset that behaved spectacularly years ago in Skype - loud and clear - but as of now with Seven and 10 only produces low levels (something like 1-2 notched over a max of 10) and a lot of noise. I had to record this voiceover with my smartphone, keeping the phone in one hand and a pillow between me and the laptop to muffle the sound of the fans. I also did it in the middle of the night, and my voice was naturally on the whisper side, sorry. I have an updated version of this video with a couple more figures (not important) and corrections on the name Helmholtz and the formula overlay frame, but I don't think I will re-upload it anytime soon. I will be more careful with the audio levels (and try to improve my pronounciation) in the next videos. Try using headphones, they can solve the whispering problem. As for the mumbling, I need to work on that 😢. If the headphones won't do the trick, try the subtitles or better yet the transcript. I still have a couple of typos to catch, but they should help. Oh, and thanks for the feedback.

Thanks for the detailed reply, and sorry if I was abrupt but it was frustrating to find such interesting content which I had to abort! It made sense when you explained that the audio was recorded at night. This was the only problem for me. If you are able to deliver speech at a daytime volume, any old microphone or smartphone will be adequate. The only fancy addition might be some post-production compression, but a smartphone probably has that built-in. Speech doesn't require much bandwidth, dynamic range or s/n ratio. I was unable to follow the content because of the late night whispering :-)

Imagine being a person complaining over a FREE video of this caliber.

@@heywayhighwayI'm happy not to be as daft as someone who complains about free feedback and advice.

@@heywayhighway thanks for the implicit compliment, but it is okay to give negative feedback. It helps improving content. In fact, if nobody had complained for the low audio levels I would have probably recorded the next video in the same way (low voice, phone on desk, instead of keeping the phone in my hand and shouting into it). I do appreciate corrections and helpful feedback.

Is it possible to do this from within TH-cam? Do you know? I have the whole voiceover in the form of text in a web page but I do not have an ad-free website where to upload it to. But a few months ago I had to open a WordPress account to talk to Simplenote's assistance. If I find where I put the credentials, I might upload it there. PS Thank you for pretending the problem is in your bad hearing and not in my pronunciation ;-) .

​@@copernicofelinisno no my friend, you're good! I just came home and tried with headphones, it helps a lot, especially with some of the weird mistakes in TH-cam automatic captions. I'm afraid I don't know anything about how to work with TH-cam's subtitles but I'm sure that someone will read this and can provide some information. Again thanks for the video and as my wife will confirm by how I can get startled when she is "suddenly" behind me, giving me a heart attack, my hearing really is that bad 😅

Try now: I edited the subtitles. It should be better, if the changes have been applied.

​@@copernicofelinisWhoa, that's a lot better indeed. Thank you so much, I appreciate it dearly.

I wish I could say it was a one-off typo, but I just realized I have been spelling "Helmholtz" wrong for YEARS. Possibly decades! I just had one of those moments that in the movies are shown with a progressively closer close-up of the actor's face, while the background expands around it. Thank you so very much for pointing it out. PS I commented (positively) to your video years ago. 😊

That is on the back burner. Right now I am planning to upload a video on the IEC definition of voltage, another one on what a voltmeter actually measures, and one on when probing does matter and why. Then I would like to do a video on the role of surface and interface charge in establishing the E field inside a circuit. And only after that consider the energy transfer question (which has a lot to do with surface charge and also requires making some complicated figures).

Great and yeah we need to understand fundamentals correctly. most people do just read the definitions and move on i really like this video

And also I never really understood the term "flux". The magnetic flux and mmf stuff if you can try to put it in your bucket list .

@@lokeshvirat5915 well, there are two important 'dual' quantities that can be defined on a vector field: a line integral that quantifies how much the vectors of the field are going along a given line (when the line is closed this is the circulation), and a surface integral that quantifies how much the vectors of the field go through a given surface (this is flux). The circulation is associated with the projection of the field along the (local) direction of the path, while the flux is associated with the projection of the field along the direction orthogonal to the (local) surface. There are very powerful theorems that can relate the circulation of a given field (how much goes around a given closed curve) to the flux of a related field (how much goes through the surface delimited by that closed curve). And if you look at Maxwells equations in their integral form, you will see that they relate fluxes and circulations of E and B. But this is too long a story to be told in a comment.

@@copernicofelinis I'm gonna subscribe and turn on notifications , so I don't miss on this ones

I agree 😊. This voiceover was a recording I did to find out what the length of the video would have been. I thought I could do that in 15 minutes, and it ended up being 33 minutes. Once I cut out all the pauses I had already invested too much time in it to throw it away. I'll try to improve my pronunciation and timing in the next video.

​@@copernicofelinis Apart from that, I find your explanations really clear and well backuped with figures and equations. Keep it going!

​@@Antonio-lp8hxthanks.

You should see a doctor

Thank you. I appreciate corrections. Apologies for mauling your language. 😁

@@copernicofelinis No worries! 🙂

I am well aware of these "probing done right" videos, and what they try to recover is the path integral of the coulombian electric field alone (the middle picture in the diagram with three rings). This is the conservative **part** of the electric field that admits a scalar electric potential. The problem with this approach is threefold: 1. It describes only "half" of the physical system: one should also give the vector potential A to give the complete picture. 2. It forces you to consider a different physics for loops that link a variable flux and loops that do not, making the physics schizophrenic. In particular one has to invent reasons for the failure of Ohm's law inside every coil. 3. Voltmeters measure the difference on voltage, not the difference in electric potential; it is only in the conservative case that these two quantities coincide. I will show this in the next video. So, no, that is not good probing because they put their probes in the middle of the changing magnetic flux region, and because the quantity they recover tells only half of the story.

@@copernicofelinis "It describes only "half" of the physical system:" No, they simply are measuring correctly. "It forces you to consider a different physics for loops that link a variable flux and loops that do not" That is not the case either - they simply show how to measure correctly. "Voltmeters measure the difference on voltage, not the difference in electric potential" Aaaaaannnddd you might want to refresh your memory on that one - the definition of voltage for example - cause that IS the difference in electrical potential. You have not addressed anything but instead are making up ridiculous claims. By now it seems more like you subsrcibed the a false notion of physics and are now becoming more and more reluctant to accept that you were wrong, to the point of throwing a tantrum.

@@ABaumstumpf let's see: what is the potential difference between point A and the top terminal of the smallest resistor, according to Mabilde? How many volts? And what is the resistance of the copper conductor for that same path? Show me how Ohm's law work in this case.

@@copernicofelinis "Show me how Ohm's law work in this case." You are the one making the false claims. You are claiming that the most well known theories of electrondynamics are wrong - the burden of proof is on you. What Mabilde has down was simply showing that Lewin had an error in his measuring-setup and if corrected all his spurious claims fall apart. And now you are getting emotional cause you made the same mistake.

@@ABaumstumpf what setup? Look at the diagrams of the two half circuits where I show the computation of voltage as E x length. There are NO PROBES. The reason the voltmeters in the video show the correct values as computed from first principles is because the probing there is correct. But the values of those voltages are computed without the need of any probes. Oh, BTW, Ohm's law works perfectly fine with my values of voltage. No need to invent a different physics. Mine is the physics taught at Berkeley and MIT. ( Invoking the authority principle is a good thing in a TH-cam video because it allows viewers to know that what they see is not some 'original research' but is at least based on real science).

Each voltmeter shows the line integral of the TOTAL electric field along the path that goes through its probes and its internal resistor. This voltage is the same as the voltage along the tested branch of circuit only if there is no changing magnetic field linked. Each voltmeter is connected to both branches of the circuit: one branch will form a loop with the probes that does not link the core, the other will form a loop that does link the core. The voltage shown by the voltmeter will be equal to the voltage along the branch that does not form a loop around the core. All of this will be shown in the next video. What is important to notice here is that the field is a composition of two components and that this resultant is almost zero inside the conductor portions of the ring (in accordance with Ohm's law).

@@copernicofelinis The low resistance only reduces the Coulomb component along the conductor, it does not reduce the Faraday electric field along the conductor.

​@@woodcoast5026 no. The coulombian component is almost equal to the induced component . They are both around 1.1V/m, their difference is the negligible 0.0002 V/m that is compatible with the high conductivity of copper. The coulombian electric field (almost completely) obliterates the induced electric field in the conductor.

@@copernicofelinis That does not "obliterate" the Faraday field.The Faraday field is still there. The two components are still there forming the two components of the composition .

@@woodcoast5026 yes, that is correct, but the operation of superposition is something that resides in our heads only. What atoms and electrons 'see' and 'feel' is just the resultant field which is nearly zero. And it is nearly zero because the coulombian field has nearly almost entirely canceled the induced field in the copper. The same happens in the electrostatic case: the electrostatic field in a conductor is exactly zero because the external field has been obliterated by the field component generated by the displaced surface charge. Even more interesting: the interior of the conductor *never* gets a chance to 'see' or 'feel' the external field, because the surface charge kills it before it can penetrate the conductor.

It was an April fools joke lol

It was a joke lmao 😂

Just a joke bro :P

The conclusion is that Copernico went back in time so far that he disappeared. He built a time machine.

Toroidal transformers are everywhere, like parsley. Here it is used as a means to generate a sinusoidally changing magnetic flux. If I had had a powerful enough magnet I could have attached that to my grandma's sewing machine and used that instead.

@@copernicofelinis so could you theoretically use a sinusoidal alternating magnetic field to replace the moving magnets of a generator? Like make a solid state generator that use 2 alternating sinusoidal magnetic fields at 90* to eachother.

​@@freemanrader75 I believe the point of a generator with moving magnets is that to convert mechanical energy (used to move the magnets) into electrical energy. If we remove the magnets and replace them with an electrically generated variable field, we end up with a transformer.