How Does Faraday's Law Explain EMF Induction Outside a Solenoid?

[Per Oni]

That should give the same answer, but it's a different picture from that of the OP. Faraday's law works for two different cases (and the second is even a whole class, related to relative motion!).
- What the OP refers to is the picture of constant magnetic flux lines, but of varying intensity.
- You seem to picture moving flux lines, such as when a magnet is moved towards a conductor.

- http://en.wikipedia.org/wiki/Faraday's_law_of_induction

Yeah I realize very well what the op is asking:

BUT! As my friend correctly stated, the only place where the B-field is non zero is inside the solenoid itself. …-dB/dt would have to be zero for all points in space except inside the it. And that means no electric field can possibly have been induced from the varying magnetic field in the solenoid …

Many years ago I was pondering exactly the same question, which is: how can there be an E field in a conductor when there’s no dB/dt in the conductor? I was familiar with the concept that when magnetic field lines cut a conductor an electric field is set up in that conductor. But in case of a transformer (apparently) no lines ever cut the conductor. I therefore developed my own visualisation of a traveling B field. Of course I realized very well that to apply Faradays law for transformers no such picture was required.

At Q-eerus, I’m not sure what the content of your last post is saying.

 

[Q-reeus]

At Q-eerus, I’m not sure what the content of your last post is saying.

And I was guessing to some extent in answering your #59. Any particular part(s) of #60 not clear?

 

[harrylin]

Yeah I realize very well what the op is asking:
Many years ago I was pondering exactly the same question, which is: how can there be an E field in a conductor when there’s no dB/dt in the conductor? I was familiar with the concept that when magnetic field lines cut a conductor an electric field is set up in that conductor. But in case of a transformer (apparently) no lines ever cut the conductor. [..]

Yes, and same for me. I thought to have now answered that most clearly in post #52...
So I'll repeat: The induced E-field spreads out all around the solenoid.
(and sorry for the shouting )

 

[Per Oni]

And I was guessing to some extent in answering your #59. Any particular part(s) of #60 not clear?

In terms of the domain's magnetic contribution to the -d∅/dt emf, all of it's field lines contribute [strictly true only if the core is driven to saturation]. So on that basis alone there is for your model a deficit to explain - no?

I don’t get this question .

@ harrylin: Yeah ok I do hear you. I know what you are saying, no problem. Faradays law says so. I knew that years ago.

 

[harrylin]

[..]
@ harrylin: Yeah ok I do hear you. I know what you are saying, no problem. Faradays law says so. I knew that years ago.

If you knew that Faradays law (although I think in fact Maxwell's) says that a varying B-field induces an E-field all-around (and not just at a distance), then why did you ask how can there be an E field in a conductor at that place?

 

[Per Oni]

If you knew that Faradays law (although I think in fact Maxwell's) says that a varying B-field induces an E-field all-around (and not just at a distance), then why did you ask how can there be an E field in a conductor at that place?

Because at that time I wasn’t happy just to cram that law in my head without any visualisation.
My dilemma was: when considering motional emf we insist on the conductor cutting magnetic flux (or visa versa). No flux cutting = no emf.
Next step was: forget all such considerations when dealing with transformer emf. No flux cutting needed. No flux cutting = lots of emf.

 

[harrylin]

Because at that time I wasn’t happy just to cram that law in my head without any visualisation.
My dilemma was: when considering motional emf we insist on the conductor cutting magnetic flux (or visa versa). No flux cutting = no emf.
Next step was: forget all such considerations when dealing with transformer emf. No flux cutting needed. No flux cutting = lots of emf.

I'm afraid that I can't follow that, and perhaps suggest a personal theory, so let's not got there!

The first 6 min. of the following video going into details about applying Faraday's law may be useful:
http://www.youtube.com/watch?v=EYYNRubHIno&NR=1&feature=endscreen

 

[Q-reeus]

Q-reeus: "In terms of the domain's magnetic contribution to the -d∅/dt emf, all of it's field lines contribute [strictly true only if the core is driven to saturation]. So on that basis alone there is for your model a deficit to explain - no?"
I don’t get this question .

Well I took it you basically stated in #59 that the field lines of an aligning domain, in cutting the windings surrounding the transformer core, fully explain the induced emf. Obviously that's after summing over all domains. My point on that in #60 was this could not be so as typically only a tiny fraction of a domain's field lines exit the core region - depending on just where a domain is situated in the core of course. Vast majority of lines simply loop back on themselves 'internally' - within the core region. Nevertheless when domains are fully or near fully aligned, there is a strong *internal* reinforcement of domain flux lines to give a large net field strength B, and it's this reinforced internal value - confined to the core, that governs the driving emf = -AdB/dt (A the X-sectional core area) seen in the windings. This net core B is considerably greater in fact than that typically existing inside a randomly oriented domain. So do you now get why I said your idea had a big deficit problem?

[I take this back - it wasn't ringing quite true and now I see why. Hadn't taken into account the 'velocity leverage factor' that would apply to flux-cutting owing to a rotating domain. Few lines compensated by large relative motion of those lines. Kind of dovetails as per #60 with the notion of changing core flux being seen as a magnetic monopole displacement current - if one substitutes 'true magnetic dipoles' for Amperian loop currents. It's clearer then how motion of monopoles could achieve the flux-cutting needed to explain emf - particularly evident in long straight solenoid case. All such being fictitious entities but equally able to give consistent results re observed emf. Still, best to stick with that actual source of emf is overwhelmingly owing to -dM/dt, M owing to intrinsic moments.]

 

[Per Oni]

I'm afraid that I can't follow that, and perhaps suggest a personal theory, so let's not got there!

When considering motional emf we insist on the conductor cutting magnetic flux (or visa versa). No flux cutting = no emf.

When dealing with transformer emf, no flux cutting is needed. No flux cutting = lots of emf.

Which of these 2 sentences can you not follow (perhaps both)? Those sentences have nothing to do with a personal theory!

@ Q-eerus: I think it will be better for us to start a new thread about magnets. This is getting a bit too far removed from the OP.

 

[Q-reeus]

@ Q-eerus: I think it will be better for us to start a new thread about magnets. This is getting a bit too far removed from the OP.

Maybe. But anyway I have erred - see edit in #68

 

[Per Oni]

I want to give example of the strangeness of having 2 different flux cutting laws.

Imagine a horse shoe electromagnet (no permanent magnet) and switch on the current supply. Now approach this magnet with a search coil. During the approach you will be able to see the voltage output of the coil on an oscilloscope. This voltage and its magnitude can be explained using the flux cutting law. No problem here, I’m absolutely fine with that.

Next, keep the coil in place and switch of the current. Depending on the time constant of the electromagnet circuit there will again be a voltage displayed on the scope. But now we not only have to accept that there’s no flux cutting involved but we also have to accept that this field somehow mysteriously disappears. I’m not fine with that at all. (Never was).

@ Q-eerus. I’m sure you can teach me a lot about magnets but again I’d like to talk in a different thread.

 

[harrylin]

Which of these 2 sentences can you not follow (perhaps both)? [..]

I did not understand what you meant with "No flux cutting = lots of emf."
If you simply meant that there is lots of E around dB/dt, then that corresponds to Maxwell's explanation of Faraday's law (if I have the history right!).

 

[Q-reeus]

Imagine a horse shoe electromagnet (no permanent magnet) and switch on the current supply. Now approach this magnet with a search coil. During the approach you will be able to see the voltage output of the coil on an oscilloscope. This voltage and its magnitude can be explained using the flux cutting law. No problem here, I’m absolutely fine with that.

Next, keep the coil in place and switch of the current. Depending on the time constant of the electromagnet circuit there will again be a voltage displayed on the scope. But now we not only have to accept that there’s no flux cutting involved but we also have to accept that this field somehow mysteriously disappears. I’m not fine with that at all. (Never was).

If that horseshoe electromagnet is purely air-core, then agree flux-cutting approach fails in 2nd instance. If it is iron-core, then as per my revised position in #68, it is actually possible to apply flux-cutting method to the core contribution by summing for each elemental dipole's rotating field lines. A cumbersome methodology though and not in accord with the modern approach of simply applying Faraday's integral law in either case. In the first instance, it pays to remember that emf = -d∅/dt (total derivative), not merely the -∂∅/∂t (partial derivative) applying in the purely time-changing flux case of 2nd instance.

@ Q-eerus. I’m sure you can teach me a lot about magnets but again I’d like to talk in a different thread.

Something we haven't covered in the past? OK but I'm nearing exhaustion on this topic. Well there is one very interesting and sure to be controversial aspect I'd like to cover down the track, but not any time soon.

 

[Per Oni]

If it is iron-core, then as per my revised position in #68, it is actually possible to apply flux-cutting method to the core contribution by summing for each elemental dipole's rotating field lines. A cumbersome methodology though and not in accord with the modern approach of simply applying Faraday's integral law in either case.

Ah, I do understand your point of view a bit better after reading this. Anyway, (at least for us 2) the mystery remains for an air core. And yeah I’m also done on this subject.

@ harrylin Yes that is indeed what I meant.

 

[Per Oni]

Saying on Sunday I was done with this subject, but as often happens some thoughts keep on lingering.
So, today I started Googling a bit and this is what I found:

http://en.wikipedia.org/wiki/Faraday's_law_of_induction#Proof_of_Faraday.27s_law

Halfway down the page:

This step implicitly uses Gauss's law for magnetism: Since the flux lines have no beginning or end, they can only get into the loop by getting cut through by the wire.

Gauss’s law is a general law and must therefore apply to transformers. So this law completely vindicates what I was arguing all along, namely that magnetic flux cannot just appear/disappear without cutting the coils.

Everyone, thanks for all the help I got reaching this result. You were ace!

 

[Q-reeus]

"This step implicitly uses Gauss's law for magnetism: Since the flux lines have no beginning or end, they can only get into the loop by getting cut through by the wire."
Gauss’s law is a general law and must therefore apply to transformers. So this law completely vindicates what I was arguing all along, namely that magnetic flux cannot just appear/disappear without cutting the coils.

Don't want to spoil your day Per Oni, but if you check the context of that wording carefully (in the drop-down 'show' part), it will be seen to apply only to the 2nd term on the RHS - the motional d∅/dt emf part, not the 'transformer' ∂∅/∂t part, in agreement with my comments in #73. As I wrote then, you can use a flux cutting approach even for transformer part, but not on the basis of 'aggregate flux line motion', because the aggregate is clearly zero in transformer cores. Only field intensity varies with time - there is no net 'motion' at all. Thinking about it some more, even in the air core case, by summing over individual conduction charge motions, flux cutting can work there also, but again, not on any aggregate flux motion basis. The simplest and preferred approach is to always apply emf = -d∅/dt; the total derivative of threading flux in a given frame.

 

[Per Oni]

Q-reeus. You are one of only a handful around here I’ve got respect for.

But can you explain to me when a circle, no beginning no end, can enter another such circle without cutting a circumference? But no, so far you have not spoilt my day. In fact I’m looking forward to be proven wrong, it can only improve my understanding. But please just answer the above question.

 

[Q-reeus]

Q-reeus. You are one of only a handful around here I’ve got respect for.

Shucks Per Oni - I feel chuffed!

But can you explain to me when a circle, no beginning no end, can enter another such circle without cutting a circumference?

I will take it the first mentioned circle here represents a line of flux circulating within a transformer core. And the second circle represents a conducting winding wrapped around that core, right? Correct me if wrong, but I will assume so. Well this is where 'aggregate' comes into force. Suppose we have an air-core toroidal transformer here, with inner primary winding generating that core flux, the outer secondary windings (one of which is the 'second circle') having a small radial gap separating them from the primary windings. It is simply a consequence of applying the Biot-Savart expression for flux owing to a current element, then integrating over all such elements comprising the solenoidal current circulating in the primary windings, that all flux lines are confined to a region encompassed by the primary windings. In principle none intersect the secondary - although owing to inevitable manufacturing limitations, a small amount of leakage is inevitable - but small is the word. That can easily be checked physically using e.g. a magnetometer.

So with AC currents flowing, all that can happen is for flux lines to appear and disappear periodically in the core, without ever moving in and out of the core. The lines aren't real and, according to vector summation over contributing moving source charges, simply 'come into being' within the core, as mere indicators of the ever changing continuum field strength. Field lines are an artifice - they represent strength and direction of a continuous field that owes it's existence to flowing source charges or magnetized media.

What's considered fundamental is not 'flux line cutting' or even 'changing threading flux' - these are associations, not ultimate causes of transformer action. The true cause is always motion of charge, plus that of magnetized media (intrinsic electron magnetic moments + atomic orbital moments) if present. Work fundamentally from the field definitions:

E = -∇V - ∂A/∂t, B = ∇×A, and in turn the definitions for scalar potential V and vector potential A given by the Lienard-Wiechert expression http://en.wikipedia.org/wiki/Li%C3%A9nard%E2%80%93Wiechert_potential#Definition_of_Li.C3.A9nard-Wiechert_potentials
They always work, even in ultra-relativistic situations. Must go :zzz:
[Edit: Customarily we add Lorentz force expression F = q(E+u×B) to above. That allows the motional emf part to be calculated without resort to 'flux-cutting' as such.]

 

[Per Oni]

I will take it the first mentioned circle here represents a line of flux circulating within a transformer core. And the second circle represents a conducting winding wrapped around that core, right? Correct me if wrong, but I will assume so.

Yes correct.

You have avoided answering this question (asked 2x):

“Can you explain to me when a circle, no beginning no end, can enter another such circle without cutting a circumference?”

OK we both know that the real answer to my simple question is: no, it is not possible.

Well this is where 'aggregate' comes into force. Suppose we have an air-core toroidal transformer here, with inner primary winding generating that core flux, the outer secondary windings (one of which is the 'second circle') having a small radial gap separating them from the primary windings. It is simply a consequence of applying the Biot-Savart expression for flux owing to a current element, then integrating over all such elements comprising the solenoidal current circulating in the primary windings, that all flux lines are confined to a region encompassed by the primary windings.

The Biot-Savart law doesn’t deal with changing currents. And it is only when the current changes that we have a dB/dt in the coils. So this part of your answer is not valid.

In principle none intersect the secondary - although owing to inevitable manufacturing limitations, a small amount of leakage is inevitable - but small is the word. That can easily be checked physically using e.g. a magnetometer. So with AC currents flowing, all that can happen is for flux lines to appear and disappear periodically in the core, without ever moving in and out of the core.

Same answer as my last one.

The lines aren't real and, according to vector summation over contributing moving source charges, simply 'come into being' within the core, as mere indicators of the ever changing continuum field strength. Field lines are an artifice - they represent strength and direction of a continuous field that owes it's existence to flowing source charges or magnetized media.

This is another answer I was expecting: “field lines are not real!”
Except when we need them to explain energy, force, momentum of moving magnetic fields they are real enough. We can’t have it both ways.

Furthermore I really wonder why Maxwell went through the trouble of including magnetic fields in his equations. If you can show me that we can do away with all the Maxwell equations dealing with magnetism and we can fully replace them with purely electrical static and moving fields then you will have made inroads to me believing you. Note that these alternative equations need to have all the proper vectors, dots, crosses etc.

Have fun!

 

[Q-reeus]

You have avoided answering this question (asked 2x):
“Can you explain to me when a circle, no beginning no end, can enter another such circle without cutting a circumference?”
OK we both know that the real answer to my simple question is: no, it is not possible.

It's not possible provided the first circle is a real entity that must somehow be conserved - i.e. not allowed to materialize/vanish. But I'm saying that's in fact how one needs to treat magnetic field lines generated by a time-changing current. Because the field itself changes sign in keeping with an AC source current! Hence must appear and disappear cyclically. How else do you imagine it all works? Very different to the case for say lines of E owing to a static charge, where a conservation law - Gauss's law, forbids any appearing/disappearing trick. That does not apply to B field whose source - *moving* charge, can be turned on or off, one-way then that, at will.

The Biot-Savart law doesn’t deal with changing currents. And it is only when the current changes that we have a dB/dt in the coils. So this part of your answer is not valid.

Biot-Savart is typically applied for steady currents, but nothing prevents it's application to time-varying situation. Only when very high frequencies are involved need we take extra care to allow for phase delay. Transformers are typically nowhere near that regime. It's just a mathematical fact that cancellation of field occurs everywhere except inside solenoidal core region. You can check out any of the many sites that will derive field for a long straight solenoid. And toroid is similar.

This is another answer I was expecting: “field lines are not real!”
Except when we need them to explain energy, force, momentum of moving magnetic fields they are real enough. We can’t have it both ways.

Fields are real. Field lines are not, but they are very handy to use for many situations. But there are limitations. If you really believe those lines of B have to cut through the secondary windings of say a toroidal transformer, explain why a compass needle for instance undergoes no deflection when a steady or slowly-varying current runs through the primary. Don't believe me? Ask around - contact some transformer manufacturers/sellers. They make it their living to know the business of how these things work.

Furthermore I really wonder why Maxwell went through the trouble of including magnetic fields in his equations. If you can show me that we can do away with all the Maxwell equations dealing with magnetism and we can fully replace them with purely electrical static and moving fields then you will have made inroads to me believing you. Note that these alternative equations need to have all the proper vectors, dots, crosses etc.

Not getting your drift here. Maxwell gave us an extremely important set of relationships between fields, and between fields and sources of those fields. Check out the Ampere-Maxwell eq'n here. Notice how those moving charges enter the picture?

Have fun!

After all this? You've rained on my day bud.

 

[harrylin]

[..] “Can you explain to me when a circle, no beginning no end, can enter another such circle without cutting a circumference?”
OK we both know that the real answer to my simple question is: no, it is not possible. [..]

I know a magician trick like that.

But seriously, In the OP I don't see two circles cutting each other.
Instead, I "see" for this case how very weak magnetic fields (see post #1) form right through the circle. If you visualise "field lines" by adding iron powder, then you will see the field lines take more shape inside the wire loop without laterally cutting through the wire loop.

And more importantly (and I stressed this before in #36), those weak changing magnetic fields have the opposite effect of the induced electric field which is according to Maxwell responsible for the EMF in this case. If you don't believe this, please make a drawing and you will see for yourself.

 

[Per Oni]

How else do you imagine it all works?

As you can see I'm working on it . Halfway there.

That does not apply to B field whose source - *moving* charge, can be turned on or off, one-way then that, at will.

Nope not thru. B fields need energy to come in existence. B fields are just as real as all other stuff containing energy, such as E fields, atoms, your shirt etc. If B fields are not real nothing is real. Anyway how are you going to stop a charge from moving? Impossible!

Fields are real. Field lines are not, but they are very handy to use for many situations. But there are limitations. If you really believe those lines of B have to cut through the secondary windings of say a toroidal transformer...

In that case do you believe they cut through a coil in case of motional emf?

 

[Per Oni]

I know a magician trick like that.

When I wrote that question that picture appeared also in my mind. Once upon a time they were quite popular but now they are seen as boring in comparison with what’s around now. (Walking on the Thames etc.)

 

[Q-reeus]

Q-reeus: "That does not apply to B field whose source - *moving* charge, can be turned on or off, one-way then that, at will."
Nope not thru. B fields need energy to come in existence. B fields are just as real as all other stuff containing energy, such as E fields, atoms, your shirt etc. If B fields are not real nothing is real.

Not making sense here. When did I even suggest B fields aren't real, or don't have energy? Arguing against a phantom. I said *lines* aren't real - big difference!

Anyway how are you going to stop a charge from moving? Impossible!

Perhaps you should explain what your notion of alternating current entails. You can arrange to avoid that twice every cycle, current = moving charges has zero net speed!?

Q-reeus: "Fields are real. Field lines are not, but they are very handy to use for many situations. But there are limitations. If you really believe those lines of B have to cut through the secondary windings of say a toroidal transformer..."

In that case do you believe they cut through a coil in case of motional emf?

Already covered in #53 - you know the answer.
Now, please do contact transformer designer/manufacturers, and see if any agree with your idea of how a transformer works. Evidently clear by now; flux lines expanding in and out of the core, and cutting the secondary (or even primary) windings of a toroidal (or any 'iron-core') transformer. Be prepared for some laughter! :zzz:

 

[harrylin]

When I wrote that question that picture appeared also in my mind. Once upon a time they were quite popular but now they are seen as boring in comparison with what’s around now. (Walking on the Thames etc.)

Hehe yeah. But what about the serious content of my reply??

 

[Per Oni]

Perhaps you should explain what your notion of alternating current entails. You can arrange to avoid that twice every cycle, current = moving charges has zero net speed!?

What I was thinking was that electrons still have the Fermi speed. But forget that and yes you are correct here.

This is my point of view as per #50:

Note that when dI/dt exist there will be a wave front spreading out from a conductor having the same speed as the speed of light for the medium in which the conductor is located (think antenna). Perhaps you can see therefore that my picture is not that far fetched.

To add to this from Wikipedia #75:

Since the flux lines have no beginning or end, they can only get into the loop by getting cut through by the wire.

Let’s look at your points of view #78:

So with AC currents flowing, all that can happen is for flux lines to appear and disappear periodically in the core, without ever moving in and out of the core. The lines aren't real and, according to vector summation over contributing moving source charges, simply 'come into being' within the core, as mere indicators of the ever changing continuum field strength. Field lines are an artifice - they represent strength and direction of a continuous field that owes it's existence to flowing source charges or magnetized media.

And #80:

…Gauss's law, forbids any appearing/disappearing trick. That does not apply to B field whose source - *moving* charge, can be turned on or off, one-way then that, at will.

You admit that a magnetic field is real and contains energy (#84), but also according to you, the field just appears in the core. Also according to you: Gauss’s law does not forbid any appearance trick in the case of magnetic fields. Can you tell me how that energy got there? Just at will? Chemical energy? Time to wake up!

 

[Per Oni]

Hehe yeah. But what about the serious content of my reply??

Thanks for your reply but I just don’t want to run 2 virtually identical discussions.

 

[harrylin]

Thanks for your reply but I just don’t want to run 2 virtually identical discussions.

There is only one discussion.

 

[Q-reeus]

This is my point of view as per #50:
Note that when dI/dt exist there will be a wave front spreading out from a conductor having the same speed as the speed of light for the medium in which the conductor is located (think antenna). Perhaps you can see therefore that my picture is not that far fetched.
To add to this from Wikipedia #75:
Since the flux lines have no beginning or end, they can only get into the loop by getting cut through by the wire.

That last bit I covered back in #76 - applies to motional not time-changing component. (And btw, it really is no service to the other readers when there is no mention the above quoted bit from Wikipedia article is 'buried' inside a drop down box one has to click on, and is not part of a normal read of that page.)
Your first bit from #50 is fine if restricted to considering just the contribution from a single current element, as I have acknowledged earlier and you should remember such things. Far from being an essential perspective though, it leads to a faulty notion of needing to have real flux lines cutting things. Just aint so. Transformers.

Let’s look at your points of view #78:
So with AC currents flowing, all that can happen is for flux lines to appear and disappear periodically in the core, without ever moving in and out of the core. The lines aren't real and, according to vector summation over contributing moving source charges, simply 'come into being' within the core, as mere indicators of the ever changing continuum field strength. Field lines are an artifice - they represent strength and direction of a continuous field that owes it's existence to flowing source charges or magnetized media.
And #80:
…Gauss's law, forbids any appearing/disappearing trick. That does not apply to B field whose source - *moving* charge, can be turned on or off, one-way then that, at will.

You admit that a magnetic field is real and contains energy (#84), but also according to you, the field just appears in the core. Also according to you: Gauss’s law does not forbid any appearance trick in the case of magnetic fields. Can you tell me how that energy got there? Just at will? Chemical energy? Time to wake up!

There is no conflict at all: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/indeng.html Energy is supplied by the currents which are the field's source.

Now I have to pull you up on a serious matter - misrepresenting my actual position a number of times now. Above is just the latest example. I preceded 'appear and disappear' with 'So with AC currents flowing'. Having previously emphasized a number of times that currents and/or magnetized media are the real source of both E and B fields, there is no excuse to slant things as you have done here. I will not accuse you of being deliberate in this - in your mind there may be a faulty interpretation process going on. Inadvertent or not, stop and think carefully before putting into print claims of what I have said or believe, which keep turning out to be false.

Getting back to what I consider your fixation with the absolute need to explain Faraday's law entirely in terms of flux-line cutting. Please, take on-board the fact that, for an infinitely long solenoid, or quite finite toroid, there is zero magnetic flux-line density except within the core region. The resolution to the apparent paradox of closed flux-lines in infinite solenoid case is that the equivalent fictitious 'poles' at each end are infinitely removed, therefore the return flux lines exterior to the solenoid core are infinitely diluted. And the problem does not exist at all for toroid geometry. So again, to put paid to your notion of flux lines expanding in and out of the core region, acknowledge this necessarily implies a quite strong and detectable B field must exist exterior to the core. Well just put it to the test. I guarantee you will be disappointed. Again, I urge you to contact those whose livelihood depends on knowing just how transformers actually behave.

 

[Per Oni]

Energy is supplied by the currents which are the field's source.

Totally agree.
Do you in turn agree that this energy is locked up in the toroid? Consider the case where we have a constant dc current in the primary.