Peltio

And if the values of the voltages do not depend on the position of the source of magnetic induction (not linked in the first case, linked in the second), then (apart from the sign) they are using the IEC/ISO definition of voltage. Their use of 'potential difference' to equate voltage is a misnomer because it is NOT the scalar PD.

And to be clear: the two examples I have made are just a battery powered resistive circuit and an AC powered resistive circuit. ALL textbooks on circuit theory, and elementary physics deal with them. ALL of them.

LOL, even Sommerfeld uses it, in its Theoretical Physics Course (I believe it dates back to 1949). - Incidentally, I hate using chats because I can only access them on one PC, and I always forget which post they are hidden into, hence the delay in answering.

To make a few names of EM textbooks that make the use of this version of voltage (equal to the IEC and ISO voltage, sign apart): Purcell "Berkeley Physics, electricity and magnetism", Popovic "Introductory Engineering Electromagnetics", Ramo Whinnery Vanduzer "Fields and Waves in Communication Electronnics", Demarest "Engineering Electromagnetics"

All the textbooks of circuit theory implicitly use voltage as the line integral of the total electric field. Otherwise, they should give values of 'potential differences' that depend on the position of external, not linked, sources of magnetic induction.

No? So, what version of voltage do they use? 'Yours' or ISO's (sign apart)?

The examples I have shown are of circuits where KVL applies. Using your definition of voltage, you would have significant voltages along near-zero resistance wires, Ohm's law not obeyed, and AC voltages in pure DC circuits. Have you ever seen any circuit theory book use that definition of voltage when analyzing a lumped AC circuit or a DC circuit near a source of EMF (possibly from another part of the circuit in the same system)?

I have added a couple of hand sketched examples here: physics.stackexchange.com/questions/446454/… (I had better examples done weeks ago, but I can't find them, so I jotted them down, lest more weeks pass - there could be some menial error of sign or value, but the gist is there).

Induced fields are everywhere on a circuit board sitting next to a transformer and yet all design procedures ignores them when the sources of magnetic induction are not linked by any mesh of a circuit. Why? Because what they use is voltage defined as the line integral of the total electric field - what is used by ISO and IEC only differs for a sign and the direction of the path. Using scalar potential difference would require to rewrite all circuit theory textbooks.

Because with 'your' definition of voltage, that is what happens - the math is crystal clear. I have yet to find one, and I have read dozens of them. How many engineers design circuit considering the effect of non-linked EMF because they can affect the scalar potential differences in their circuits? Can you name one?

How many circuit theory books can you name where it is said that a DC circuit that does not link any source of magnetic induction can have AC 'voltages' (aka potential differences) when they are positioned just near a transformer? Can you name at least one?

@JánLalinský well, you should consider it a problem because it goes against the whole of circuit theory. It means you cannot use 'your' voltage to analyze circuits without taking into account the position and the strength of all nearby sources of magnetic induction. That is, if you have circuit formed by a battery and two resistors near a transformer, you can't use Ohm's law, nor you can tell what the voltages 'across' each resistor is without mapping the induced field and then integrating along the specific path followed by the circuit. Imaging having to precisely measure a voltage...

@JánLalinský consider a battery powered voltage divider sitting next to a toroidal transformer. If the circuit does not link the core, one would expect the circuit to only involve DC voltages. But using 'your' definition of voltage, the 'voltages' in the circuit will be mixed AC+DC, while the current would still be DC. This is why using the scalar potential difference as definition of voltage is not the best idea when you are in the presence of sources of dB/dt. Using voltage as the line integral of the total electric field will give the correct results expected by circuit theory.

I feel compelled to point out that 'your' definition of voltage as scalar potential difference would yield values that depend on the position of nearby sources of magnetic induction dB/dt EVEN WHEN such sources are not linked by the circuit. Try it out.

Found one of my answers where I express this: physics.stackexchange.com/questions/733491/…

The answer on SE does NOT imply there is a 'partial voltage' along the wires of the loop. Quite the contrary: if the resistance is negligible the voltage (as the line integral of the total electric field E = Ecoulombian + Einduced ) is essentially zero. The introduction of a finite small resistance won't change the discussion if not by a handful of millivolts. But yes, voltage has two components: the scalar potential difference delta phi, and the induced voltage - and failing to recognize this composition can cause confusion. I suggest using different symbols for scalar potential and voltage.

I don't do chats, so I'll just add that you are ignoring all the examples I have made. Please compute 'your' voltage in a battery powered circuit (battery + R) when there are no external sources of magnetic field , and when there is an infinitely long solenoid with changing dB/dt outside of the circuit, in two different positions. You will see that 'your' voltage will be different because the induced electric field will change the surface and interf. charge distribution.

@JánLalinský the electric field outside an infinitely long solenoid is tangential and has a magnitude that decreases as 1/r. The only way to get non-zero circulation along a closed path it to go around the coil. Locally, the curl is zero in any neighborhood of any point outside the perimeter of the coil. This is where the magic is. Decompose an arbitrary path in radial and tangential components and you will see why you have zero circulation. This is why in lumped circuit theory, if you don't go inside or around magnetic components, IEC voltage has the same value on every path.

@JánLalinský you need to look closer at the fields. Do it with the tangential electric field of an infinite solenoid and you will see that unless your path goes around the source, the contribute of the induced field will be so as to give zero circulation everywhere. The surface and interface charge in the circuit will alter the total electric field inside and outside the circuit itself to give that behavior that you seem to think is not possible. I am making a series of videos on this but I don't want to mix my identities. Let me just say that outside of the infinite coil curl E = 0.

@JánLalinský here, have a look at the figure of the coil in this answer: electronics.stackexchange.com/questions/506590/… - voltage is the same along all paths that do not go inside the inductor. Why? because any two paths the do not go inside and do not link the magnetic region will delimit an area where there is no dB/dt, so the circulation along its closed boundary will be zero. (I will delete this comment in a few hours - this comment list has become too long)

@JánLalinský what you are missing is that the charge distribution on the wires and at the interfaces between materials is influenced by the position of the sources of the changing magnetic field. It's the induced electric field that does not depend on the charge distribution, so when you move an external source of dB/dt, the surface and interface charge will redistribute to give the same IEC voltages in the circuit, but different partition into scalar electric potential ('your' voltage) and the induced voltage (changed with the position of the source). I really need to finish all those figg.

@JánLalinský that is incorrect. IEC voltage does NOT depend on the path if you stay in a simply connected region of space (I had comment limit in chars) that contains the circuit and does not contain the magnetic components. Even when there are varying magnetic field external to the components and this region in which the circuit sits. Please verify it is so, because it is. 'Your' voltage - or if you will the decomposition of IEC V into phi and induced voltage - DOES depend on externally (NOT linked) variable magnetic fields. Try with battery operated circuit (E and R) near a a long solenoid

@JánLalinský yes, that's correct: the IEC voltage is useful if we don't go inside lumped magnetic components. And that is the basis of lumped circuit theory: the components are black boxes and we interact with them via the voltage at the terminals (what Haus & Melcher call 'terminal voltage') and there are no changing magnetic fields linked by the circuit (and that's enough). On the other hand, 'your' voltage, the IEC scalar potential requires that there be no dB/dt in the whole universe except for inside the magnetic components of your circuit. IEC voltage does not depend on external dB/dt.

@JánLalinský while I agree with you on the first part (if we agree to call 'potential of E' the 'scalar potential phi of Ec'), you don't need to specifiy a path for the IEC voltage IF your entire circuit is in a region of space that does not link any relevant time-varying magnetic field (and the relevant magnetic field is confined inside lumped components). The IEC voltage between two points has the same value along infinite paths joining those points if it sits in that region. That's what makes engineer pretend KVL is still working.

One can express the electric field into the sum of its irrotational and solenoidal components. In that sense the irrotational part admits a scalar potential and some authors use the difference in scalar potential as the definition of voltage. I strongly disagree with this choice because voltage so defined has several disadvantages. Voltage defined as the line integral of the 'total' electric field, on the other hand becomes path dependent and it is the sum of scalar potential difference and induced voltage. Just to make the OP aware of different conventions.

Do you know what a line integral is?

@Dale look up my other answer on the difference between voltage and PD, linked in my answer to this question: physics.stackexchange.com/questions/667777/… (when I'm done with the pictures I will add the direct links in my answer here, as well)

@JohnDoty with Rleads=0 it's what you said the voltmeter measures. In order to have KVL hold, McDonald is advocating using the scalar potential difference, instead of voltage. Which is a legitimate position if you turn definitions around, but it's not a practical one because voltmeters do not (always) measure scalar potential difference. But they always measure voltage, as defined by the International Electrotechnical Commission.

@JohnDoty McDonald says: "the voltmeter as modeled above reports Vmeter = Io (Ro + Rleads) which equals the line integral $\int_1^2E.dl$ (along meter leads), along the path of it's conductors." It's on page 2 of "What does an AC voltmeter measure". While on p. 10 of "Lewin circuit paradox" he advocates using the term voltage drop only for the scalar potential difference. It's his original point of view. I use the definitions of voltage and PD given by the IEC and with these definitions KVL does not always hold.

@JohnDoty yes, I was aware of McDonald's essay, but notice that he is advocating using voltmeters that measure the scalar potential difference. In reality, though, voltmeters measure voltage that, in the presence of a changing flux, is different from scalar PD. The voltage measured by a voltmeter is the path integral of the total electric field on the path set by its probes and internal resistance. And with this definition of voltage KVL does not always hold. Case in point: a single secondary coil - voltage across the coil is the secondary voltage, while voltage along the coil is zero.

@FlatterMann a transformer is an example of lumped component that, when part of a lumped components circuit, can be thought as obeying 'extended KVL' (which is the 5+3-8=0 example made by Lewin). Not all circuits can be modeled by a collection of lumped components. The fact that other can does not remove the exceptions, and the OP might be interested in those.

@FlatterMann no, not in general. If you can devise a circuit path that does not link the changing flux (like in basics answer below), then you can hide Faraday's law inside the (lumped) magnetic components and pretend they 'drop' a voltage. But there are unlumpable circuits ( like Lewin's ring) that make this impossible.

See for example: "Energy flow from a battery to other circuit elements: Role of surface charges" by Manoj K. Harbola 2010 American Association of Physics Teachers. DOI: 10.1119/1.3456567

it is quite normal to use this kind of idealization. Infinite parallel planes for capacitors (to neglect fringe effects), infinitely long parallel current-carrying wires, ... infinite ground planes... Sommerfeld is using that idealization to bring symmetry to the problem. It also uses a return surface that is infinitely thick, IIRC. :-) And he was no spring chicken. As a matter of fact, the results he finds are in line with modern day simulations of finite wires.

If you can get hold of Sommerfeld's Lectures on Theoretical Physics, it is possible that the problem he solves at p. 125 in the Electrodynamics volume, "Detailed treatment of the field of a straight wire and a coil", might help you visualize the electric field.

Ok, just to make sure I'm doing it right: the parameters specified in your assumptions (Δt, v , b, hbar , kx ,ky ) are all constants during the calculation, right?

Yes, I understand what you need. And it shouldn't be difficult to implement. I will try to write something tomorrow, now that is seems you have fixed your syntax. (I don't understand why you deleted the explanation of the steps needed you had wrote in TeX, it was fine). When (if?) I have some working code, I'll ask to reopen this question.

Please use Mathematica syntax, and make sure your code works. You used sqrt instead of Sqrt, {} instead of [], \pi instead of Pi, and so on. Write your code in plain text, no funny characters, and make sure you can compute actual values for some instances of the variables.

Herman, write at least the Mathematica code for the definitions of matrices M1 and M2 and the procedure that receives f[t] and M1, M2 and returns f[t+dt]. Then I can show you how to fit it into a Nest or NestList command.