Talk:Faraday's law of induction/Archive 4

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"Counterexamples" section, and Hughes & Young book

There is objection to the "counterexamples" section - see [1] and [2]. Supposedly in Hughes & Young, The Electromagnetodynamics of Fluid (1965), there is some formulation of Faraday's law (i.e. EMF = derivative of flux) which is valid for homopolar generators etc. I'm willing to believe that such a formulation might exist, and if it does, it should probably be in the article. But I can't find this formulation in the book so far. Does anyone know what page it's on? Or better yet, can anyone just tell me what is this formulation of Faraday's law? --Steve (talk) 18:19, 23 April 2017 (UTC).

At least for the homopolar generator, the formulation of Faraday's law as the complete time-derivative of the magnetic flux through the circuit will return the correct EMF, if when you push the derivative through the integral you include the term for non-stationary boundary (see Jackson Classical Electrodynamics (3rd ed.) pp. 208–211 for a general case). References that I know of that specifically address the homopolar generator are Thomas Valone The Homopolar Handbook p. 7 and Heinz Knoepfel Magnetic Fields: A Comprehensive Theoretical Treatise for Practical Use p. 324 - the latter re-imagines the disk as a ring with a single conducting spoke. --FyzixFighter (talk) 19:27, 23 April 2017 (UTC)

For infinitely-thin wires, we have the statement "EMF around a circuit is the time-derivative of flux through it", which is succinct and correct. I don't think there's anything which is just as succinct and correct for thick conductors, other than "just use the Lorentz force and Maxwell-Faraday equations". I've read Jackson and I found the Homopolar Handbook page 7 on google books. I couldn't find anything like that. Here's the best I can come up with right now:

Take a closed path within the conductive material, through which current can travel. Allow that path to change by getting dragged along with the local material velocity. The EMF around this path is equal to the time-derivative of flux through the path.

It's less succinct than the infinitely-thin-wire version. More importantly this is not really 100% correct. For example, take a homopolar generator with a slip ring. If you "allow the path to change by getting dragged along with the local material velocity", then the path immediately stops being a closed loop, because the local material velocity is discontinuous at the slip ring. OK, so I'm going to change it to make it more widely applicable:

Take a closed path within the of conductive material, through which current can travel. Allow that path to change by getting dragged along with the local material velocity. Oh, and also, you are allowed to add new path segments parallel to the local material velocity, if it helps keep the path closed. Anyway, the EMF around this path is equal to the time-derivative of flux through the path.

The statement is now more convoluted but at least it is usable for homopolar generators. But it still doesn't apply to the animated "counterexample" in the article: There's no path that both gets dragged along with the local material velocity and also stays closed for more than one instant, even if you add path segments parallel to the local material velocity, if I'm not mistaken. And there are other things that can go wrong too. Like what happens if the "material" is an electrolyte with different ions flowing in different directions? Do we need an extra statement on what "local material velocity" is in this case? Anyway, taking all this together, I'm a bit stumped, which is why I like the article's current approach, which is to say that Faraday's law is "EMF around a circuit is the time-derivative of flux through it" and that the law is inapplicable to thick conductors, so please use Maxwell-Faraday and Lorentz force instead. But maybe there's a way, that's why I'm eager to see how Hughes & Young do it, or anyone else for that matter. --Steve (talk) 12:50, 24 April 2017 (UTC)

Here is Fred Young again.

See on page 50 section 2.6 of aforementioned book where the familiar form of Faraday's law is derived and explained in detail. On page 58 is Example 2.7 entitled "The Faraday disc generator". It is shown how to use Faraday's law for the homopolar EMF calculation.

When Bill Hughes and I wrote the Electromagnetodynamics of Fluids in 1960 volume II of the Feynman Lectures on Physics had not been published or we would have explained how Faraday's law could have been used to verify Feynman's measurement of zero EMF in his Fig. 17-3. In my book "Magnetically Induced Electromotive Force" (ISBN 978-1-4797-3789-5) published by Xlibris on 15 October 2012 it is shown how to use Faraday's law to calculate the EMF of rocking conducting plates in a uniform magnetic field. See section 6.13 'EMF generated by plates rocking in a uniform magnetic field'. Here it is important not to take a short cut though the moving conductor to evaluate the flux used in Faraday's law. Youngfj (talk) 16:25, 24 April 2017 (UTC)

I’ve been staring at the rocking plates and I notice that: the plates move in the opposite direction of the path, the path moves a lot but the plates move very little, and Feynman says, in the text near the figure, “v X B is very small and there is practically no EMF.” He did not say zero EMF. It looks like this is the same case as the homopolar generator. The small EMF is associated with the small motion of the material that is dragging the electrons along with the conductor. Constant314 (talk) 17:52, 24 April 2017 (UTC)

Constant314: For the rocking plates, no one is claiming that the EMF is exactly zero, just that it is very much smaller than the total time derivative of flux through the circuit (as defined in the caption), so that the two things are not equal. This is not "the same case", as a homopolar generator. I would say it is the exact opposite! For a homopolar generator, the total time derivative of flux through the circuit (as defined in the caption) is zero but the EMF is large. But in both cases, the two are not equal. --Steve (talk) 02:40, 25 April 2017 (UTC)

Thanks Fred, I got the book from the library and read the section you referenced. It seems the main equation used for analysis is (2.6-13):

${\text{EMF}}=\int _{S}{\frac {\partial \mathbf {B} }{\partial t}}\cdot d\mathbf {S} +\oint \mathbf {V} \times \mathbf {B} \cdot d\mathbf {l}$ (2.6-13)

where S is any path and "V is the velocity of the segment of the conductor at the point corresponding to dl... It is very important to notice that (1) V is the velocity of the conductor ... not the velocity of the path element dl and (2) in general, the partial derivative with respect to time cannot be moved outside the integral since the area is a function of time".

I 100% agree that this equation is correct in every circumstance including homopolar generators and rocking plates and whatever. If someone had asked me last year how to calculate EMF in situations with thick moving conductors, I would have immediately written down this exact equation.

The only question is whether or not equation (2.6-13) should properly be called "Faraday's law" or whether it should be called "a combination of the Maxwell-Faraday equation and Lorentz force". The article currently takes the latter perspective. According to the article right now, "Faraday's law" by definition is a relation between EMF and the total time derivative of flux, and therefore (2.6-13), while a lovely and important equation, should not be called "Faraday's law".

If we want, we can change the wikipedia article to follow Hughes & Young's terminology: "Faraday's law" is a relation of the form "EMF = something", and for thin wires the "something" is total time derivative of flux, and for thick conductors the "something" is the complicated expression in Eq. (2.6-13).

The terminology surrounding Faraday's law is very much inconsistent in the literature. For example, Griffiths and Feynman use the terms "Faraday's law" and "flux rule" to refer to what this article calls "Maxwell-Faraday equation" and "Faraday's law". Still, we should try to follow common usage to the extent possible. So, has anyone seen the Hughes & Young usage where (2.6-13) is called "Faraday's law", anywhere other than that book? I don't think I've seen it, but that's just me.

My slight preference right now is to put (2.6-13) into the article (in the section under discussion), but not call it "Faraday's law". I'm going to do that right now. We can always change it again later. :-D --Steve (talk) 02:40, 25 April 2017 (UTC)

@Sbyrnes321: Just to clarify, are you saying that what we should call "Faraday's law" is ${\text{EMF}}=-{\frac {d}{dt}}\int _{S}\mathbf {B} \cdot d\mathbf {S}$. If so, then I don't see the difference between that and 2.6-13, at least if you properly move the total time derivative into the integral using the identity (per Jackson) ${\frac {d}{dt}}={\frac {\partial }{\partial t}}+\mathbf {v} \cdot \mathbf {\nabla }$. I don't think it's a matter of infinitely thin versus thick conductors, it's a matter of do you mean a partial derivative inside the integral or a total derivative outside the integral when you say "time-derivative of flux through the circuit" in the formulation. --FyzixFighter (talk) 02:51, 25 April 2017 (UTC)

Yes I am saying we should call "Faraday's law" ${\text{EMF}}=-{\frac {d}{dt}}\int _{S}\mathbf {B} \cdot d\mathbf {S}$. In which, case you cannot use the law to get the correct answer for the EMF of a homopolar generator. With S constant and B constant, the RHS is obviously zero, and the LHS is obviously nonzero.

Here's why I think you are getting confused. (2.6-13) is always true. But the "v" in (2.6-13) refers to the velocity of the material, not the velocity of the path S. In a certain case--infinitely-thin wires--the two are the same. In that case (and only that case), you can do a mathematically transformation of the RHS of (2.6-13) into a total time derivative of flux, following the steps in the "Proof of Faraday's law" show/hide box in the article. --Steve (talk) 15:25, 25 April 2017 (UTC)

While I agree that is the most common form of Faraday’s Law, I don’t see why it does not apply to the homopolar generator. If you replace the brush with s slip ring, nothing changes in the “paradox” and if you replace the solid wheel with a spoked wheel with very thin spokes separated by vanishingly small insulated barriers, then in each small increment of time, each spoke sweeps out a small increment of area which gives you just right amount of B · dS to account for the voltage.Constant314 (talk) 21:55, 25 April 2017 (UTC)

The statement

The EMF around an arbitrary closed loop S (moving or stationary) equals the time derivative of the flux of B through S (Statement X)

is not a true law of physics, because it has counterexamples. The homopolar generator is a particularly simple one. I can spin the wheel and take B and S to be fixed in time—nothing in Statement X says I can't—and then the EMF is (experimentally) nonzero while the time derivative of flux through S is obviously zero. Therefore, it is an experimental fact that Statement X is false. Does everyone here agree with that?

If we're on the same page so far, then the question is how to edit Statement X to make it true. I can't think of any way to edit Statement X to make it simultaneously (1) true, (2) able to calculate the EMF generated by any arbitrary configuration of thick moving conductors, and (3) of the form that looks kinda like "EMF = total time derivative of flux".

For example: If I replace "arbitrary closed loop S" with "loop of infinitely thin wire S", then Statement X would become true, I claim. (Do you have a counterexample? Note that the phrase "loop of thin wire" excludes slip rings and much else.) It also becomes unfortunately narrow, offering no prediction whatsoever about homopolar generators etc. So that edit satisfies (1) and (3) but not (2).

Another example: If I edit Statement X by replacing it with (2.6-13), then it becomes true. This meets conditions (1) and (2) but not (3).

I suspect that if any statement exists that satisfies (1), (2), and (3), then it is so hopelessly complicated and convoluted that we would all be much better off just forgetting about condition (3) and using (2.6-13). --Steve (talk) 02:26, 26 April 2017 (UTC)

I'm sorry Steve, but I think we are going to have to agree to disagree on this, and I don't see us doing anything but going back and forth on it. I don't think for the homopolar generator that the RHS of Faraday's law is zero, at least when the disc is rotating, based on my reading of Jackson. Specifically, I think that equation 5.137 comes into play because the circuit elements along the disc have a non-zero instantaneous velocity. Statement X is only false when the total time derivative is incorrectly applied to moving circuit elements. The path of integration is static but the circuit elements, the little elements of conductors that make up the circuit, are not stationary, so (2.6-13) naturally falls out of using Jackson's 5.137 with Statement X. More importantly, we have at least two reliable sources that start with Statement X and specifically solve the homopolar generator. So, based on sources, we cannot say that Statement X is false for the homopolar generator. --FyzixFighter (talk) 03:38, 26 April 2017 (UTC)

What are the "two reliable sources that start with Statement X and specifically solve the homopolar generator"? I don't see that in Jackson (which doesn't discuss the homopolar generator AFAICT), nor in Hughes & Young (which uses (2.6-13) rather than Statement X). BTW, Feynman and Griffiths are two gold-standard references that say that Statement X is false for homopolar generators, and indeed discuss the point at length.

You say "for the homopolar generator...that the RHS of Faraday's law is [not] zero...when the disc is rotating, based on my reading of Jackson". Jackson neither wrote down Statement X nor applied it to a homopolar generator, so we need to use our own brains for this. :-D Read Statement X again. In Statement X, S is an "arbitrary closed loop (moving or stationary)". There is nothing in Statement X that says that the loop S needs to move following the motion of the material. The word "circuit element" is not part of Statement X. There is no tricky interpretation related to the time derivative in Statement X. It's just a time-derivative of a scalar quantity, defined the same as in your intro calculus textbook. Specifically: S is static (in the lab frame), B is static (in the lab frame), and the wheel is spinning steadily. What is the flux is the flux through S right now? Maybe it's 7e-4 webers. What was the flux through S five minutes ago? 7e-4 webers. What will be the flux through S next month? 7e-4 webers. So, what is the time-derivative of flux? It's zero. Do you understand what I'm saying here? I find this extremely straightforward, and I am trying hard to understand where you're coming from. If the time derivative is not zero, do you think the flux is increasing or decreasing? If we run this for a month, is the flux always increasing, i.e. growing ever larger and larger? Or what?

I think you might have in mind a different statement than Statement X, perhaps a statement where S is not just any arbitrary closed loop in space but has to move in some relation to the motion of the material. If so, can you please write this statement down? (Maybe something like the text I wrote earlier in this conversation, under "Here's the best I can come up with right now"?)

For the record, I wrote the Faraday's law of induction#Proof of Faraday's law show/hide box text, which is equivalent to Jackson (5.137-8). So don't worry that I am unfamiliar with this total time derivative thing! :-D --Steve (talk) 13:10, 27 April 2017 (UTC)

I love reading the Feynman Lectures and I like to use them as a reference. But there is a problem. They were transcribed from lectures. Sometimes Feynman speaks rigorously and sometimes he speaks casually. Sometimes he makes intentionally incorrect statements and then corrects them later. Sometimes he leaves the correction as an inference for the reader. If he says that you cannot solve this problem with that equation does he mean that no one has found a way to solve the problem or that it has been rigorously proved that you cannot solve the problem. Feynman was writing in the 1960s. Maybe a problem with no known solution then has a known solution now. In both the homopolar generator and the moving plates, Feynman uses the word circuit with quotation marks. What does he mean? Does he mean that the so called circuit is not really a circuit? Constant314 (talk) 00:16, 28 April 2017 (UTC)

I disagree that there is ambiguity in Feynman's views about this, but if you prefer, use Griffiths instead, that was written as a textbook in which every sentence is supposed to be correct. Also, the issue is not "that you cannot solve the problem with that equation". It is incredibly easy to "solve" the standard homopolar generator problem starting from statement X. The issue is that, when you do so, the answer you get is wrong, i.e. inconsistent with experiments. :-P If you know a way to edit Statement X so that it doesn't give wrong answers, please write it down here! (Just please define for us any potentially ambiguous terms like "circuit". The word "circuit" is not part of my Statement X but you are welcome to use it if you like.) --Steve (talk) 18:47, 30 April 2017 (UTC)

Griffiths is much less ambiguous; lets go with that. Griffiths' argument is "it's not even clear what the flux through the circuit means." He might as well said, it doesn't work because I cannot figure out how to do it. When I use the spoke model and apply it to

${\text{EMF}}=-{\frac {d}{dt}}\int _{S}\mathbf {B} \cdot d\mathbf {S}$

I get

${\text{EMF}}=-{\frac {B\omega }{2}}r^{2}$

Is that the wrong answer? Constant314 (talk) 18:02, 1 May 2017 (UTC)

It sounds right, but the problem is where "the spoke model" came from. For something to be properly called a law of physics, it must be totally unambiguous how to apply it in any possible situation in which the law makes any prediction at all. So if you want "EMF =-d/dt ∫B·dS" to be a valid law of physics applicable in all situations including homopolar generators, you need to write some explanatory text to go with it (I expect that it will particularly focus on what S needs to be and how S needs to move). This explanatory text needs to be so specific that any intelligent person reading the text will know that "the spoke model" is the only correct way to apply the equation for a homopolar generator (even if the generator itself does not actually have spokes!). So, can you please write that text for us? Again, this text that makes it obvious that one must use "the spoke model" and not any other conceivable model such as the "S remains fixed" model. --Steve (talk) 22:04, 1 May 2017 (UTC)

I don’t think I can write the text you want without committing WP:SYN. I can offer two ways out. First, we could go back to calling it “counter examples” with the quotes, or we could say that in the case of moving thick material it is easy to misapply the equation and get the wrong result. The counter examples then become examples of common misapplication. Constant314 (talk) 14:40, 2 May 2017 (UTC)

Don't worry about WP:SYN, this is the talk page, not the article. When you say "it is easy to misapply the equation", what equation are you talking about? In this whole conversation, I have been looking and looking for an equation (and by "equation" I include also precise definitions of terms) which is simultaneously (1) true, (2) able to calculate the EMF generated by any arbitrary configuration of thick moving conductors, and (3) of the form that looks kinda like "EMF = total time derivative of flux". Nobody so far has written down such an equation: Not Hughes & Young, nor Jackson, nor anyone in this conversation. (I think I came closest with my text a while back labeled "Here's the best I can come up with right now".) Well then, maybe such an equation simply does not exist? Or if you think it does exist, can you write it down please? I bet we could find a citation if we knew exactly what we were looking for.

I personally like the equation of Feynman and Griffiths, who say that there is a law "EMF = -d/dt ∫B·dS for thin wire loops, where S is the path of the wire loop". This is a true and correct law of physics, it satisfies (1) and (3) (but not (2)), it has good mainstream sources, and it is perfectly adequate to equip readers to perform any possible electromagnetism calculation. (...because you can always use the Lorentz force law and/or Maxwell-Faraday equation to work out what happens with thick conductors, as in Hughes & Young (2.6-13), and actually lots of other textbooks do this too if I recall.) So, I propose to go back to my last version. I have good textbook sources for everything in that section, if I recall. If you don't like that version, what about it should be changed? --Steve (talk) 01:56, 3 May 2017 (UTC)

Like I said before, I don't see either of us convincing the other of our position. Jackson does give Statement X in equation 5.141 (with 5.134 where he defines the EMF as the LHS of 5.141). From my perspective, the ambiguity is the meaning of S, or rather C the circuit which bounds the surface. Is it simply a path of integration or is tied to the conductor/medium in which it resides. S isn't some arbitrary surface, it's the surface bounded by the circuit, so even if S is static and B is static, if C is not static (ie includes media that is moving) then I argue that Jackson 5.137 applies. When evaluating Statement X for the homopolar generator, why can't I use Jackson 5.137 to evaluate the total time derivative?

I see two possible ways to tweak Statement X for clarity (call it Statement X'). One way would be say that the time derivative on the right is a total time derivative calculated using the convective derivative. With this caveat, I see (2.6-13) popping out naturally from Statement X' without having to invoke the Lorentz force law. Another would be to go the other route, and say "EMF = -dΦ/dt" where Φ is the flux through the circuit. "Circuit" now is clearly in Statement X'' instead of hidden in the definition of S and we can then address what the proper meaning of circuit might be.

Griffiths is good, but I would not say it is perfect. I get why he says that Faraday's experiment #1 classically isn't the same phenomenon as experiments #2 and #3, but he does say that all three cases are subsumed by his universal flux rule (essentially Statement X). I disagree with handwave he does to go from 7.14 to 7.15 - somehow, magically the total time derivative becomes a partial derivative. Derivatives don't always work that way. Jackson is much more explicit and rigorous in how the total time derivative turns into a partial derivative. As noted by Constant314, Griffith doesn't say the flux rule can't be used for the homopolar, just it's somewhat ill-defined. The spoke model deals with it somewhat, but if we altered the setup so B only existed along a radius from the axis to the sliding contact, we'd have a well defined current path, the spoke model would fail, but the convective derivative would not.

I don't mind changing the section heading back to "counterexamples" - these are "counterexamples", or more properly paradoxes, put forward for Faraday's Law by reliable sources. But I would not agree with saying that they cannot be explained using Faraday's Law (Statement X). I see a number of sources that disagree with that: Valone on page 7 says, in the context of the homopolar generator, "The emf is then negative of the change in flux: closed curve line integral of E·dl = -d/dt ∫B·dS" then citing a previous work by D.R. Corson applying the divergence theorem to get "curl E = -dB/dt + curl (vxB)" which is essentially (2.6-13). That's one source that jumps from Statement X to 2.6-13 without going through Lorentz or a spoke model. Hansjoachim Bluhm's "Pulsed Power Systems: Principles and Applications" also starts with Statement X in 3.48 and sees no problem using Jackson 5.137 to get the correct expression of the loop voltage of a homopolar generator. In "Intermediate Electromagnetic Theory" by Joseph V. Stewart, he says "This example of Faraday's Law [the homopolar generator] makes it very clear that in the case of extended bodies care must be taken that the boundary used to determine the flux must not be stationary but must be moving with respect to the body" (p 396). --FyzixFighter (talk) 04:00, 3 May 2017 (UTC)

Thanks, don't give up yet, I think we're making progress. :-D

I don't understand Statement X'. Can you help me? When I see ${\frac {d}{dt}}\int _{S}{\vec {B}}\cdot d{\vec {S}}$, I understand that in the usual freshman calculus way, i.e. there is a function $f(t)\equiv \int _{S(t)}{\vec {B}}(t)\cdot d{\vec {S}}$, and I am taking the derivative of that function, i.e. $\lim _{\Delta t\rightarrow 0}(f(t+\Delta t)-f(t))/\Delta t$. How do I stick a convective derivative in here? f is not a function of space and time, just a function of time. In other words, d/dt is outside the integral, so if we want to use ${\frac {d}{dt}}={\frac {\partial }{\partial t}}+\mathbf {v} \cdot \mathbf {\nabla }$, then we wind up with (among other things) the expression $(\mathbf {v} \cdot \mathbf {\nabla } )\int _{S}{\vec {B}}\cdot d{\vec {S}}$. But this is a gradient of something which is not a function of spatial coordinate, so I can't get a nonzero correction term this way. Am I missing something? Can you help me fill in the details?

I don't think Jackson makes any mistakes but I think the section is a bit ambiguous in parts. Here is how I read it. I think E' is always by definition the E-field in the rest frame of dl. I think that (5.136) (∫E'·dl=dΦ/dt) is universally true (and equivalent to (2.6-13)), where C is any arbitrary geometrical path in space and S is any arbitrary surface in space whose boundary is C. I think that (5.134) (EMF=∫E'·dl) is not universally true, but is true when C is a thin-wire circuit. (Thin-wire circuits are special because the rest frame of dl is also the rest frame of the atoms in the metal wires.) It's not written explicitly, but I think Jackson is really only talking about thin-wire circuits from the beginning of the section right up until p210 where he says "the circuit C can be thought of as any closed geometrical path in space", and that paragraph is basically a transition from real (wire) circuits to this new broader context, and after that transition, you can keep using (5.136), but you have to stop talking about EMF. And that's what he does! He never mentions EMF again after the suggestion that C can be reinterpreted as an arbitrary path in space, as opposed to a real wire circuit.

In (5.137), as I understand it, C is a curve in space, S is a surface whose boundary is C, and the equation (5.137) is only true if v is defined as the velocity of C. To be explicit: If the velocity of C is different than the velocity of the material, well then v still better be the velocity of C, or else equation (5.137) is false. Do you agree? If not, I can fill in the math details omitted by Jackson, and/or give some worked example calculations. :-D

I like your quote "in the case of extended bodies care must be taken that the boundary used to determine the flux must not be stationary but must be moving with respect to the body". I read this as a straightforward declaration that Statement X is false! In statement X, the boundary is any arbitrary curve in space: No restrictions whatsoever! But your quote says that "the boundary used to determine the flux" cannot be just any old curve. Maybe it's kinda like my formulation above under the text "Here's the best I can come up with right now"...? :-D --Steve (talk) 01:44, 6 May 2017 (UTC)