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Is there something missing in Faraday's law or just in Wikipedia's explanation?

Faraday's law of induction is confusing because for one matter it is incomplete. For another the way “magnetic flux linked to the circuit” is undefined. For a third it is by no means responsible for the working of all electrical generators. Aren’t thermocouples or some piezo crystals electrical generators?

A permanent magnet could move all day in the vicinity of a stationary conductor without inducing the slightest EMF in that stationary conductor. Inducing EMF in a conductor depends on the relative direction of that movement. The following experiment should demonstrate this:

Consider two permanent magnets the north pole of one parallel to and facing the south pole of the other. The magnetic flux existing between these two surfaces is among other constant parameters dependent on the distance between these two surfaces. Reducing this distance would certainly change the magnetic flux thus satisfying the requirements of Faraday's law. Should there be a stationary conductor just in the middle of this changing field no EMF is induced in it thus disproving to Faraday’s law.

The contradiction is probably due to the second point of confusion namely just how is the “magnetic flux linked to the circuit”.

AdrianAbel (talk) 08:12, 29 April 2008 (UTC)

I would say "magnetic flux through the circuit", which is perfectly unambiguous in the case that the circuit is a loop of thin wire, which is really the only case that (this version of) Faraday's law should be said to apply to. I agree that "linking" isn't the right word. You're right about "electrical generators"...is there a more specific term for "electrical generators involving spinning things and magnets and wires"? Or, it could just say "most electrical generators".

through the circuit" is correct.

I am an electrical engineer specialising in the design of electrical generators. flux linkage is a commonly used term in this field. The flux linking a circuit is the integral of the flux passing through the surface of a closed loop, normal to that surface. In an unsaturated iron cored coil this will be found by the formula NBA where N is the number of turns surrounding the core, B is the flux density in the core and A is its cross-sectional area. The rate of change of flux linkage gives the emf. The more general formula is $\lambda =\int _{S}B\cdot dA$. In this case through is also correct if you take the meaning that through means passing normal to a surface defined by a closed loop in some region of space. —Preceding unsigned comment added by 129.215.122.1 (talk) 10:51, 6 January 2010 (UTC)

AdrianAbel (talk) 17:52, 1 May 2008 (UTC)

In the "experiment", could you specify what you mean by "conductor" (A loop of wire? A flat planar conducting sheet?) and how you know that "no EMF is induced"? I'd suspect there would be an EMF, but I couldn't say for sure unless you're more specific. Thanks!! :-) --Steve (talk) 16:35, 29 April 2008 (UTC)

A loop of electrical conducting wire such as a copper wire connected at its ends through a resistor of finite resistance. The loop fits through a hole just in the middle of the magnets. Any EMF generated in this wire while moving the magnets closer and farther away from each other must be measurable across the resistor with a voltmeter. Is that specific enough or would you rather see a drawing?

AdrianAbel (talk) 17:52, 1 May 2008 (UTC)

It is a lot to ask that a short definition cover all nuances of "linking". The section Flux through a surface and EMF around a loop attempts to fill in some details. Later examples, I believe, make the notion very clear. The treatment of surfaces that vary in time, as happens in the drum generator or the Faraday disc, require some extended discussion. I'd suggest that the definition is correct, but it does require the later discussion.

As for generators: any device that converts one form of energy to another is a generator. I've revised this phrase - take a look. Brews ohare (talk) 18:15, 29 April 2008 (UTC)

The current "many forms of electrical generators" is also an improvement. And for a definition the above suggested "through" the circuit is considerably more appropriate than a word like "linking" with so many nuances.

AdrianAbel (talk) 17:52, 1 May 2008 (UTC)

Hi Adrian: The word "through" is in common use with a huge variety of meanings. I suspect it sounds better just because the reader can imagine whatever meaning they like best, while the unusual word "linking" forces the reader to ask "What does that mean?" Unfortunately, the concept is not straightforward, so the latter reaction is closer to appropriate, I believe. In any case, a lot of examples are necessary before it sinks in, whatever word you choose. Brews ohare (talk) 02:33, 2 May 2008 (UTC)

Probably so. But I consider changing the original wording from Faraday from through to linking as a crime. The perpetrator is probably Saiduko. I would appreciate your correcting it in Wikipedia. If it's not done within a few weeks I'll change it myself and give more reliable sources.

AdrianAbel (talk) 11:42, 2 May 2008 (UTC)

Hi Adrian: I'm not at all clear about the "crime" involved here. Searching the term flux linkage with the qualifier Faraday produces 3330 hits (303 on Google books). However flux through + Faraday while requesting no linkage or link produces 13,000 hits (629 on Google books). So linkage is in the minority, but it's pretty common, especially with authors/educators.

It’s a crime because Wikipedia's article is about Faraday's Law of Induction and Farady never used such an expression as “linking”. Also what Wikipedia proclaims the law states, Faraday never stated. Depending upon the interpretation of the word “linked” it could even negate his law entirely as my above experiment would show.

If you’re still not convinced consider the following. If I would give you two open bicycle chains and a master link and ask you to link them together you could probably manage it in a jiffy. Then I would hand you a permanent magnet and ask you to now link the bicycle chains to the magnetic lines of force emanating from it. After first deciding that I’m not crazy you’d probably wonder where to start. In other words "Linking" in the classical sense means to or within a chain with identical or similar members. Internet links providers, for example.

Google.com found about 64 hits on the key “Faraday’s law of induction”. Many were repeats or childish. A single one used the term “linking” who’s author I assume got it from Saiduko or vice versa. If Saiduku’s work was written originally in Japanese the error could well be in the translation to English. If his book wasn’t so expensive I would check that. I could give you about 10 others references found by Google.com that use exclusively the term “through”. But it’s not really necessary. Farady’s original findings in “Experimental researches in electricity” are available in copy for 50$ or 35£. If you should have access to the Library of Congress there’s a copy in their rare books department in the Franklin street That library has not yet put it on line. Excerpts of it are available from other sources on line. In reading them you’ll find how meticulously Faryday recorded his findings. He uses the word “through” and "cutting" often as well as “introduce” and “insert”. It’s so clear that anyone could repeat it at any time.

My suggestion is to exchange the word “linked” to “cutting through”. Then add the sentence: This law is derived from Faraday’s observations first reported in 1831 in his “Experimental researches in electricity”. If you want to leave a reference to Henry, no objections. But please remove the link to Saiduko so that that chain of confusion gets cut off.

AdrianAbel (talk) 17:30, 3 May 2008 (UTC)

In this connection, I've found reference to "Neumann's law" of 1845 (apparently Faraday's law often is referred to as the Faraday-Neumann law): When the magnetic flux linked with a coil (or circuit) is changed in any manner, then an emf is set up in the circuit such that it (the emf) is proportional to the rate of change of the flux-linkage with the circuit. Electro-Magnetism: Theory and Applications -If this statement is historically accurate, there appears to be a long tradition of "flux linkage", eh? Care to advance why you think use of linkage is a "crime" and an "error"? ;-)

Hello Brews. The use of "linkage" in Neumann's law in 1845 surprises me. It could well have been written originally in German and mistakenly translated. I see a lot of such translation errors. But in any case he takes a paragraph and a drawing to define exactly what he means. Wikipedia does not.

AdrianAbel (talk) 17:30, 3 May 2008 (UTC)

Do you have a suitable exact quote stating the law that can go in the quotation box with a "more reliable" attribution? Brews ohare (talk) 13:36, 2 May 2008 (UTC)

No. Not even Faraday has one. He just reported his observations. I didn't even see an equation in his report. But if you just change it as I suggested it would be a big improvement.

AdrianAbel (talk) 17:30, 3 May 2008 (UTC)

OK, I've eliminated linking. Brews ohare (talk) 20:02, 3 May 2008 (UTC)

Thanks. AdrianAbel (talk) 16:29, 6 May 2008 (UTC)

The current version of the article misleads the reader. In Defense of my previous adjustment to the article...Faraday's lab work helped him to visualize what was taking place physically. While he verbalized the relationships expressed by the math, he would not have used math symbols to express it. The current article implies that Faraday expressed his ideas using the language of calculus which was beyond his extensive lab based background. Maxwell, following Faraday's path, would have used this calculus based expression to translate Faraday's written idea into mathematical expressions shown throughout this article. On page 217 of Cantor's (1991) biography of Michael Faraday...Totally lacking from Faraday's Diary and his scientific papers were mathematical formulae and algebraic expressions. Unlike so many of his contemporaries, Faraday did not use mathematics as a language and a manipulative tool. See also two sections in Faraday article, Electricity and Magnetism and Faraday cage.CUoD (talk) 18:54, 22 January 2010 (UTC)

Clarification for the reader of the role that Faraday and Maxwell played in this chain of events.CUoD (talk) 15:25, 23 January 2010 (UTC)

A simple question....no emf is generated from a constant (ie non changing) magnetic field right? Then why do we sometimes see screened dc power cables? —Preceding unsigned comment added by 87.194.19.201 (talk) 12:08, 10 February 2010 (UTC)

A few simple answer(s)... (1) the current through a DC power cable generally does vary, (2) DC power supplies might be noisy, (3) screening works both ways, keeping noise from getting in as well as getting out. Alfred Centauri (talk) 14:21, 10 February 2010 (UTC)

Other forms of induction?

Just an idea I would like to hear your opinions of, especially from those who knows this law well. Okey, we are used to change the flux by either moving the magnetic field source and the coil relative to each other, or to change the field directly by using more or less current. But what about changing the magnetic permeability of the medium between them? Shouldn't we by doing this be able to alter the magnetic flux and thus induce a voltage in the coil? Also are there any special laws or rules that cover this aspect of magnetic induction, something to read on? Thanks. --Nabo0o (talk) 12:44, 16 September 2009 (UTC)

NaboOo, this is an excellent question! Note that we are talking the relevant Maxwell"s Law here because Faraday's Law is false and cannot be used to determine anything. I believe that it does not constitute an intrinsic change in the magnetic field because it does not necessarily affect the entire field. Mike La Moreaux (talk) 21:04, 16 September 2009 (UTC)

NaboOo, see See Engineering Electromagnetic Fields and Waves by Carl T.A. Johnk for a discussion of how the permeabilities will affect the B field (and a generally accesible introduction to all aspects of electromagnetic theory). —Preceding unsigned comment added by 129.215.122.1 (talk) 11:09, 6 January 2010 (UTC)

Nabo0o, Yes, an interesting question. You could try setting up a stationary circuit in a steady state magnetic field inside a container, with no power source, and only a galvanometer which can be read outside the container. Then fill the container with mercury so as to totally submerge the circuit. You would then see if the changing magnetic permeability caused any deflection of the galvanometer. David Tombe (talk) 00:49, 17 September 2009 (UTC)

One could argue that a transformer core is an example. You wouldn't use the word "permeability", since it's a ferromagnet and therefore not linear. But certainly the current in one winding changes the magnetic properties of the core, which in turn induces a voltage in the second winding. (There's also a direct effect but it's much smaller.) So that's a little bit like what you're saying I think. :-) --Steve (talk) 07:42, 17 September 2009 (UTC)

You would still use the word permeability, we assume for simplicity that permeability is a scalar value, but in fact it is a function of the local magnetic field and properties of the material. However, in the case of the transformer, it will have been designed so that the permeabilty is effectively a constant in the range of operation it is to be used, it will not be changed by the current in the coils. It's also not really like what he is saying, the core simply ensures that two coils have the same magnetic field passing through their centres, as one has less turns than the other the voltages (or emfs) induced are different. —Preceding unsigned comment added by 129.215.122.1 (talk) 11:29, 6 January 2010 (UTC)

@David Tombe, I think I'll try something safer at first run, wouldn't want to risk my health just because of a little experiment (I'm unexperienced at handling mercury). @Steve, wouldn't you say it is the magnetic current, or the flowing flux inside a transformer which induces the current, and not the change of its magnetic permeability? I already thought of a simple way to test this. Okey imagine this: Have one magnet (neodymium is best), placed some centimeters from a coil, where the magnet's north pole is pointing into the center of the coil (the coil does also have a solid core). The medium between them is air, and so has roughly a relative permeability of 1. If I then was to slide in between them a steel plate with a relative permeability of 700, which almost completely filled the gap between the two, I would imagine that that should affect the field somehow. Maybe what I am trying to alter here is the magnetic reluctance, because I want to give the field a shorter path into the coil, if you get that. So, if we can induce voltage and thus current in this untraditional way, is there any formulas or laws which can be of used to determine the actual induced voltage? I realize that wikipedia may not be the place to discuss engineering and such, but perhaps this could belong in the article, as yet another way to generate electricity? --Nabo0o (talk) 13:12, 18 September 2009 (UTC)

Nabo0o, You could try that. The steel plate would indeed do something to the field lines. But then of course it would be the field lines that would change in the vicinity of the coil, whereas you were really asking about the issue of changing the permeability, which is why I went for the total immersion option. Having said that, I think that even the total immersion option would suffer from the same problem as your hard shield option.

At any rate, you are basically asking if Faraday's law is curl E = -dB/dt, or if it is curl E = -μdH/dt. In other words, you are asking whether the μ is inside the brackets or outside the brackets. I don't know. It's an interesting question. My guess is that it is outside the brackets. I can't see how you could actually have a pure varying μ situation. David Tombe (talk) 13:53, 18 September 2009 (UTC)

NaboOo, upon reflection, I would like to revise my previous comment. I am now not sure of the answer. What counts is not just the change in flux linkage; what counts is the intrinsic change in the flux. The question is whether the entire flux has to change or just a region of it. If a region of the magnetic field is concentrated by a change in medium, does this constitute an intrinsic change? Does it change the total flux at all? And if the medium of the entire field is changed, does this constitute an intrinsic change? I believe that it does. Mike La Moreaux (talk) 21:44, 18 September 2009 (UTC)

The flux linkage is simply the integral of the flux density passing normal to the surface enclosed by a loop. The emf in a closed loop is given by the rate of change of flux linkage, therefore changing the flux is the same thing as changing the flux linkage.

That is interesting.... Some tests should definitely be done :) --Nabo0o (talk) 21:57, 18 September 2009 (UTC)

I just want to apologize for the lack of clearness in my question. What I really wanted to know was if there existed other ways to alter the flux density in an area, other than to either increase the current, or to move the magnetic source or coil relative to each other. Can we alter other magnetic parameters and as so create a change in the local magnetic flux? My first thought was then to alter the reluctance, because we would then be capable alternating the path which the magnetic field traveled. This could be a way to increase the efficiency of a generator, I even think I have heard of some patents doing just that....--Nabo0o (talk) 13:07, 20 September 2009 (UTC)

Nabo0o, If it were possible to change the magnetic permeability of a medium, that would change the concentration of the lines of force. That may then cause EM induction. But I can't think of any way of changing the permeability of a medium. David Tombe (talk) 09:33, 21 September 2009 (UTC)

Sorry this is too long ago for a reply, but the answer is of course; then we change the medium.

That was also my proposal, although it was not my idea originally. Just wanted to hear your views about it :) --Nabo0o (talk) 17:10, 9 March 2010 (UTC)

Nabo0o, You might be interested in Piezoelectricity. David Tombe (talk) 20:23, 22 September 2009 (UTC)

So if we have two turns in the secondary coil and the voltage in the scondary coil is two volts (dc) then the rate of change of flux is 1 webber? Confused.com —Preceding unsigned comment added by 87.194.19.201 (talk) 16:05, 15 February 2010 (UTC)

Enough of talk of falsity

I've been watching the conversation go on and on and on. While I have no care about whether or not Mike's points have been addressed (though I have been following it slightly), Mike, you are in violation of WP:NOTFORUM and WP:OR. Until such time that you can come back and have at least one peer-reviewed paper published (I suspect you'll need more than simple thought experiments), this is not the place to argue whether Faraday's law is incorrect or not.

Once that is done, feel free to come back. But until such time, I think this discussion should end, as this isn't the place. I will ask for action to be taken beyond this warning if necessary. --Izno (talk) 04:57, 24 September 2009 (UTC)

Izno, you may well have a point about the discussion not belonging in this discussion page. But why are you addressing your complaint to me? I merely pointed out the fact that Faraday's Law is false. I certainly believe that it is an important point about an article that treats Faraday's Law as a given. It is certainly not original research, as Nobel laureate Richard Feynman stated what amounts to the same thing in his "Lectures on Physics." It was also pointed out by someone else in the previous discussion of this article. The discussion really began when others challenged me in this discussion page rather than on my talk page. Mike La Moreaux (talk) 00:06, 25 September 2009 (UTC)

• TRUE: "Faraday's law can be (and often is) misstated or misunderstood, in which case it's false".
• MIKE'S ORIGINAL OPINION: "Faraday's law is always false, no matter how it's phrased or formulated".

Feynman definitely doesn't make the second statement. You should re-read that chapter more carefully. To my knowledge no one believes this but you. Izno is right, this has gone on enough. If you really still believe the second statement, then I'm done trying to convince you otherwise.

(On the other hand, I'm very happy that you appreciate the first statement, and if you ever realize your error about the second statement, I'd be happy to discuss with you how to better incorporate the first statement into the article, where I don't think it's explained especially well.) :-) --Steve (talk) 05:48, 25 September 2009 (UTC)

To Steve's first point, a review of reliable sources would actually reveal the subtleties for why the mentioned scenarios are not valid counter-examples of Faraday's law. For starters I would recommend Griffiths' "Introduction to Electrodynamics" (3rd ed) pp.301-303, Jackson's "Classical Electrodynamics" (3rd ed) pp.208-211, and for the historical aspect Darrigol's "Electrodynamics from Ampere to Einstein" pp.36-37,139,199-201,377-378 (there's a lot more in Darrigol, but I think these are the pages that touch most directly on points brought up). Suffice it to say that when Faraday's law is stated as curl E=-(partial)dB/dt (ie limiting Faraday's law to refer to electric fields induced by changing magnetic fields and not the motional, magnetic emf), it requires that the circuit/conductor in question be fixed (see the Jackson and Griffiths). Examples where the conductor is moving, like the homopolar motor and the unwinding secondary coil of a toroidal transformer with a slip-ring, are outside the assumptions used to get to that differential form (see Jackson for the full proof). However, in both of these examples, Faraday's 1832 summary of the law (Darrigol, pp.36-37), which says that the emf is caused by the cutting of magnetic curves (field lines), is still true. --FyzixFighter (talk) 08:27, 25 September 2009 (UTC)

I find myself here being in full agreement with FyzixFighter. It would perhaps help if a short note were added to the introduction to clarify this area of confusion. It might take the form of 'One of Maxwell's equations is a restricted version of Faraday's law that only caters for the time varying aspect of electromagnetic induction'.

I have shown the mathematical link between the restricted version and the full version that includes motionally induced EMF. I showed it a few sections up. But unfortunately it steps on the toes of modern terminologies by using the relationship F/q = E in the equation E = vXB. Anyway FyzixFighter, you really meant to say 'homopolar generator'. I made the same mistake further up too. I think that Mike's confusion stems from his inability to see the vXB force as an integral part of a full total time derivative Faraday's law. As for Mike's experimental scenarios, I admit that I didn't follow the details too closely and so I wasn't able to check them out. But I doubt very much if they are counter examples. I think that Mike is assuming too much when he says that the magnetic field will cancel outside the toroidal transformer. Where an EMF is induced, there will always be some magnetic field, and there will either be a vXB induced or a -(partial)dA/dt induced. David Tombe (talk) 11:59, 25 September 2009 (UTC)

I have responses to each of the three comments above, but I will post them on your respective talk pages. Mike La Moreaux (talk) 20:10, 25 September 2009 (UTC)

FyzixFighter (and Mike), we should be clear that the topic under discussion is the statement "the time-derivative of the magnetic flux through a circuit equals the EMF around that circuit, even if the circuit is moving or deforming". That's what I'm talking about above, and that's what Mike wants us to purge from the article. But in fact, it's a true and uncontroversial statement...as long as you define the terms "circuit" and "moving" and "deforming" in a rigorous way. See here for a proof, for example. It's a true statement, it's a famous statement, it's a useful statement, so it has every reason to be stated (accurately) on wikipedia. :-)

On the other hand, FyzixFighter, the statement curl E=-(partial)dB/dt is uncontroversial, everyone here believes it including Mike. The only issue is moving circuits. :-) --Steve (talk) 02:49, 26 September 2009 (UTC)

The restricted applicability of Faraday's law

I think I now know what Mike is getting at, and it may well be the same point that Steve was getting at earlier. Even the total time derivative expression at the head of the main article has restricted applicability in electromagnetism, even though it caters for the v×B force.

I assume we all agree that the law as stated at the top of the main article caters for two principles,

(1) $\mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}$

(2) $\mathbf {F} =q\mathbf {v} \times \mathbf {B}$

We further agree with the fact that these two principles account for all instances of electromagnetic induction, including the homopolar generator. But what Mike, and maybe Steve too, have been saying is that Faraday's law 'as stated' does not accurately account for the induced EMF in the homopolar generator. Faraday's law applies to a strict geometry involving closed circuits of thin wires cutting through magnetic lines of force, and the homopolar generator lies outside the scope of this law. That of course under no account means that Faraday's law is false. David Tombe (talk) 09:19, 28 September 2009 (UTC)

David, by George I think you've got it. I have just a couple of clarifications. First, I believe that it is not at all clear that the article makes the restriction on Faraday's Law that you, and Steve, have mentioned. Second, no cutting of magnetic lines of force is necessary for transformer emf, as is demonstrated by the normal alternating current toroidal transformer, although you may have just been referring to the motional emf aspect of Faraday's Law relative to the discussion of the homopolar generator. Mike La Moreaux (talk)

Mike, I removed the reference to the homopolar generator in the introduction because it is too controversial, and it doesn't need to be mentioned there. I think that Steve was unto this issue a while back, but there was a misunderstanding between Steve and myself. Steve wanted to state in the introduction that Faraday's law applies to two different laws in electromagnetism. I didn't want the introduction to state it in that explicit manner. As far as I am concerned, the law that is stated in the introduction is the original Faraday's law. Even if it doesn't cater for every scenario involving electromagnetic induction, it is still Faraday's law. There is however a problem which I have traced to Oliver Heaviside. In fact, I was reading his 1889 paper last week. Heaviside refers to curl E = -(partial)dB/dt as Faraday's law, when in fact it is merely a differential (curl) equation that deals with one of the principles that is involved in Faraday's law. This is largely the basis upon which Steve wanted to claim the existence of two Faraday's laws, whereas I wanted to leave that issue until further down the article and point out that one of the modern Maxwell's equations is commonly called Faraday's law, but that it doesn't cater for motionally induced EMF. During that debate, I drew attention to equation (D) in Maxwell's original eight equations as being the best and most general equation for induced electromotive force. Equation (D) is E = -gradψ -(partial)dA/dt + v×B.

Anyway, it now seems that even aside from those issues, there is the additional issue that even the full Faraday's law in the introduction does not cater for every instance of electromagnetic induction. I tried to visualize the homopolar disc in terms of spokes, but even then I got lost trying to match it with Faraday's law as stated. But I don't agree with you that Faraday's law is wrong, or that the introduction needs to be changed. This is a matter which needs to be elaborated on in the main body of the article.

As regards the flux linkage in the toroidal transformer, I can see how the linkage may arise in the AC scenario when the secondary coil is wrapped tightly, side by side with the primary coil. But in your DC scenario with the unwinding secondary coil, I can't see how there can be any flux linkage. In this case, are you not extrapolating Faraday's law beyond its jurisdiction? The secondary coil certainly wraps around the region where the magnetic field is contained, but it doesn't connect with it. I would always have taken it that Faraday's law assumes direct flux linkage. David Tombe (talk) 03:44, 29 September 2009 (UTC)

David, flux linkage is also referred to as the threading of magnetic field lines through a closed path. The field lines do not have to touch the path. If the number of field lines intrinsically varies with time (rather than just the number threading the path), an electric field is produced. It is this field which is responsible for the emf. This electric field can extend beyond the magnetic field. Mike La Moreaux (talk) 19:03, 29 September 2009 (UTC)

Mike, OK. But are you saying that when the secondary coil is unwound when there is a DC current in the primary coil, that you would be expecting to get an induced EMF, but that in actual fact you don't. Whereas if there is an alternating current in the primary coil, then an EMF is induced in the secondary coil even if the secondary coil is far out from the primary coil? David Tombe (talk) 20:34, 29 September 2009 (UTC)

David, yes on both accounts. Mike La Moreaux (talk) 21:52, 30 September 2009 (UTC)

Mike, OK. That is interesting. Let's deal with the dynamic AC scenario first. Clearly there is a flux linkage between the primary and secondary coils. EM radiation, which is a propagation of changing magnetic field, flows from the primary to the secondary coil and discharges on the secondary circuit so as to induce an electric current. Even though the solenoidal field lines are totally confined to the ferromagnetic core in the steady state situation, the dynamic state seems to cause this magnetic field to warp and to extend with the EM radiation to the secondary circuit. The AC scenario is pure $\mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}$ stuff. It is not $\mathbf {F} =q\mathbf {v} \times \mathbf {B}$ stuff.

Your DC scenario is definitely not $\mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}$ stuff. But I might have expected your DC scenario to also produce an EMF. I might have expected that the process of unwinding the secondary coil may have taken the form of a kind of screwing effect that would have screwed the energy from the magnetic field across to the unwinding secondary coil. But you say that that doesn't happen. The question then remains why not? I think that these are aspects relating to the mysteries of EM induction that are not explained by Faraday's law. If no EMF is generated in the secondary in your DC scenario, then clearly this is because there is no $\mathbf {F} =q\mathbf {v} \times \mathbf {B}$ force induced, which in turn is because the magnetic field is entirely inside the ferromagnetic core, and the unwinding wire does not therefore move in the magnetic field. Once again, I don't see this as a breach of Faraday's law. I believe that this scenario lies outside the jurisdiction of Faraday's law because there is no direct contact flux linkage. The fact that the AC scenario can create such direct flux linkage is beside the point. Whatever happens in the dynamic AC situation that allows the energy to get screwed from the core to the secondary coil does not seem to kick in in the DC situation, and the magnetic field remains purely solenoidal inside the core. That's how I would see it. What about if we tried your unwinding scenario with the secondary coil very tightly wound and interlaced with the primary around the core? Would that make any difference?

On thinking more about it, the two aspects of electromagnetic induction listed above do differ physically in some important respects. The time varying kind involves actual EM radiation and a warping of the magnetic field lines. The motionally induced kind does not involve EM radiation. It is about motion through an existing magnetic field and that may be the big difference between the AC toroid and the DC toroid. In the latter, the magnetic field does not leave the core. David Tombe (talk) 11:17, 1 October 2009 (UTC)

David, it is all so simple. Transformer EMF is due solely to an intrinsically time-varying magnetic field. It has nothing to do with motion, even the apparent motion of the expanding or contracting magnetic field. The changing magnetic field creates an electric field that is not necessarily coextensive with the magnetic field. If a coil is linked by the magnetic field, it is necessarily in the electric field. This has nothing to do with electromagnetic radiation. Faraday's Law purports to include both transformer EMF and motional EMF by using the ordinary derivative. While this ploy works for some cases, admittedly the usual ones, it does not work for all cases because there is actually no basis for the so-called law, at all. Therefore Faraday's Law fails to give an EMF for some cases where there is one and gives an EMF for some cases where there is none. The secondary coil is definitely flux-linked regardless of whether there is AC or DC in the primary. In the AC case, there is an EMF in the secondary regardless of whether it is unwound or not. In the DC case, there is no EMF even when the secondary is unwound. When the secondary coil is unwound, the flux linkage decreases in either case. But in the DC case, neither transformer EMF nor motional EMF applies. Faraday's Law, being defective, however, specifies an EMF. The case where the primary and secondary windings are interlaced really muddies the water. The secondary would be partially in the magnetic field of the primary and the field would probably be very complicated. There might be some motional EMF due to the unwinding, but I doubt that it would be the amount predicted by Faraday's Law. Mike La Moreaux (talk) 00:13, 2 October 2009 (UTC)

Mike, When $\mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}$ is involved, there will be EM radiation. That equation is a constituent component of the EM wave equation. The changing magnetic field will propagate at the speed of light. In the AC transformer, energy in the form of a propagated change of magnetic field will flow from the primary to the secondary. This means that the simple solenoidal field lines in the core that would exist in the steady state will necessarily be re-arranged in the dynamic state, and the effect will extend to the secondary coil. It is this definite linkage between the two coils in the dynamic state that makes it fundamentally different from the DC steady state scenario. I can't explain to you why EM radiation flows from the primary to the secondary, but my guess is that in doing so, it is the path of least resistance for the pressure as the magnetic field builds up around the primary on each cycle.

In the steady state case, the iron core absorbs all of the magnetic field and there is no magnetic field outside the primary coil. Hence the secondary coil beyond will not be in the magnetic field, and when we move it, nothing will happen. I don't see this as being a breakdown of Faraday's law. I see this as being a case of no actual flux linkage. I know that your secondary coil totally encircles the region where the magnetic field exists, but it is not touching it. Faraday's law 'as stated' may appear to apply to that scenario. Is that what you are saying? In other words, Faraday's law 'as stated' breaks down for your DC toroidal scenario?

Yes, I see what you are saying now. Your DC toroidal scenario breaches Faraday's law 'as stated'. I would however object to that assertion on the grounds that where it may break the letter of the law, it doesn't break the spirit of the law because the magnetic field is not touching the secondary coil. David Tombe (talk) 01:02, 2 October 2009 (UTC)

David, yes there will be EM radiation, but it has nothing to do with induced EMF in our discussion. Again, any EMF in the secondary is due to the electric field, not the magnetic field. Linkage between a magnetic field and a closed path, a circuit in this case, has nothing to do with whether the magnetic field touches the closed path or not. The essential requirement is that it threads the closed path, meaning that there is a net penetration of the surface whose boundary is the closed path. Mike La Moreaux (talk) 19:46, 2 October 2009 (UTC)

Mike, Let's deal with the DC situation first. I see exactly what you are saying. The magnetic field in the core goes through the circuit loop, the unwinding motion changes the flux linkage, but no EMF is actually induced. You think that this is a counter example to Faraday's Law. I see why you are saying that, and you may be correct as far as the letter of the law is concerned. But the magnetic field is totally inside the ferromagnetic core, and the secondary coil does not touch that magnetic field, so I would say that in spirit, this scenario lies outside the jurisdiction of Faraday's law. OK, yes it's a counter example, but in my view it is a very pedantic counter example. It's more on the level of a conjurer's counter example.

Now your attitude to the DC scenario is very much influenced by your attitude to the AC scenario in that you think that the linkage situation is similar. I don't think that it is. I think that in the AC situation, the linkage is very real. As the current builds up in the primary coil, energy expands both into the magnetic field in the iron core and also right across to the secondary coil. The changing magnetic field is propagated to the secondary coil at the speed of light, and that is of course in the form of EM radiation with the frequency of the AC supply.

You have said that it is the electric field which causes the induced current in the secondary. That is true. But that electric field is intricately intertwined with a magnetic field and they are both alternating with the frequency of the AC source. That $\mathbf {B}$ field and that $\mathbf {E}$ field are joined at the hip, and mutually orthogonal. The $\mathbf {B}$ field satisfies $\mathbf {B} =\nabla \times \mathbf {A}$, and the $\mathbf {E}$ field satisfies $\mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}$. (note that the $\mathbf {E}$ field is not the electrostatic $\mathbf {E}$ field from Coulomb's law).

No such actual linkage arises in the DC scenario. Your unwinding coil is merely unwinding in a region of zero magnetic field. (zero as far as the circuit is concerned. There is always some magnetic field everywhere, even if it is only the Earth's magnetic field). No substantial $\mathbf {v} \times \mathbf {B}$ force is induced because the $\mathbf {B}$ field from the primary electric circuit doesn't extend out to there. David Tombe (talk) 00:00, 3 October 2009 (UTC)

David, at any low frequency, like 60 hertz, the EM radiation will be at an extremely small level. It is a completely separate phenomenon from transformer EMF. The electric field induced by transformer EMF is not intricately intertwined with the magnetic field. There is no magnetic field, due to cancellation. Mike La Moreaux (talk) 00:45, 3 October 2009 (UTC)

Mike, Just a question. Does this AC scenario only work if the secondary coil is wound tightly around the core side by side with the primary coil? If so, the energy transfer effect may be direct, hence skipping the interconnecting EM radiation stage. David Tombe (talk) 14:20, 3 October 2009 (UTC)

David, no, the secondary coil does not have to be at all tightly wound. It can be as loose as one desires, thousands of miles in extent. Check out physics textbooks on this, transformer EMF, given by one of Maxwell's Laws. Just avoid any that mention the cutting of the wire of the winding by expanding or contracting magnetic flux lines. They are incorrect. Mike La Moreaux (talk) 18:56, 3 October 2009 (UTC)

Mike, In that case, it seems that you are perceiving the induced EMF in the secondary coil to be induced at a distance from the primary coil without any involvement of an intervening magnetic field. I would see that induced EMF as being connected to a magnetic field through the equations $\mathbf {B} =\nabla \times \mathbf {A}$ and $\mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}$, with the effect having been propagated from the primary coil at the speed of light, and in a cyclic manner with a frequency of 60Hz or whatever. In other words, the energy that has flowed from the primary coil to the secondary coil is electromagnetic radiation in its most basic form. The question mark then hangs over the issue of the magnetic field in the space between the two coils if the circuit magnetic field is entirely contained within the ferromagnetic core. There still nevertheless has to be some kind of magnetic field in the space between the two coils for that magnetic field to be changing in the EM radiation. And there always will be a magnetic field present, even if it is just the Earth's magnetic field. And it is those equations of EM radiation that link to the so-called Faraday's law of Maxwell's equations.

Your argument seems to be that since the circuit magnetic field is entirely within the core, then the magnetic field does not have to come out and touch the secondary coil, even in the AC scenario. Hence your DC scenario, which also involves no magnetic field in the region of the secondary coil, is a counter example to Faraday's law because no EMF is induced when the secondary coil is being unwound, and hence changing the flux linkage.

The way I see it is that all electromagnetic induction involves the two principles,

(1) $\mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}$

(2) $\mathbf {F} =q\mathbf {v} \times \mathbf {B}$

The full Faraday's law at the beginning of the main article involves both of those two principles, and it applies to most scenarios involving electromagnetic induction. In my view, your AC toroidal scenario does involve contact between the secondary coil and a changing magnetic field, but although your DC toroidal counter example seems to break the letter of Faraday's law, it is not a sufficient basis to state that Faraday's law is wrong. However, it may make an interesting section in the article. I cannot however guarantee that such a section would not be reverted by somebody, and so I suggest that if you wish to add a mention of this fact in the article, that you have a source to back it up. David Tombe (talk) 20:27, 3 October 2009 (UTC)

Again, the induced EMF of the transformer EMF type is not dependent upon EM radiation. The fact that an electric field is produced by a time-varying magnetic field is just a given. There is no known mechanism. That is why it is a law. In the AC version of the toroidal transformer, there is a much reduced magnetic field outside of the primary winding. It cannot account for the magnitude of the EMF in the secondary. Mike La Moreaux (talk) 20:37, 4 October 2009 (UTC)

Mike, You seem to have this idea that the EM induction mechanism is not related to EM radiation. You are overlooking the fact that the two mechanisms are joined at the hip. Equation (1) above is an integral part of the EM wave equation. Where we have time varying EM induction, we will have EM radiation, and vice-versa. The EM radiation is not some mere side effect, or energy leakage factor. The EM radiation is the very EM induction process itself. And outside of the primary coil, it doesn't matter what magnitude the magnetic field is. The issue is about the magnitude of the rate of change of the magnetic field. In the dynamic state, the solenoidal field line pattern breaks down. The magnetic lines of force will be constantly breaking and re-joining in the dynamic state. But one thing is sure, and that is that the secondary coil will be in direct contact with a changing magnetic field.

Think of the iron core as a storage tank. When the primary circuit is switched on, in the absence of any secondary coil, energy will be radiated from the primary circuit and stored in the tank. If however, we also have a secondary coil, the energy will flow to both the tank and the secondary coil. When the steady state is reached, the energy flow will cease and we will be left with a steady pool of solenoidal energy in the tank. If we now unwind the secondary coil, nothing happens because the secondary coil is not immersed in the tank of magnetic energy, and hence no v×B interaction occurs. You see this as being a breach of Faraday's law. I see it as being a very cheap breach of the letter of Faraday's law. Quite frankly, I don't think that Faraday's law applies to this latter DC scenario because the secondary coil is not in the magnetic field in the steady state. In the dynamic state however, the secondary coil is immersed in a changing magnetic field. David Tombe (talk) 02:51, 5 October 2009 (UTC)

David, at say 60 Hz, the efficiency of the primary coil as an antenna is extremely small, since the wavelength of the radio transmission is thousands of miles long. So the EM radiation is indeed a kind of leakage, and in this instance, a very tiny one. Your explanation seems particular to you. Mike La Moreaux (talk) 01:31, 6 October 2009 (UTC)

Mike, In the AC scenario, energy goes from the primary coil to the secondary coil. If that energy flow does not, as you claim, constitute EM radiation, then what does it constitute, and what speed does it flow at? The EMF that is induced in the secondary coil satisfies the equation $\mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}$. It doesn't matter how small the underlying magnetic field is that surrounds the secondary coil. The important factor is the rate of change of that magnetic field. Think of $\mathbf {E}$ as an acceleration and $\mathbf {A}$ as a velocity. It's possible to have a very large acceleration in conjunction with a very small velocity. You seem to think that there is negligible magnetic field at the secondary coil. Maybe in the steady state. But all I can say is, that when an AC current flows in the primay, and an EMF is induced in the secondary, there will be a changing magnetic field in the vicinity of the secondary coil with a magnitude commensurate with the relevant equation of EM induction. So we will have a net flow of energy through space, going from the primary coil to the secondary coil, in conjunction with a cyclically changing magnetic field. We know that the EM induction equation $\mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}$ is an integral component of the EM wave equation, and so I can't see how you can deny the involvement of EM radiation in the EM induction process for the AC scenario. The entire EM induction process IS EM radiation.

And it's because you can't see the changing magnetic field in the AC scenario that you think that the DC scenario breaches Faraday's law, because you have convinced yourself from the AC scenario that the secondary coil doesn't have to actually touch the magnetic field. That is where you are going wrong. A changing magnetic field touches the secondary coil in the AC scenario, but in the DC scenario the secondary coil is not immersed in either a steady state magnetic field (apart from the nominal background magnetic field), or a a changing magnetic field, and so nothing happens and Faraday's law does not apply.

Since writing the above, it has occurred to me that your understanding of EM radiation seems to be focused on the far field scenario. Transformers are of course near field stuff. The secondary coil is submerged right inside the near field. David Tombe (talk) 09:57, 6 October 2009 (UTC)

David, for a given frequency, the rate of change of the magnitude has to be proportional to the magnitude. It is a fact that for a toroidal winding the external magnetic field is greatly reduced. You do not have to take my word for it. See the article titled "Transformer" and look for "Toroidal cores" under "Cores." The magnitude of the external magnetic field is independent of whether there is AC or DC. I do not know where you came up with your ideas about induction and the necessity of the magnetic field touching a conductor. Maxwell's Law certainly does not require it. Mike La Moreaux (talk) 19:45, 6 October 2009 (UTC)

Mike, Are you then saying that the EMF is induced in the secondary coil as a consequence of some mysterious kind of instantaneous action-at-a-distance, without the involvement of a changing magnetic field? I am saying what I am saying because it follows directly from the principles of EM induction. And I might add that frequency has got nothing to do with any of this. It only takes the magnetic field to be changing for the EMF to be induced. That change doesn't have to be cyclical. And the change can be in conjunction with a magnetic field that began at close to zero in value. David Tombe (talk) 03:48, 7 October 2009 (UTC)

David, there is an involvement of a changing magnetic field. It is in the core but does not have to extend out to where the secondary coil is located. A changing magnetic field produces an electric field. It is this electric field that is solely directly responsible for the EMF in the secondary coil. I agree with everything else you say. Mike La Moreaux (talk) 19:58, 7 October 2009 (UTC)

Mike, I'll agree with you that in the steady state situation, the magnetic field is entirely inside the core of the primary, and not touching the secondary coil. But according to you, when the dynamic situation arises, the magnetic field in the core changes and induces an EMF in the secondary coil, at-a-distance, without touching the secondary coil.

Think carefully about what you are saying. First of all, think about the changing magnetic field inside the core. Try and imagine the solenoidal lines of force in the steady state situation and then try and imagine what is happening to them in the dynamic state. In the dynamic state, magnetic field lines break and re-join with other magnetic field lines. (In radial field lines, this can happen too when sources and sinks are swapped in the dynamic state.)

Anyway, can you imagine the breaking and re-joining of these magnetic field lines occurring in the absence of some effect propagating beyond the primary?

And if there is no 'changing effect' propagating beyond the primary, what is the basis of the energy flow from the primary to the secondary? What speed would it flow at?

How can you address these questions without contemplating the idea of a changing magnetic field propagating from the primary to the secondary, fully in line with $\mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}$ at any point in the path, and hence in line with both the Faraday's law of Maxwell's equations, and with the EM wave equation?

But I can see also what you are thinking. You are thinking that since there was no existing magnetic field in the region beyond the primary, that the changing magnetic field in that region would not be of a very great magnitude. It's an interesting issue and it has got me thinking. If there was no secondary coil present at all, would there still be this outgoing radiation in the dynamic state? I would suspect 'yes'. I would also suspect that the presence of the secondary coil actually dramatically increases the amount of this energy flow outwards from the primary. This then gets us into the mysterious issue of electric current knowing what lies ahead. It gets us into the mysteries of what happens when we first switch on an electric circuit and why the electric field crosses the space between the 'in' wire and the 'out' wire.

There are alot of unknowns in this, but I don't see any breaches of Maxwell's equations or of the full Faraday's law in the AC scenario, and I only see a technical 'letter of the law' breach of the full Faraday's law in the DC scenario. David Tombe (talk) 01:25, 8 October 2009 (UTC)

David, only the electric field need propagate beyond the primary winding, and I assume that it does so at the speed of light. The presence or absence of the secondary coil has no essential effect. One can choose any one of an infinite number of closed paths in the electric field which thread the hole in the toroidal primary winding (the donut hole), and each one will have, in this special case of a toroidal primary winding, the same EMF. (That is, of course, assuming no magnetic field external to the primary winding, which is an approximation.) Of course, if the closed path threads the hole more than once, as in a winding, that EMF must be multiplied by the number of turns. There are never any breaches of Maxwell's Laws, and my counter example to the full Faraday's Law only applies in the DC case. Mike La Moreaux (talk) 18:59, 8 October 2009 (UTC)

Mike, You're predicting a propagating electric field that doesn't have a magnetic field component? What mathematical function describes this electric field? As far as I am concerned, the only propagating electric field that could be possible in the circumstances is the one described mathematically by the equation $\mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}$. It therefore automatically involves an orthogonal magnetic field joined at the hip. David Tombe (talk) 00:42, 9 October 2009 (UTC)

David, I was talking about just the establishment of the electric field, not its existence. If the magnetic flux in the core changes at a uniform rate, then the electric field it produces will not vary with time and therefore have no orthogonal magnetic field. Mike La Moreaux (talk) 21:30, 9 October 2009 (UTC)

Mike, But that electric field can only be established by a changing magnetic field. The two are inseparable, and the maths says that when the induced electric field arises that it must also be changing in time. Don't forget about the sister equation $\nabla \times \mathbf {B} =\mu _{0}\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\$.

As far as I am concerned, the primary is a transmitter and the secondary is a receiver of EM radiation. Energy in the form of EM radiation flows at the speed of light from the primary to the secondary. OK, in the steady state, the magnetic field is entirely within the primary core. It forms neat solenoidal rings of force inside the core. But as soon as we have an AC situation, the solenoidal lines all break up and re-arrange and there will be an outward expansion effect to the secondary coil. David Tombe (talk) 08:17, 10 October 2009 (UTC)

David, no, in the AC case, as well as the DC, in the case of a toroidal coil, the external magnetic field is greatly reduced by symmetrical cancellation. Now in a case which is neither AC nor DC, if the flux increases at a uniform rate, the electric field induced will be constant. Thus, its time derivative will be zero, and, according to your equation, the curl of B will be zero, and thus there will be no external magnetic field due to the electric field. Mike La Moreaux (talk) 23:57, 10 October 2009 (UTC)

Mike, Just because the curl of a magnetic field is zero doesn't mean that the magnetic field itself is necessarily zero.

Anyway, you are focusing on a particular hypothetical situation in which the induced EMF is constant. I'm not sure that such a situation could easily be brought about in practice, and it certainly doesn't arise in connection with a toroidal transformer, whether we have AC, or whether we consider the situation in which the current in the primary is increasing or decreasing. It looks theoretically possible just from the maths, on the basis of having a flux linkage being made to change at a constant rate. But can you think of a way of bringing that about in practice?

Time varying EM induction always relates to an induced EMF that is itself also time varying. Even in motionally induced EMF, I doubt if we could ever generate a constant EMF.

If it were possible to induce a constant EMF from a changing flux linkage, it would shatter the entire linkage between EM radiation and EM induction. The concept simply doesn't tie in with the wider inter-relationships between $\mathbf {E}$ and $\mathbf {B}$. What looks theoretically possible based on the single equation $\mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}$ in fact turns out to be impossible in practice when the wider inter-relationships are considered.

This is important because you have made yourself believe in the possibility of an EMF being induced at a distance without any direct contact with a magnetic field. You are trying to convince yourself that in the AC toroidal scenario that no magnetic field is touching the secondary coil. But there is. A changing magnetic field is breaking out from the primary core in the dynamic state and radiating to the secondary. But because you have convinced yourself that there is no direct contact magnetic field involved in the AC scenario, and yet still believe that Faraday's law is upheld, you have then convinced yourself that Faraday's law doesn't require the secondary to have direct contact with a magnetic field. And because you have convinced yourself of that fact, you have then subsequently convinced yourself that Faraday's law breaks down in the DC scenario. I on the other hand, don't see Faraday's law as having broken down in the DC scenario because I insist that the magnetic field must be touching the secondary coil in order for Faraday's law to apply, and we are both agreed that in the DC scenario, the magnetic field doesn't touch the secondary coil.

Yes, I think that the joker in the pack has been your belief that we could ever actually generate a constant EMF in practice, in relation to the principles of electromagnetic induction. David Tombe (talk) 04:09, 11 October 2009 (UTC)

David, I believe that your problem is that you approach physics from the math rather than the other way around. Math is used to support physics rather than the inverse. Relying on the math rather than using physical reasoning can lead to problems. Yes, just because the curl of the magnetic field is zero, it does not imply that the magnetic field is zero, but in this case that we are discussing, any magnetic field resulting from the electric field is specified solely by the curl of the magnetic field, so if there is no curl, there is no magnetic field. Just another example of knowing the physics leading to the correct interpretation of the math. Whether or not producing a constant EMF is difficult in practice or not is hardly the point. We are discussing the theoretical implication. I do not know why you say that it certainly does not arise in connection with a toroidal transformer. It would just require a ramp function for the current in the primary. I do not know why you say that time-varying EMF induction always relates to an induced EMF that is also time-varying as neither the math nor the physics require it. You see my point exactly when you say that inducing a constant EMF from a changing flux linkage would shatter the connection between EM radiation and EM induction. That is exactly what I was demonstrating with that particular situation. The lack of the necessity of the touching of the secondary by the magnetic field is not directly related to the discussion of Faraday's Law. It is strictly a function of the relevant Maxwell's Law. It is in my college physics textbook. It would seem that we are never going to agree on this matter. We are coming from different directions, entirely. We have gotten off the subject of Faraday's Law and into a discussion of transformer EMF, alone. If you want to continue, and I do not hold out much hope of it being productive, why don't you post future comments on my talk page. Mike La Moreaux (talk) 20:41, 11 October 2009 (UTC)

Mike, In the past I've been accused of both extremes. I've been accused of being a maths junkie, and I've been accused of not considering the maths. In actual fact I have made strenuous efforts to keep the maths on track with the physics, and I've been a fierce critic of those who have been carried away by maths to the extent that the maths has long gone off the rails and lost any connection with the physics that it is supposed to be describing. As regards Maxwell's equations, few have made as much effort as I have to highlight the connection that these equations have with physical reality.

Anyway, we have now agreed on the exact point of dispute. You believe that it is possible to have an induced EMF that is constant. I don't. The maths may allow for it in theory when we consider the $\mathbf {E} =-{\frac {\partial \mathbf {A} }{\partial t}}$ equation in isolation. But when we consider the other Maxwell's equations and put it all together, we end up with the EM wave equation. I assume that you have seen the derivation of the EM wave equation. Modern textbooks use the Heaviside-Maxwell curl equations, one of which is the one that Heaviside called Faraday's law, and the other is the Ampère-Maxwell law. Maxwell himself derived the EM wave equation slightly differently, but the underlying principles are the same. The overall picture is that the dynamic state involves both a dynamic $\mathbf {B}$ field and a dynamic $\mathbf {E}$ field. There is no allowance for a static $\mathbf {E}$ field with that formula. We can of course have a static $\mathbf {E}$ field in connection with Coulomb's law but that has got nothing to do with EM induction. I know you have suggested that increasing the current in a wire in line with a special ramp function might result in a steady induced $\mathbf {E}$, but I can guarantee that it won't work. It is not in the nature of things for it to work. At every point in space where $\mathbf {B}$ is changing, the associated $\mathbf {E}$ will be changing too. They are cyclically inter-related and they are always in phase with each other.

If you were correct, then it would be possible to have a flow of energy from a primary to a secondary without the involvement of EM radiation. What would the energy flow mechanism then be? David Tombe (talk) 07:47, 12 October 2009 (UTC)

David, I am afraid that you and I just have a different understanding of induction. I believe it to be independent of EM radiation. I also certainly believe that a constant induced EMF is possible and that it does not require the magnetic field to be present where the electric field is. I have enjoyed discussing this with you but think that I will stop, now. Perhaps we will discuss something else in the future. Mike La Moreaux (talk) 00:27, 13 October 2009 (UTC)

Mike, Sorry for not getting back to you earlier. And yes, I think that the discussion at has indeed run its course. I did learn alot from it, and I'm grateful to you for raising those issues. You introduced some very interesting case scenarios. I can see that our main clash lies in the fact that I see the energy transfer from the primary to the secondary, in the case of time varying induced EMF, as constituting EM radiation. You on the other hand don't. I don't of course see EM radiation as being involved in the motionally induced EMF as per v×B. (Maxwell actually eliminated v×B when he derived the EM wave equation in 1864)

It was a very interesting discussion, but I couldn't help sometimes thinking that you are perhaps too focused on trying to find flaws in Faraday's law. Sometimes the concept of 'conjuring tricks for physics audiences' flashed through my mind. I think that you may just about have a technical breach with the DC toroidal scenario, but I think that even that has been largely influenced by your belief that the (changing) magnetic field doesn't actually directly touch the secondary coil in the AC scenario. I think that it does.

I gave further thought to your idea of the ramp function generating a steady EMF. I still don't think that it could ever be possible in practice, because it would ultimately break down on the microscopic scale. But let's just say for the sake of argument that it is possible. In the AC scenario, we have a time varying induced EMF and hence no obstacle to the applicability of the EM wave equation. So is it not highly unlikely that EM theory is suddenly going to crash, just because we increase the current in the primary at a very particular 'ramp function' rate? David Tombe (talk) 04:01, 15 October 2009 (UTC)

David, OK, one more comment. You have summed things up well. I believe that I should also sum up by returning to my original complaint. Virtually all the textbooks and encyclopedias seem to treat Faraday's Law as describing a physical principle, namely that a flux change due to an intrinsic change in the flux , or due to motion, induces an EMF in a circuit. This is false. In every instance where the flux changes due to motion, the EMF is actually motional (v x B). The associated flux change is just along for the ride. It is like guilt by association. The change in flux happens to accompany most cases of motional EMF, even though they are independent. In the case of the homopolar generator, the two are separated, in effect, because there is no flux change, just pure motional EMF, and Faraday's Law fails completely. This is not to disparage Faraday. When he formulated Faraday's Law it was an entirely reasonable thing to do. Later on, however, the subject became better understood with Maxwell's Laws and the expression for the Lorentz force. Faraday's Law at that point was obsolete except as an engineering convenience, yet modern physics does not seem to recognize the fact; Feynman seems to have been ignored. Mike La Moreaux (talk) 20:24, 16 October 2009 (UTC)

Mike, I would agree with you that Maxwell's equation (77) of the 1861 paper, which became equation (D) of the original eight Maxwell's equations in the 1865 paper, is indeed a superior law to Faraday's law, in that it unequivocally caters for all aspects of electromagnetic induction. It unequivocally states the full forms of both kinds of induced EMF.

However, the full version of Faraday's law does nevertheless cater for the two aspects that are involved in Maxwell's equation (77). I think that the main problem with Faraday's law in your mind is that perhaps it uses archaic terminologies which make it difficult to correctly apply to some of the more obscure scenarios.

The Lorentz force is interesting because it is essentially a re-write of Maxwell's equation (77). But we must remember that in Maxwell's equation (77), the v×B term is exclusively an EMF that drives an electric current. With the Lorentz force, the v×B force can also apply to the mechanical force on a current carrying wire. In Maxwell's 1861 paper, the motor version of the v×B force is dealt with at equation (5), and I don't think that I ever found an equivalent to equation (5) in the original eight Maxwell's equations in the 1865 paper.

Hence the Lorentz force brings back both the equation (5) and the equation (77) versions of the v×B force all in one. It was Heaviside who lost the v×B force with his 1884 re-formulation. People have forgotten that Maxwell had already derived it, and so they now credit v×B to Lorentz.

And finally, Maxwell did actually have an equation, at equation (54), which Heaviside referred to as Faraday's law. It is not exactly the same as the full Faraday's law. It is a differential curl equation, and as you know, it appears in the modern Maxwell's equations, but it didn't appear in Maxwell's original eight equations. The Heaviside/Maxwell/Faraday curl equation (54)(1861) only caters for the time varying aspect of EM induction. David Tombe (talk) 03:12, 17 October 2009 (UTC)

Second Statement of the Law in Lead Section and Changes to Its Last Word

It’s true that some succinct statements of physical theories aren’t necessarily readily understandable by laypeople. But in my opinion, scientists should strive to make them as understandable as is feasible without reducing their descriptive power.

In this particular case, it is clear to me that the sentence, “The EMF generated is proportional to the rate at which flux is changed” is a correct common-language equivalent of the mathematical statement given below in the article: ‘’|ε| = |dΦ_B/dt|’’ (Note that it is shown graphically in the article rather than typed out as I have done here.)

I also find that the replacement of the final word, “changed”, with “linked” in this sentence does not really make the sentence a better description of the phenomenon in question. In fact it probably confuses the reader.

I cannot find in a brief search of Wikipedia itself, or of various online dictionaries, that the term “to link” has a special meaning in a math or science context that would render this sentence more comprehensible to me. But if I have simply failed to find it and someone more familiar with the subject than I would like to explain it to me, please do so.

I also cannot find (via a relatively brief Google search) an independent source for the statement, "The EMF generated is proportional to the rate at which flux is linked":

http://www.google.com/#q=%22The+EMF+generated+is+proportional+to+the+rate+at+which+flux+is+linked%22&hl=en&safe=off&start=30&sa=N&filter=0&fp=459800ffd7291a29

Accordingly I’m inclined to assume that the final word of the statement was simply a typographic error by the sentence’s original author, and I am restoring the edit (00:19 UTC 16 May 2010) that altered the final word to “changed”.

More info which might help to clarify why I think what I do about this matter:

The statement in question (the sentence ending in “linked”) was written on 15 March 2009 by an IP user: http://en.wikipedia.org/w/index.php?title=Faraday%27s_law_of_induction&action=historysubmit&diff=277498352&oldid=276680618

The same IP address was used within several minutes to modify this article as well as the closely related Electric field article: http://en.wikipedia.org/wiki/Special:Contributions/88.97.205.98

From inspection of all these edits (i.e. those starting at 20:59 UTC 15 March 2009 and ending at 22:12 UTC on the same date), I think the editor was the same between all of them, and that he was making good faith edits and furthermore generally knows what he’s talking about with respect to electrodynamics.

A relatively juvenile edit about an hour earlier on the article Romeo+Juliet might be cause for suspicion. But the time difference is great enough that IMO it’s probable that the IP was reassigned to another user between the two edits, and that the user responsible for the silly film edit is not the same person as is responsible for the edits to this and the Electric field articles. Of course if it can be shown that the address in question was statically assigned at the relevant time, then it would suggest (but not be dispositive of) the opposite conclusion.

Riyuky (talk) 17:12, 4 June 2010 (UTC)

Given that neither version is referenced, I propose to take it out entirely.- Wolfkeeper 22:33, 30 May 2010 (UTC)

Obscure explanations of equations

Could someone please explain more about the equation |E|=N|d(phi)/dt especially about d(phi) and dt because right now it is only geared towards extremely experienced physics professors, and not towards those who may need it more. Thanks — Preceding unsigned comment added by HBO134 (talk • contribs) 03:57, 10 March 2011 (UTC)

Faraday's Law is False!

Actually, there are two versions of Faraday's Law. One is one of Maxwell's Laws. It is responsible for transformer emf. The principle of it is that an intrinsically time-varying magnetic field produces an electric field. An intrinsically time-varying field is one which varies as a whole with time. The law is in the form of an equation which gives the emf of a closed path (not necessarily conducting) in the electric field when the time rate of change of the magnetic flux is known. There is a widespread misconception that the magnetic flux lines of an increasing or decreasing magnetic field cut a conductor present and induce an emf. In fact, the conductor does not have to be in the magnetic field to have the emf. This can be shown by the toroidal transformer. Assume that the primary winding is wound on the toroidal (donut shaped) core and that the secondary winding is wound over the primary winding. Because of the geometric symmetry of the primary winding, its magnetic field is essentially confined to within the primary winding and cancels itself outside it. Therefore, the secondary winding is not in the magnetic field of the primary winding! Yet the toroidal transformer works. The secondary winding is, however, linked by the magnetic field of the primary winding. This demonstrates that the apparently moving magnetic flux lines of an increasing or decreasing magnetic field are not really moving in the same sense as those of a moving magnet.

The other, more common, version of Faraday's Law purports to combine that one of Maxwell's Laws with motional emf. This version is that one of Maxwell's Laws with the partial derivative changed to the ordinary derivative. It is false! Motional emf is just the direct application of the definition of the magnetic field. The two principles involved are independent and cannot be combined into a single term of an equation. Richard Feynman pointed out that Faraday's Law does not always work. He gave a counter example, and there are others. A physical law has to work in every instance; otherwise, it is not a law. While Faraday's Law works in most instances, it is not because it is based upon a valid principle. It is a rule of thumb and a convenience rather than a law.

Because Faraday's Law is false, it cannot be used in an argument. There is a lot of nonsense in areas related to Faraday's Law which can easily be cleared up once it is realized that Faraday's Law is false. For instance, it is known that it is just the relative motion between a conductor and a magnet that is relevant in motional emf. This shows that the magnetic field of a magnet of a magnet does move with the magnet, because knowing that moving the magnet produces an emf in the conductor and knowing that Faraday's Law does not apply because it is false, any variation in the magnetic field with position is irrelevant. Mike La Moreaux (talk) 21:30, 6 September 2009 (UTC)

Mike La Moreaux, This issue has been well discussed in the past. There is only one Faraday's law and there are two aspects to it. See the section two up from this one. David Tombe (talk) 10:20, 9 September 2009 (UTC)

David Tombe, did you read beyond the first sentence of my comment? I agree that there should be just one version of Faraday's Law. There is no reason to give more than one name to an existing law. But the textbooks seem to give the two versions, although there is a lot of confusion about it, as with everything relating to Faraday's Law.

As to there being two aspects of it, it is kind of silly to speak of two aspects to something which is false. Faraday's Law does include every case of transformer emf and most of the cases of motional emf, but most is not good enough. The law is false. It is purported to be based upon the two valid principles but is, in fact, based upon no principle at all. It cannot be derived from any valid principle. The fact that it gives a correct answer in a case of motional emf is just coincidence. Mike La Moreaux (talk) 00:32, 11 September 2009 (UTC)

"The EMF around an infinitely-thin wire loop is the derivative of the magnetic (B) flux passing through it". Do you agree that this is a true law? If not, what is a counterexample? (Feynman's counterexamples are things that are not wire loops, as I recall.) --Steve (talk) 05:40, 13 September 2009 (UTC)

Steve, no I do not believe that it is a true law. It would be if the derivative were the partial derivative, thus making it one of Maxwell's Laws. A counter example would be a single-layer coil wound on an iron core with a constant magnetic flux passing through it lengthwise. One end of the coil is connected to a slip-ring whose brush is connected to the return circuit, the other end of which is connected to the other end of the coil. Allow the coil gradually to be unwound. The flux will obviously decrease, but there will be no emf because neither motional emf nor the relevant Maxwell's Law applies. Mike La Moreaux (talk) 21:29, 13 September 2009 (UTC)

Mike, The establishment believe it to be true, and it fits perfectly with my own views. I have done alot of research on the topic. Are you sure that the counter examples that you have in mind are not due to a confusion of terminologies as regards the two aspects? Please describe one of your counter examples. If Faraday's Law breaks down, it will send shock waves right through the linkage between electromagnetism and the speed of light. In my view, that linkage is beyond negotiation. David Tombe (talk) 10:40, 13 September 2009 (UTC)

David, ironically, a counter example is Faraday's disk dynamo or homopolar generator. Assume that the magnetic flux is constant and uniform over the entire disk. This is a steady-state case; therefore there is no variation of the flux with time. In addition, at least to a first approximation, the magnetic flux lines are parallel to the plane of the electric circuit, and thus there is no flux linkage, anyway. Mike La Moreaux (talk) 21:29, 13 September 2009 (UTC)

Mike, But I would have assumed that since the disc rotates on its magnetic axis, that the radial electric current in the disc would move through the disc at right angles to the magnetic field, and that the F = qvXB force would be induced to act on the current. The rotation would then be due to the reaction force. I think that the problem here is not so much Faraday's law itself, so much as a lack of understanding of the physical nature of the magnetic field, and how the magnetic field is embedded in the ferromagnetic material. The electric current keeps to its path when the rotating system is constrained. Therefore, what is the rotating disc pushing off against? There must be an aspect of the interior ferromagnetic field that is firmly embedded in the ferromagnetic material. David Tombe (talk) 06:27, 14 September 2009 (UTC)

David, you do not seem to be very familiar with the homopolar generator. The disk is copper and the source of the magnetic flux is external to the disk with the flux lines parallel to the axis of the disk. Also, please see the other counter example that I described in my response to Steve on 13 September. I see that 72.64.33.173 in his/her comment dated 16:35, 4 February 2009 agrees with me that Faraday's Law does not always work and is therefore not a true law. Mike La Moreaux (talk) 20:31, 15 September 2009 (UTC)

Mike, in slip rings and disk dynamos, current passes through something that is not a thin wire. A disk is not a thin wire. Once again: "The EMF around an infinitely-thin wire loop is the derivative of the magnetic (B) flux passing through it". Name your counterexample. :-) --Steve (talk) 06:57, 14 September 2009 (UTC)

David, you can dispense with the slip ring. If an infinitely-thin is used, twisting would not be a problem. Where did you come up with such a restrictive version of Faraday's Law? An infinitely thin wire precludes any practical application. Mike La Moreaux (talk) 20:31, 15 September 2009 (UTC)

Steve, At any rate, I don't think that Faraday's law breaks down, even in the homopolar motor. David Tombe (talk) 11:55, 14 September 2009 (UTC)

Oh, I'm with Mike and Feynman on that. Faraday's law has to be stated correctly in order to be correct. Slip rings and homopolar motors would be a counterexample if you state the law too generally. --Steve (talk) 16:41, 14 September 2009 (UTC)

Steve, So you're saying that Faraday's law breaks down in the homopolar motor. I don't agree with that. Please explain clearly what aspect of Faraday's law breaks down. David Tombe (talk) 07:49, 15 September 2009 (UTC)

Sorry, I meant homopolar generator. Mike and I are talking about this: [1] --Steve (talk) 20:39, 15 September 2009 (UTC)

Mike, I came up with this "restrictive version of Faraday's Law" by looking at any textbook that proves the law (based on Maxwell's equations and the Lorentz force law), and what the assumptions are that are stated at the start of the proof. The proofs I've seen assume an infinitely-thin wire. Therefore the law is true with an infinitely-thin wire. Maybe it's true more generally, but that's at least a starting point...something that I hoped you could agree is a true law.

See [2] for example...not quite the general case but it proves the hard part. --Steve (talk) 20:48, 15 September 2009 (UTC)

Steve, I was not able to access your second link because I am not a subscriber. I believe that the counter example I gave you disproves your restricted version of Faraday's Law. Do you understand it? Mike La Moreaux (talk) 20:51, 16 September 2009 (UTC)

Steve and Mike, Yes of course, we are talking about the generator and not the motor. It is only in the context of EM induction that the F = qvXB force becomes part of Faraday's law. OK, in that case then you are no doubt talking about the Faraday paradox, am I correct? When the magnet rotates on its magnetic axis, the magnetic field doesn't rotate, and hence there can be no induced EMF as per E = -(partial)dA/dt. But we can still of course have an induced EMF from F = qvXB. I'll get back to you in a minute when I have examined the geometry of the situation. OK. I've looked. Isn't it just a simple case of the F = qvXB force? Where exactly does Faraday's law breakdown? David Tombe (talk) 06:09, 16 September 2009 (UTC)

David, I was not referring to the Faraday paradox. The homopolar generator is a pure example of motional emf uncontaminated by any change in flux. Because Faraday's Law only addresses a change in flux, it gives the incorrect result of zero emf, thus disproving Faraday's Law.

Steve's first link is to the relevant passages in Richard Feynman's "Lectures on Physiscs." Feynman understood the subject better than anyone else that I have read. The thing is that he uses the term "Faraday's Law" to refer to one of Maxwell's Laws and uses the term "flux rule" to refer to what we are calling Faraday's Law. He calls it a rule rather than elevating it to the status of a law. He states that the flux rule does not always work. If he had referred to it as a law he might very well have stated that it is false. He does contradict himself, however, when he states that the flux rule is an accurate statement and then proceeds to give exceptions to it.

Since you brought up the subject of the Faraday paradox, I direct you to the last two sentences of my original comment of 6 September. Do you really believe that a magnetic field can be unattached to its source? I believe that somewhere in the discussion attached to the Wikipedia article on the Faraday paradox someone explains the mystery by invoking the effect of the magnetic flux on the return circuit. I do not believe that there is any essential difference in operation between the rotational setup in the article and the linear version, namely a magnet and a conductor with relative motion between them in a straight line. In the latter case, it is just the relative motion between them that is relevant. An emf is generated if only the magnet moves, thus proving that the magnetic field moves with the magnet. Mike La Moreaux (talk) 20:51, 16 September 2009 (UTC)

Mike, Yes of course you are right that this has got nothing to do with the Faraday paradox. I realized that shortly after I wrote the section above. And yes I see what you mean about there being no apparent flux change in relation to the rotating disc. Nevertheless, we can still view the disc in terms of having an infinite number of spokes, and that will bring everything back into line with Faraday's law. And at any rate, if we express Faraday's law purely in mathematical terms, then the motionally induced EMF will be catered for by the v×B term. I agree that there has been alot of semantic confusion in the literature over what exactly constitutes Faraday's law. I have always taken it to be anything to do with electromagnetic induction and which then fits into E = -(partial)dA/dt + v×B, which in turn leads to curl E = -(partial)dB/dt -(v.grad)B = -dB/dt. But there is the problem that the modern Maxwell's equations (Heaviside versions) list Faraday's law as curl E = -(partial)dB/dt, hence ommitting the v×B effect. To say that the homopolar generator actually contradicts Faraday's law is somewhat pushing it a little in my opinion. It perhaps breaks the law literally when it is expressed in terms of 'flux change', but I don't see it as breaking the spirit of the law, or of breaking the mathematical formulation of the law.

Now moving on to the Faraday paradox, I would see a magnetic field as moving with a magnet where translational motion is concerned, or where rotational motion is involved, other than about the magnetic axis. I haven't yet examined the wiki-article on this topic in depth. Can you please elaborate on the point that you were making above. David Tombe (talk) 00:38, 17 September 2009 (UTC)

David, the Faraday's Law that we have all been discussing is just the ordinary derivative -dB/dt. If there is no flux change then the derivative is equal to zero and the emf is zero.

As to the Faraday paradox, I do not understand why you single out rotation about the magnetic axis. Mike La Moreaux (talk) 20:36, 17 September 2009 (UTC)

Mike -- I assume you're referring to: "a single-layer coil wound on an iron core with a constant magnetic flux passing through it lengthwise. One end of the coil is connected to a slip-ring whose brush is connected to the return circuit, the other end of which is connected to the other end of the coil. Allow the coil gradually to be unwound. The flux will obviously decrease, but there will be no emf because neither motional emf nor the relevant Maxwell's Law applies." No, I don't understand this, particularly how you get rid of the slip ring, and what the return circuit looks like. I'm sorry! Can you try again with more details and more words? :-) Thanks! --Steve (talk) 02:46, 17 September 2009 (UTC)

Steve, allow the wire from the end of the coil (and core) to extend out in a line along the axis of the core in the direction away from the core for some finite distance and then to go to the return part of the circuit, whose description is unimportant. Let it be a galvanometer. Slip-rings are used to avoid the twisting of a wire. In this case, however, it is not a concern because, as an infinitely thin wire, it cannot twist. As a refinement of the example, let us assume that the core does not rotate, but rather that the coil slips around it as it is unwound. Mike La Moreaux (talk) 20:48, 17 September 2009 (UTC)

I'm afraid I'm still confused. So far I have: There's a coil of wire wrapped 10 times around a cylinder of magnetized iron (magnetization parallel to the core axis). This coil has two ends, and each end is connected to a straight wire that travels along the edge of the core, parallel to the axis of the core, for a mile after the core ends. Then these two straight wires attach to each other.

To unwrap it, you rotate one of the straight wires around the core ten times, while holding the other straight wire fixed. There will be some "slack" within the coil, but you can put that into a tight wind somewhere so there's no need to worry about it.

Is that right? If not can you please correct me? Thanks! --Steve (talk) 21:09, 17 September 2009 (UTC)

Steve, sorry that I was not clearer. I would put in a drawing, but I would have to learn a lot of new techniques first, and I do not want to make you wait. Only one end of the coil wire extends beyond the end of the core in a straight line in line with the core axis. The other end of the coil wire is pulled at right angles to the axis of the core to unwrap the coil by letting it slip around the core.

It occurred to me that if you are going to use such a restricted version of Faraday's Law because it can be proved, you might as well just use the standard version and state that it always works for those cases to which it properly applies. Mike La Moreaux (talk) 00:20, 18 September 2009 (UTC)

Yeah, a pencil and paper or chalkboard would be great, but I think we can do this with just text. :-) I think I'm almost there.

So there's a coil of wire, plus two straight wires (one for each end): One ("Straight Wire A") goes 1 mile perpendicularly away from the axis of the core, the other ("Straight Wire B") goes 1 mile parallel to the axis of the core. Then there's a Sqrt(2)-mile-long straight wire ("Straight Wire C") connecting the other end of Straight Wire A to the other end of Straight Wire B. Straight Wire C never goes anywhere near the magnetic field, so we don't need to worry about it. Right so far?

Now, if both "Straight Wire A" and "Straight Wire B" are anchored in position, then there's no way to unwrap the coil. It's locked at both ends. So one or both of those has to rotate around the core. Does "Straight Wire A" rotate around the core, or does "Straight Wire B"? Or both? :-) --Steve (talk) 01:59, 18 September 2009 (UTC)

Steve, the coil is not locked. It is free to rotate or slip around the core. By pulling on Straight Wire A you can unwrap the coil. The other end just rotates. 66.19.21.168 (talk) 03:05, 18 September 2009 (UTC)

Mike, Let's split this into three. I'm as confused as Steve regarding this coil. I'll keep out of that discussion for the meantime and watch how it unfolds. As for the Faraday paradox, let's leave that until afterwards. Let's now get back to the homopolar generator. You say that -dB/dt is zero in the homopolar generator? I would say that it is not zero. The total time derivative expression dB/dt expands into (partial)dB/dt + (v.grad)B. The (v.grad)B term is a consequence of taking the curl of v×B. Every element of that rotating disc is moving through the magnetic field and generating a v×B force. David Tombe (talk) 03:27, 18 September 2009 (UTC)

David, of course, in actuality there is a motional emf. But according to the verbal statement of Faraday's Law, namely that the emf in a circuit equals the time rate of change of the magnetic flux linking the circuit, which is reflected in the equation emf = -dB/dt, there is no emf because there is no change of flux. I do not understand how any expansion can alter this. vxB is independent of the flux linkage of the circuit. Mike La Moreaux (talk) 21:26, 18 September 2009 (UTC)

Mike -- So you unwrap the coil by pulling lengthwise on Straight Wire A. As the coil unwraps, Straight Wire B travels around and around the core. (At any given time we keep Straight Wire B parallel to the core axis.) Correct? --Steve (talk) 08:56, 18 September 2009 (UTC)

Steve, not quite. Straight Wire B, once it is past the end of the core, lines up with the axis of the core and therefore does not move. It continues one mile out past the end of the core and would only twist, if it could twist. Since it is infinitely thin, there is no difference between twisting and not twisting. Mike La Moreaux (talk) 21:33, 18 September 2009 (UTC)

OK...What is Straight Wire B before it is past the end of the core? Are you implying that Straight Wire B isn't actually a straight wire? Or is the coil of wire right at the very end of the core? --Steve (talk) 23:06, 18 September 2009 (UTC)

Steve, yes, the one end of the coil is near the end of the core. Straight Wire B is straight as soon as it is a little past the end of the coil. Mike La Moreaux (talk) 00:07, 19 September 2009 (UTC)

Mike, That's the point that I was making higher up. You are looking at a wheel spinning in a magnetic field and the whole thing looks as if no lines of force are being cut. And so you are taking the wording of Faraday's law and claiming that Faraday's law has broken down. I was saying further up, that in doing so, you are pushing it a little bit. My argument was that Faraday's law is not being broken because each element of the wheel is moving across the lines of force and inducing the v×B force.

Let's say that E = -(partial)dA/dt + v×B as per equation (D) in Maxwell's original eight equations of 1864. Maxwell didn't use a Faraday's law in his original eight equations. Equation (D) was Maxwell's equation for electromagnetic induction.

Now let's take the curl of equation (D). We get,

Curl E = -(partial)dB/dt - (v.grad)B

The Faraday's law that appears in the modern (Heaviside) Maxwell's equations does not include the (v.grad)B term. Hence if we are talking about that particular version of Faraday's law, then you are correct in saying that it does not explain the homopolar generator. Indeed, that is a major part of the reason why Steve wanted to claim last year that there are two laws that bear the name of 'Faraday's law'. Steve was technically correct. But he was overlooking the wider picture. The equation directly above reduces to,

Curl E = -(total)dB/dt

which is the full Faraday's law including the v×B force. But it is never seen in modern textbooks written in that form. I agree with you that this is a matter which causes alot of confusion due to the fact that different aspects are taught at different stages in the education system and that the final overview equation, curl E = -(total)dB/dt is never seen in the textbooks. In fact, even when it appears at equation (54) in Maxwell's 1861 paper, I think that Maxwell only had a partial time derivative in mind. David Tombe (talk) 09:10, 19 September 2009 (UTC)

David, you state that the full Faraday's Law, including the vxB force, is curl E = -dB/dt, but in the homopolar generator it is equal to zero because the flux does not vary with time. We know that there is a motional emf, but Faraday's Law gives none for this counter example, thus disproving the so-called law. Mike La Moreaux (talk) 19:52, 19 September 2009 (UTC)

OK Mike, I think we're finally on the same page. At the end of Straight Wire B is a little segment of wire ("Wire Segment C") that goes from the coil to the core axis. As the coil unwinds, Wire Segment C moves -- one end stays put on the core axis, but the other end moves around the core as the coil unwinds. There's a qvXB force in Wire Segment C. So in your example, as the thing unwinds:

• The flux changes
• There is an EMF.

Just like Faraday's law requires. It's not a counterexample, it's an example. :-) --Steve (talk) 18:29, 19 September 2009 (UTC)

Steve, yes there will be a tiny emf, but we can make it arbitrarily small by reducing the diameter of the core. The emf will approach zero, thus disproving Faraday's Law. Mike La Moreaux (talk) 19:42, 19 September 2009 (UTC)

I don't think you can make the EMF arbitrarily small. There's a huge magnetic field at the end of the core, going out in all directions. Shrinking the core won't help. Maybe you could make Wire Segment C shorter, but it would pass through a larger field.

But you don't have to take my word for it: This can be easily settled by you actually calculating it. You should write down (mathematically with coordinates) what the wire path is as a function of time, calculate B everywhere, integrate the vXB to get the EMF, and compare it to the flux change. I am 100% confident that if you do that calculation (correctly), you'll find that the two are equal. (It's a mathematical certainty...I've shown you generalized proofs.) The calculation would be a bit onerous, but you can make simplifying assumptions. For example, keep the wire where it is, but replace the core with an infinitely-thin core (infinitely-thin cores have a much simpler formula for B). You could also make the core semi-infinite, so you don't need to worry about the other end, and then formula for B is even simpler.

I do hope you do this, you would find it educational. If you end up with integrals that you can't evaluate, I can plug them into Mathematica if you want. :-) --Steve (talk) 01:05, 20 September 2009 (UTC)

Steve, consider this variation. Let Straight Wire B extend to a point one mile out beyond the end of the core and on the axis of the core. This means that Straight Wire B will form a tiny angle to the axis of the core and describe a very narrow cone as it revolves.

Actually, the generalized proofs are irrelevant, because we both agree that the homopolar generator disproves Faraday's Law, in general. Faraday's Law purports to include both motional and transformer emf without adding anything extra. These two principles are independent and very different from each other.

Transformer emf does not require a conductor, requires a closed path, and involves an electric field which produces electric forces on the electrons, if present. Motional emf does require a conductor, does not require a closed path, does not involve an electric field, and produces magnetic forces on the electrons. It is not possible to combine two such different and independent principles in one term of an equation. Faraday's Law is an imperfect attempt to do so. Mike La Moreaux (talk) 19:09, 20 September 2009 (UTC)

Mike, If I were to use the partial time derivative version of Faraday's law that is one of the modern Maxwell's equations, then you would be correct. The EMF induced by the homopolar generator would not be accounted for by this restricted version of Faraday's law because it does not cater for the v×B effect. Are we agreed that the EMF that is induced in the homopolar geneator is induced by the v×B force? And if so, is your question ultimately about whether v×B is a part of Faraday's law or whether it sits outside Faraday's law? Are you aware of the physical distinction between a total time derivative and a partial time derivative? In fact, reading over this thread again, it would seem that you have got your understanding of total time derivatives and partial time derivatives completely reversed. It's the partial time derivative version, which is one of the modern Maxwell's equations that fails. You seem to think that it is the total time derivative version that fails. David Tombe (talk) 12:54, 20 September 2009 (UTC)

David, we are agreed that the emf that is induced in the homopolar generator is induced by the vxB effect. The vxB effect, or motional emf, is not perfectly included in Faraday's Law. The only independent variable in the equation for Faraday's Law, curl E = -dB/dt, is dB/dt. Because in the homopolar generator the flux is constant and equal to zero, Faraday's Law gives zero for the emf. It has nothing to do with whether the derivative is ordinary or partial. Mike La Moreaux (talk) 18:43, 20 September 2009 (UTC)

Mike, once again, here is a mathematical proof of Faraday's law for infinitely-thin wires, and there are others. It's pretty standard. You think you have a counterexample involving infinitely-thin wires, and I think you don't. If you actually do the calculation, which is maybe 20 minutes of work if you know what you're doing, you will have either convinced yourself that the theorem is true after all, or you'll have an important result that you could easily get published in a physics pedagogy journal like American Journal of Physics. I would be honored to be your coauthor. I hope you put in the little bit of extra effort to follow through and do this. --Steve (talk) 22:43, 20 September 2009 (UTC)

Steve, how about this, a variation on the toroidal transformer. Take a toroidal core and tightly wind it with a primary winding. Instead of alternating current, let it carry a constant current. Then loosely wind a secondary winding on top of it with one end free and the other again attached to a slip-ring which loosely slides around the primary winding. The free end and the brush on the slip-ring are connected to the external circuit. Because of geometric symmetry, there is no significant magnetic field outside of the primary winding. Therefore the thickness of the secondary winding wire and the size of the slip-ring are irrelevant because they are not in the magnetic field. The magnetic flux still links the secondary winding. Now let the secondary winding be gradually unwound. The flux linking it will be gradually reduced to zero, but there will be no motional or transformer emf, thereby disproving Faraday's Law. Mike La Moreaux (talk) 23:53, 21 September 2009 (UTC)

Mike, OK so it's down to the issue of whether or not the v×B force is included in Faraday's law. Well it's certainly not included in the partial time derivative version that is associated with Heaviside and which appears in modern textbooks as one of Maxwell's equations. But contrary to what you say, the v×B term is included in a total time derivative version of the same equation. The issue of partial time derivatives and total time derivatives is crucial in this argument. A total time derivative splits into a partial time derivative and a convective term. The v×B force is that very extra convective term. So while you may have found a counter example based on the restricted form of Faraday's law that we see in modern Maxwell's equations, you have not found a counter example to the more general form of Faraday's law that appears in the introduction to the article. Maybe Steve was right after all as regards the need to highlight this issue in the introduction because it clearly does cause alot of confusion.

On your debate with Steve, I am inclined to agree with Steve. I haven't as yet made my final judgement, but based on a few readings of the debate, I don't think that you have found a counter example to Faraday's law. David Tombe (talk) 09:26, 21 September 2009 (UTC)

David, let us take this one step at a time. Do you agree that that there is no flux change in the homopolar generator? Mike La Moreaux (talk) 00:00, 22 September 2009 (UTC)

Mike, There's a flux change. Every element of the rotating disc cuts lines of force and generates a v×B force. This accounts for the non-zero value of the dB/dt term. The total dB/dt splits into (partial)dB/dt + (v.grad)B. The non-zero value of (total)dB/dt is coming from the (v.grad)B component, which is actually the curl of v×B. David Tombe (talk) 07:39, 22 September 2009 (UTC)

Mike, your toroidal example has the problem that the secondary winding cannot be gradually unwound--due to topology. Imagine you have a rubber band that is wrapped around a donut so that it passes once through the hole. Now try to remove the rubber band, without cutting the donut and without cutting the rubber band. Obviously it's impossible.

OK, now replace the rubber band with a wire and the donut with the core. All the slip-rings in the world won't help: The winding number of the wire with respect to the torus is always an integer, so it can only change in discrete, discontinuous steps, unless you cut open the core to slide the wire through.

Look, you don't need me to keep pointing out these problems: I can tell you exactly what I'm doing. Imagine the lines of magnetic field flux. If a bunch of flux lines pass through your thin-wire-loop, and then later on, fewer flux lines pass through your loop (with constant B), then there must be some part of the loop which has been passing through the flux lines, bringing them from the inside of the loop to the outside. That's the part of the loop where the vXB force will be. Every time you think you've come up with an infinitely-thin-wire counterexample to Faraday's law, just ask yourself: How are these flux lines passing out of the loop? Draw a picture or whatever. Then you'll know where the vXB EMF is.

Or you can do an explicit calculation. Or you can double-check this derivation and realize that you shouldn't even try to come up with a thin-wire counterexample because there never will be any. :-) --Steve (talk) 07:50, 22 September 2009 (UTC)

Steve, I believe that you are wrong about there having to be an integral winding number. The use of a slip-ring allows a fractional winding number. As the coil is unwound there will be a smooth, continuous decrease in the flux. No part of the loop passes through the flux lines because they are all contained within the toroidal primary winding. Mike La Moreaux (talk) 20:42, 22 September 2009 (UTC)

Mike, I should also add that you seem to think that there is no magnetic field if we wire a coil such that the loops are alternately in opposite directions. That is not true. There will still be a magnetic field. It will have a very complex corrugated texture and it may give the impression of summing to zero for large scale motion. But it will still exist, and changes in the electric current will lead to changes in the flux linkage of a secondary wire in the vicinity.

It is pointless trying to find counter examples to Faraday's law. Faraday's law is the least controversial of all the equations in electromagnetism. It is a staple component of the EM wave equation. There is however alot of confusion as to what actually constitutes Faraday's law, and its link with the v×B force. Only two nights ago, I was reading some original paper of 1889 by Oliver Heaviside. Heaviside clearly uses the name 'Faraday's law' for a restricted partial time derivative version of what I would consider to be Faraday's law. The restricted version of Heaviside does not invoke the v×B force. Maxwell's original papers, to the best of my memory don't even mention Faraday's law by name. Maxwell's EM induction equation is E = grad(phi) -(partial)dA/dt + vXB, although the Heaviside one does seem to appear at equation (54) in Maxwell's 1861 paper. You can see equation (54) on this link [3]. I don't think that Maxwell links it to Faraday, but I haven't read it for a while. It is clearly the Faraday's law that appears in the modern Maxwell's equations. Even though it has total time derivatives, I think that Maxwell is only treating them as partial time derivatives. This is the equation that Heaviside used. David Tombe (talk) 09:37, 22 September 2009 (UTC)

David, I do not know where you got the idea that the primary winding has loops that are in alternately opposite directions. It is a straightforward winding. All the flux is confined to the core. The geometric symmetry results in the cancellation of the magnetic field external to the primary winding. That is why toroidal transformer are used; they reduce interference to nearby components.

vxB does not define an electric field. Therefore, its curl is zero and it is thereby removed from the equation. The idea that Faraday's Law can be derived is as fallacious as Faraday's Law, itself. Mike La Moreaux (talk) 19:12, 23 September 2009 (UTC)

Mike, even if what you say is true, it's besides the point, because a slip-ring is not an infinitely-thin wire. --Steve (talk) 04:45, 23 September 2009 (UTC)

Steve, I fail to see the relevance of the thickness of the conductors when they are not in the magnetic field.

You know, physical laws cannot be proved; they can only be disproved. If Faraday's Law can be proved, that would just confirm that it is redundant and adds nothing to our theoretical knowledge. Motional and transformer emf cover every possible case. Mike La Moreaux (talk) 21:00, 23 September 2009 (UTC)

Mike, I made a claim involving infinitely-thin wires. You can't give a counterexample with slip rings. If the slip ring is not necessary, then change your counterexample so that the slip ring isn't there. If the slip ring is necessary, then it's not a counterexample, it's a different topic, which we can talk about later. My aim right now is to convince you that the law is true for infinitely-thin wires. When we agree on that, then we can talk about whether or not it can be made slightly more general, and how. I'm looking forward to that part, I think it would be interesting, and I'm a bit surprised that it's taking so long for you to accept the infinitely-thin wire bit.

When I say proof, I mean the following: If the Lorentz force equation is true and Maxwell's equations are true, then it's a mathematical necessity that the "Faraday's law" formulation with infinitely-thin wire loops is also true. I showed you such a proof...in fact, two of them. Have you found a flaw in the math of those proofs yet?

I fully agree that the law is redundant with Maxwell's equations and the Lorentz force law. (As is the Biot-Savart law, Coulomb's law, etc...) If you don't want to use it, you don't need to. Other people (e.g. electrical engineers) do apparently want to use the law from time to time, even though they don't have to. Therefore I'd like the law to be stated and discussed on Wikipedia, in a way that's true and correct, and not subject to Feynman's counterexamples. It sounds like you think the law should be purged from wikipedia entirely. I'm sorry but that's not going to happen, in accordance with wikipedia's content guidelines. Of course you're welcome to personally ignore the topic if you'd like. :-) --Steve (talk) 22:40, 23 September 2009 (UTC)

Steve, I have not had the time to look for a flaw in the proofs, yet. I do not expect that there would be a flaw in the math but, rather, a subtly incorrect physical premise. For example, I still do not see what difference a slip-ring would make to the proof as long as it is not in the magnetic field. I am not claiming that the proofs are false at this point. I just suspect that they may be. And even if they are not, I believe that they are irrelevant to Faraday's Law as defined at the beginning of the article. I appreciate your strategy of trying to establish the law for a restricted case and then proceeding from there, as hopeless as I believe that to be. Engineers definitely use the unrestricted version of the law. I do not want to remove Faraday's Law from Wikipedia entirely. I believe that it is important historically and as an engineering convenience. I do not believe that it belongs in modern physics. Mike La Moreaux (talk) 21:43, 24 September 2009 (UTC)

Mike, As regards your coil scenarios, I admit that I haven't been altogether clear about the details. I have found your scenarios hard to vizualize. But on the theory bits, the curl of vXB is (v.grad)B which is the convective component of a total time derivative. It follows from the standard vector identity of taking the curl of a vector cross product. As for deriving Faraday's law, Maxwell did it. You can see it at equation (54) in his 1861 paper. David Tombe (talk) 11:58, 24 September 2009 (UTC)

David, if a wire moves through a magnetic field, an emf is induced in it by motional emf (vxB). While mathematically equivalent to an electric field strength, it does not constitute an electric field and its curl is always zero. Mike La Moreaux (talk) 23:42, 24 September 2009 (UTC)

Right, another question. The basic equation V = N d(flux in webbers)/dt troubles me greatly. (sorry about the notation, only using a regular keyboard). So, in a transformer, the emf generated in the secondary coil is purely a function of number of turns and the rate of change of flux? i.e is has nothing to do with the number of turns in the primary, and nothing to do with the primary voltage??? But then V (primary) = N (primary) d(flux in webbers)/dt - so after all it has! There has to be an equation somewhere that links all this together. In a nutshell I cannot relate the primary number of turns and primary voltage with the rate of change of flux, yet they have to be linked, as the secondary emf is a function of both. So my basic question is.... if the rate of change of flux is so great (as in an ignition coil)that a massive voltage is generated, what is stopping this voltage exceeding the voltage stated by V (secondary) = (N (secondary)/ N (Primary)) * (V (Primary)? I hope you understand the question, apologies of you do not. —Preceding unsigned comment added by 87.194.19.201 (talk) 15:59, 15 February 2010 (UTC)

I have come up with a detailed analysis of the homopolar generator. Let us consider the macroscopic path of a single free electron in the disk. Since its drift speed is on the order of micrometers per second, it is dragged around with the rotation of the disk. It motion relative to the disk is radial. Its path relative to the laboratory frame of reference is a tight spiral of millions of turns. This path is stationary in the laboratory frame. Therefore, the magnetic flux linking the circuit is a constant. Thus, Faraday's Law gives an emf of zero. This disproves Faraday's Law and shows that this is a pure example of motional emf. Mike La Moreaux (talk) 00:28, 8 August 2011 (UTC)

Some people make a statement: "The time derivative of flux through any circuit equals its EMF". This statement is false. The homopolar generator is a great counterexample. Feynman explains very explicitly that this statement is false, using the homopolar generator example and another example too. Some people might call this false statement "Faraday's law" or "the flux rule" (whatever terminology they prefer). I don't call it that. I call it "over-generalized Faraday's law" or "the over-generalized flux rule". The over-generalized law is most certainly false.

On the other hand, more knowledgeable physicists and engineers say something different: "The time derivative of flux through an infinitely thin wire loop equals its EMF". This law is true: In fact, it is a straightforward and direct mathematical consequence of the laws F = qv X B and curl E = - ∂B/∂t (See Proof 1, Proof 2). I call this statement "Faraday's law" or "the flux rule", and therefore I can correctly say "the flux rule is always true". Maybe to be more specific I'll call it "the infinitely-thin-wire flux rule".

A homopolar generator is not a counterexample to the "infinitely-thin-wire flux rule": A homopolar generator is obviously not made from infinitely thin wires.

Here's where I hope this discussion will go... :-)

• First, we already agree that "the over-generalized flux rule is false".
• Second, I hope we can agree that "the infinitely-thin-wire flux rule is true". (If you have a counterexample, please write down the parametrized x,y,z coordinates of the wire as a function of time, and the formula for the B-field, then calculate the EMF using F = v X B and curl E = - ∂B/∂t, then calculate the time-derivative of flux. I promise you'll find they're equal. This isn't as hard as it sounds, and I'm happy to help!)
• Third, we can discuss whether the "infinitely-thin-wire flux rule" can be generalized at all without being over-generalized into a false statement. For example, I suspect that it's still true if the wires are not infinitely thin, but still have a negligible amount of flux passing crosswise through them. I'm not positive, and I haven't tried looking this up.
• Finally, we will know the best way to state the law in the article :-) --Steve (talk) 01:34, 8 August 2011 (UTC)

Steve,

I am glad to see that we agree so much, after all.

The statement which we agree is false is, of course, the definition at the beginning of the article.

The infinitely thin wire version is numerically correct, but I do not accord it the status of a law, but rather a rule of thumb. In this case the two laws you mention give the same value of emf. This, however, does not mean that both laws apply in any one example. This distinction may appear to be somewhat subtle or even irrelevant. It seems to me that a law has to be an expression of a scientific principle. Providing the correct answer is not sufficient. That is math but not physics. What we might call the law of motional emf only applies where motion is necessarily involved. Maxwell's third equation only applies where the strength of the magnetic field actually changes. There will be a flux change where motion is essential, but it is only along for the ride, as it were. It has no effect. It is just through a mathematical quirk or coincidence that it gives the correct value for the emf. The expression for motional emf and Maxwell's third equation are all that are required: between them they cover all cases without restriction. They are both necessary and sufficient. Faraday's Law is not necessary, but just a convenience.

I would not quibble about the thickness of the wire.

Again, I would not give Faraday's Law the status of a true law. It should be the subject of an article for historical reasons and because it is so well known, but I believe that what we have most recently discussed here should form the gist of the article. Mike La Moreaux (talk) 02:41, 8 August 2011 (UTC)

You're right that we might as well improve the definition in the article straightaway. I made some small changes...I'll do more when I have time.... I also started to make an image with one of Feynman's funny "counterexamples" to Faraday's law. I'll post it soon... :-)

The fact that motional EMF and transformer EMF are different is I think already discussed very thoroughly in the article, section 2. Of course, as Einstein explained, they're not totally unrelated, a transformer EMF in one frame of reference is a motional EMF in another frame, so there's a requirement that the formulas be consistent in a certain way.

You're saying Faraday's Law is "not necessary", because Maxwell's equations and the Lorentz force law cover everything. Correct? I'm sure you understand, the same is true of Coulomb's "law", the Biot-Savart "law", the Abraham-Lorentz force "law", etc. etc. etc. Also, Maxwell's equations themselves are "not necessary" because quantum electrodynamics is sufficient to derive them. The fact that a law is not "necessary" is not a reason to scoff at it. Virtually every law in physics can be derived from more fundamental laws! :-)

I suggest you avoid the term "rule of thumb" in this context. People usually understand that to mean "an approximation, not exactly accurate, but simple". But the flux rule is accurate. You can call it a "mathematical calculation trick" or something if you want to sound dismissive. :-)

If you agree that the infinitely-thin-wire flux rule is a "true statement", then that's good enough for me. You call it "not deserving to be called a law", you can call it "math not physics", you can call it "Stupidface", whatever pleases you. :-) --Steve (talk) 05:35, 8 August 2011 (UTC)

File:FaradaysLawWithPlates.gif -- image here...i'll add it to the article when i get a chance :-) --Steve (talk) 16:01, 13 August 2011 (UTC)