Inducing EMF Through a Coil: Understanding Flux

[Charles Link]

Just a couple more comments on the E&M topics we covered in this thread:

Back in post 111 , the magnetomotive force (mmf) was discussed, and IMO it makes for much better physics if the mmf ($NI$ ) is taught as $\oint H \cdot dl=NI$. [Edit: It should be mentioned that instead of going along the path of the wire of the coil, as the integral of $E$ that we computed to calculate the voltage, this integral goes around/through the core of the transformer, and the current is found from the amount of current that goes through a cross-section. The closed loop line integral comes from Stokes' theorem, with the area that has the loop as its perimeter defines the cross-section for the current].

Perhaps one item that arises in this case though is that the $H$ needs some clarification on exactly what it is. I tried to present $H$ with some detail starting in post 130. Without the extra detail, I think much of the magnetism subject with the $B$, $H$, and $M$ can be a real puzzle.

On another item, the formula $B=H+4 \pi M$(cgs) and $B=\mu_o H+M$ (mks) holds in all cases, even when currents in conductors are present. This necessarily results because $H$ is defined to include currents in conductors as sources, as was mentioned in post 146 above. If we can write $M=M(B)$, then we can also write $M=M(H)$, but we find in many cases that $M=M(B)$ isn't exactly the case either, largely because of the exchange interaction. With this interaction, the $M$ at nearby locations, (i.e. neighboring atoms), has an effect on the $M$ at a given location, and thereby the magnetic field $B$ is not the only factor in determining what the magnetization $M$ is at a given location.

The mathematical relationship between $M$ and $H$ remains a peculiar one at times in any case. In the case of linear ferromagnetic materials we have $M=\mu_o \chi_m H$ where $\chi_m$ can be 500 or more. Then we have cases such as the permanent magnet, where $H$ points opposite the $M$ in a permanent magnet. For a spherical shaped permanent magnet, $H=-M/(3 \mu_o)$, so we can draw the line $M=-3 \mu_o H$ on our hysteresis curve of $M$ vs. $H$ to find the value that $M$ will have for a permanent magnet with a spherical shape for that material. Why some materials make good permanent magnets while other materials (e.g. the linear ferromagnetic materials) have their $M$ return to zero when the $H$ that is applied from currents in conductors returns to zero could be worth further discussion, but I have yet to figure that part out yet. I think it is related to the way the domains are formed, but it seems to be somewhat complicated and there probably isn't a real simple answer.

In any case, I did want to present the $H$ in as much detail as possible, because it sees some very widespread use. Hopefully we did a better job of presenting it above than some textbooks do, where they might define $H$ as $\mu_o H=B-M$, and leave the reader guessing what it represents. And hopefully at least a couple readers found some of the details to be good reading. I welcome any feedback.

 

[Charles Link]

In the discussion of $H$, I want to add one more thing, and that is an Insights article I wrote a couple years ago. If you do choose to look at the "link" below, I recommend you click on the "continue reading" in the first post=I looked at the second post on this one, and the input seems to be somewhat irrelevant=the fellow never seemed to complete what he was computing.

See https://www.physicsforums.com/insig...tostatics-and-solving-with-the-curl-operator/

The Insights article discusses how to compute $H$ from $\nabla \times H=J_{conductors}$, or from $\nabla \cdot H=-\nabla \cdot M$. That had puzzled me many years ago when I was a student, and it was only years later that I figured out that the integral solutions are missing a homogeneous solution in both cases. The correct complete solution turns out to be the sum of both integral solutions, as is mentioned in the article.

The article also has a derivation of the EE's mmf (magnetomotive force) equation.

 

[Charles Link]

Since we discussed computing the magnetic poles with $\sigma_m=M \cdot \hat{n}$, and computing $H$ from the poles, the reader may find another previous thread of interest where a couple of students measured the external magnetic field $B$ and computed the magnetization $M$ of a cylindrical magnet as a function of temperature, and did an assessment of the Curie temperature $T_C$ for that material.

See https://www.physicsforums.com/threa...tionship-in-ferromagnets.923380/#post-6543010

In post 21, I also discuss how I used a boy scout compass to measure the on-axis magnetic field strength of a cylindrical magnet.

Hopefully at least a couple readers find this of interest. I've tried to make this thread into one where the reader can get somewhat of a good picture of what the formula $B=\mu_o H+M$ is all about, along with a couple different applications. I welcome your feedback.

 

[Charles Link]

In posts 130and 150, I mention Feynman's discussion of the $H$, especially in solving for the magnetization and the magnetic field in a transformer core that has an air gap. (He just uses a primary coil, but that is ok). I have an alternate solution to this using the pole method which is as follows:

There will be a magnetization $M$ in the core which can be assumed to be uniform. There will result magnetic poles with surface magnetic pole density $\sigma_m=\pm M$ at the gap. The $H$ for this scenario for small gap $d$ is well known: The poles look like infinite sheets of magnetic charge for the region in the gap, so that $H_{gap }$ from the poles is $H_{gap \, poles}=\frac{M}{\mu_o}$.

There will be an additional contribution to $H_{gap}$ from the current in the conductor coil of $\frac{NI}{l}$, so that $H_{gap}=\frac{NI}{l}+\frac{M}{\mu_o}$.

In the material $H_m=\frac{NI}{l}-\frac{M d}{\mu_o (l-d)}$, where the second term of $H_m$ keeps the part of the loop integral zero from the magnetic charges.

Then we have $B=\mu_o H_m+M$, so that we can substitute for $M$, and get the equation of the line of $B$ vs. $H_m$, as Feynman does in (36.27) .

I also can do the linear case where $B=\mu_o H_m+M=\mu H_m=\mu_o H_{gap}$, and solve for $M=(\mu-\mu_o) H_m$, after solving for $H_m$ and $H_{gap}$.

IMO, mine was almost a preferred solution to the problem, but I found there was a slight inconsistency between my solution and Feynman's. It was slight in that the two solutions agreed in the approximation that $l>>d$, but it still puzzled me where the difference was. I thought it might be in the $H$ term from the conductors, but clearly that should be $l$ in the denominator where the $l=l_1+l_2$. (I'm using $d$ for Feynman's $l_1$). It took a while, but I finally discovered the source of the inconsistency, and I will show the solution to this inconsistency in post 155 below.

 

[Charles Link]

To solve the puzzle of post 154, I first went to the surface current model of magnetostatics that says $B=\frac{\mu_o NI}{l} +M \frac{(l-d)}{l}$ for the torus with the gap, rather than simply an $M$ in the second term.

[Edit: Note that the magnetization currents from a complete torus yield the result that $B=M$. If there is a section of length $d$ missing in a total loop length of $l$, we can expect that $B=M \frac{l-d}{l}$ from the magnetization surface currents for this torus with the gap.]

(The $B_{gap}=\mu_o H_{gap}=\frac{NI}{l}+M$ that we got from the pole model above clearly has the error in the term $M$ which would result if the torus had no gap. One additional error becomes apparent if we compute $B_m=\mu_o H_m +M$ and it gives a result with an additional subtractive term of $Md/(l-d)$ and does not agree with $B_{gap}$ of the pole model here and above ).

From the surface current result for $B$ we can compute $H_{gap}=\frac{B}{\mu_o}=\frac{NI}{l}+\frac{M(l-d)}{\mu_o l}$, and from this we also get by making the loop integral from the charges term zero that $H_m=\frac{NI}{l}-\frac{M(l-d)d}{\mu_o l(l-d)}=\frac{NI}{l}-\frac{Md}{\mu_o l}$.
These results are just slightly different from our pole model result above, but they agree with Feynman's results, and it turns out to be what we needed to explain the inconsistency we got with the pole model calculation.
We can also compute $B=\mu_o H_{gap}=\mu_o H_m+M$, and we get consistency with this magnetic surface current result. It turns out the assumption that $H_{gap \, poles}=\frac{M}{\mu_o}$ is only accurate to zeroth order in $d/l$. If we assume instead that $H_{gap \, poles}=A \frac{M}{\mu_o}$, where the form of $A$ is determined by solving for consistency with $B=\mu_o H_{gap}=\mu_o \frac{NI}{l}+AM=\mu_o H_m +M=\mu_o \frac{NI}{l}-AM \frac{d}{l-d} +M$, we do in fact get that $A=1-\frac{d}{l}=\frac{l-d}{l}$, and our pole model solution then agrees with the surface current result as well as Feynman's solution.

It's a lot of detail, but I resolved what was an inconsistency that really needed an explanation, and maybe at least one or two readers will find it of interest. :)

@TSny You might find these last two posts of interest. I remember discussing Feynman's solution to the transformer with the air gap with you a couple years ago=I did a little more work on it the other day to resolve the inconsistency that I had =I welcome your feedback. :)

 

[Charles Link]

I'm a little surprised that so far I have gotten no feedback from the previous two posts, but one reason may be that there may be a very limited number these days who are familiar with the pole model of magnetostatics.

Feynman has a very good solution to the problem of the air gap in the transformer, but he does a very mathematical approach, simply assuming two different $H's$ and using $\oint H \cdot dl=NI$ for the loop of the transformer. I thought my method of solution, with the current in the conductor coil and the magnetic poles at the surface endfaces at the gap as sources of $H$ demonstrates the physics principles in more detail.

I actually spent about 3 or 4 hours on it before I figured out why my solution was giving a slightly different answer. I must have checked my algebra about fifteen times, before I noticed that the assumption of an $H_{gap}$ that is completely independent of the gap width $d$ is only good for very, very small $d$.

@vanhees71 , @alan123hk I would enjoy your feedback on this one. There's a fair amount of algebra to sift through, but I think you might find the results somewhat interesting, and I welcome your feedback.

Edit: Both methods give the result for the linear material that $H_{gap}=\frac{NI \mu}{\mu_o l+(\mu-\mu_o)d}$. I am very pleased that with the $1 -\frac{d}{l}$ correction factor to $H_{gap \, poles}=\frac{M}{\mu_o}$ that my alternative solution is now in complete agreement with Feynman's solution. :)

 

[Charles Link]

Maybe you should write an insights article or such.

I had a similar thought, but there were too many items to cover, that were more easily written up with a couple of posts. Hopefully at least a couple Physics Forums readers found a couple of the posts to be of some interest. Meanwhile we've just had two full days of mostly sub-zero weather, and have one more day with similar weather before it gets about ten degrees warmer, so it's very good to have the Physics Forums to help make the day a little livelier. Cheers. :)

 

[vanhees71]

I'm a little surprised that so far I have gotten no feedback from the previous two posts, but one reason may be that there may be a very limited number these days who are familiar with the pole model of magnetostatics.

Feynman has a very good solution to the problem of the air gap in the transformer, but he does a very mathematical approach, simply assuming two different $H's$ and using $\oint H \cdot dl=NI$ for the loop of the transformer. I thought my method of solution, with the current in the conductor coil and the magnetic poles at the surface endfaces at the gap as sources of $H$ demonstrates the physics principles in more detail.

I actually spent about 3 or 4 hours on it before I figured out why my solution was giving a slightly different answer. I must have checked my algebra about fifteen times, before I noticed that the assumption of an $H_{gap}$ that is completely independent of the gap width $d$ is only good for very, very small $d$.

@vanhees71 , @alan123hk I would enjoy your feedback on this one. There's a fair amount of algebra to sift through, but I think you might find the results somewhat interesting, and I welcome your feedback.

Edit: Both methods give the result for the linear material that $H_{gap}=\frac{NI \mu}{\mu_o l+(\mu-\mu_o)d}$. I am very pleased that with the $1 -\frac{d}{l}$ correction factor to $H_{gap \, poles}=\frac{M}{\mu_o}$ that my alternative solution is now in complete agreement with Feynman's solution. :)

I can always only repeat myself. The "pole model" and the "Amperian current model" are entirely equivalent. It's just two different equivalent methods to calculate the one and only observable electromagnetic field with components $\vec{E}$ and $\vec{B}$.

 

[Charles Link]

Just an additional comment or two on posts 154 and 155. It turns out $\nabla \times H=0$,
[Edit: When we consider it with just the magnetic poles, without currents in the conductive coil],
(i.e. $\oint H \cdot dl=0$, with Stokes' theorem, analogous to $\nabla \times E =0$), along with $B$ being the same in both regions, (in the gap and in the material), makes for the magnetized torus with a gap to be just slightly different mathematically than the almost analogous electrostatic problem of capacitor plates with surface charge density $\pm \sigma$ separated by a distance $d$.

The result is a geometric factor of $A=1 -\frac{d}{l}$ that gets applied to the $H_{gap \, poles}=\frac{M}{\mu_o}$ result, where magnetic surface charge density $\sigma_m= \pm M$, so that $H_{gap \, poles}=\frac{M}{\mu_o}(1 -\frac{d}{l})$ for the magnetized torus with a gap.

For the electrostatic capacitor plates, $E=\frac{\sigma}{\epsilon_o }$, independent of $d$ for small $d$.

 

[Delta2]

I try to read hasty the posts here but couldn't find if the following issue is addressed:

We claim that $E_s=E_i$ inside the wire of the coil so inside the wire of the coil should be $$\nabla\times E_s=\nabla\times E_i$$. But since $E_s$ is electrostatic we should have $$\nabla\times E_s=0$$ everywhere hence also inside the wire of coil

But from Maxwell-Faraday equation we have $$\nabla\times E_i=-\frac{\partial B}{\partial t}$$ also everywhere hence inside the wire of the coil.

So why it should be $$\frac{\partial B}{\partial t}=0$$ inside the wire of the coil???

 

[Charles Link]

$E_s=-E_i$ inside the wire=perhaps a minor typo.

If I interpret it correctly, you are trying to address what the OP mentions, thinking that the changing flux needs to go into the wire. Later throughout this thread we attempted to show the validity of $E_s$ and $E_i$.

If $E_s$ and $E_i$ are accepted as valid, it looks like you have made an interesting calculation about $\frac{\partial{B}}{\partial{t}}$ being very close to zero in the conductor.

The thread is an old one, and it since has had a number of closely related topics covered, especially in regards to the transformer coil, and even items such as what if we place the leads of the voltmeter across an integer number of turns plus an additional fraction of the coil? We also discussed Feynman's write-ups including his write-up of the inductor coil, where he could have supplied a little more detail when he talks about the voltage that gets measured. Feynman also does a very good calculation to compute the magnetic field $B$ in the case of a transformer with an air gap. We also discussed some details about the origins of the formula $B=\mu_o H+M$, and the inclusion of currents in conductors as a source of $H$ in addition to the magnetic poles.

 

[Delta2]

If Es and Ei are accepted as valid, it looks like you have made an interesting calculation about ∂B∂t being very close to zero in the conductor.

Hmmm, then this means that the curl of $E_i$ is negligible (inside the wire always) but $E_i$ is not necessarily negligible... Not sure if mathematically we can have a vector field with small magnitude of his curl but the field itself doesnt have small magnitude. I guess we can have.

 

[Charles Link]

Hmmm, then this means that the curl of $E_i$ is negligible (inside the wire always) but $E_i$ is not necessarily negligible... Not sure if mathematically we can have a vector field with small magnitude of his curl but the field itself doesnt have small magnitude. I guess we can have.

I don't know if it is valid to assign properties to $E_i$ using $E_s$. It is $E_i$ that we have been trying to show as being whatever it is, and staying that way, and $E_s \approx -E_i$ inside the wire. The $E_s$ comes as a result of the $E_i$, and you can expect $E_s$ to have the properties of $E_i$, and if it doesn't, we need to explain why, but not the other way around. For example, we had to explain how $\nabla \times E_s=0$ and have a non-zero value for $\int E_s \cdot dl$ when we took the path to be around one ring, where it needed to have the value $\dot{\Phi}$.

Note that IMO it is the $E_s$ that winds up also in the external path, e.g. when you connect voltmeter leads, (with the $E_i$ staying where it is), where it seems with the EMF based approach, they allow the EMF to move into another part of the circuit loop, without specifying a mechanism for that to occur.

 

[Delta2]

Hmm, I see...

Well for me an interesting mathematical problem came up as a consequence of this discussion:

Find Vector fields that have same magnitude (everywhere) but different curl.

 

[vanhees71]

Now we are again in this discussion about some artificial split of the electric field. It's really ennoying!

 

[Delta2]

Now we are again in this discussion about some artificial split of the electric field. It's really ennoying!

Not sure why you finding it annoying, it is essentially the conservative and non-conservative part of E-field, $$E=-\nabla V-\frac{\partial A}{\partial t}$$, the first term is the conservative and the second term the non conservative.

 

[vanhees71]

There is no physically sensible split into these two parts. It's gauge dependent and thus cannot be in any way intepreted in a meaningful way!

 

[Delta2]

The gauge just changes V and A, but they remain conservative and non conservative.

For me it's like the E-field that comes from charge densities and the E-field that comes from time varying B-field or time varying current density. Thus it is physically sensible very much indeed.

But ok I 've got instructions not to talk in other forums except the HW forums I hope I didn't do big evil here.

 

[vanhees71]

Yes, it is totally arbitrary how you decompose $\vec{E}$ (a vector field with a clear physical meaning) into a "conservative" and "non-conservative" part. These parts have no physical meaning at all, because they can be chosen arbitrarily.

 

[Delta2]

They have physical meaning, the conservative part comes from charge densities (time varying or not) and conserves the work, that is the work in a closed loop is zero, and the non conservative part comes from time varying B-field or time varying current densities and doesn't conserve the work.

I wont write any more here no matter if you reply or not, because I am not allowed to write anywhere except HW forums but I believe I said something sensible and usefull, even if it is kind of wrong.

 

[vanhees71]

How do you come to that conclusion? The causal sources of the elctromagnetic fields are $\rho$ and $\vec{j}$ and nothing else. It's conceptually wrong to say a time varying magnetic field is the source of some ill-defined part of the electric field, etc. Gauge-dependent fields cannot be unambiguously intepreted physically!

 

[Charles Link]

@vanhees71 Your post 171 is interesting. Even Faraday's law with $\mathcal{E}=\oint E \cdot dl=-\dot{\Phi}$ sort of treats the changing magnetic flux like a source of $E$. It doesn't give precise information though on where the $E$ is located. In any case, we've probably discussed this one more than enough.

Edit: (a few hours later) But now I do see one other problem that arises if you simply do a Faraday's law treatment of the inductor and treat it as if the EMF is simply part of the complete circuit and that you can't localize it: It essentially means that the inductor problem is the same as an uncoiled inductor, (where you have unwrapped it), but now instead for every turn in the loop you place a cylinder of changing magnetic field in the center of the large area you now have for your circuit. You need one cylinder of changing magnetic flux for every turn that you had in the inductor. I wish I could draw a diagram, but I think you might get the picture. IMO, it is much simpler and better physics to localize the induced electric field as being part of the inductor, than to have all these cylinders of changing magnetic field in the stretched out circuit.

(@alan123hk I welcome your feedback on this "Edit" above. The problem with the inductor just came up on an introductory homework post, and there was some disagreement on how to solve it. I just made the observation that if you do not allow the EMF to be localized in the inductor, it makes for some rather clumsy physics, as described in the "Edit" above. If you alternatively choose to localize the EMF, you then have the problem that the $E_{induced}$ points to the positive voltage point, instead of away from it as in the case of charged capacitor plates. The $E_{induced}$ then needs special treatment, and can't be treated like an ordinary electric field. Either way there is a dilemma. I thought you might find this of interest).

( @Delta2 you might find the above "Edit" of interest=you can at least give it a "like" or even PM me if you want, since you apparently have instructions to only post in the HW).

For something else that you (and others) might find of interest, see https://www.physicsforums.com/threads/mutual-inductance-in-a-transformer.1059168/ where I'm waiting for the OP to return to see if they find it interesting that it is the torus loop length that needs to be used in their inductance formula. Cheers. :)

 

[vanhees71]

The inhomogeneous Maxwell equations give relations between the fields but no cause-effect relations between different field components. The split into electric and magnetic field components is dependent on the (inertial) frame of reference. The cause-effect relation is between the fields and the charge-current distribution, as, in a gauge-independent way, is reflected in the retarded solutions for the fields, the co-called "Jefimenko equations" (although they are known since Lorenz in the mid of the 19th century ;-)).

 

[Charles Link]

See https://physics.stackexchange.com/q...n-inductor-if-the-voltage-through-any-conduct

The author is Jan Lalinsky.

The topic may remain one that we might never get agreement on in the Physics Forums, but it does look like other intelligent people are putting some thought into this one.

See also my "Edit" in post 172. I am not satisfied with the Faraday's law being applied to the whole circuit and saying that the EMF all of the sudden moves out of the inductor coil and into the other parts of the circuit with no mechanism. The Coulombic field, which is what Jan Lalinsky uses in his calculations, is created from charge distributions in the coil, and IMO has some merit to it. That seems to be a reasonable way to describe the mechanism of how the electric field gets transferred out of the inductor and into external parts of the circuit, including into a voltmeter.

 

[vanhees71]

It should be clear that you have to integrate over a closed loop when going from the (fundamental) local form of the Maxwell equations to the integral form. For static areas and boundaries you have
[Typo corrected in view of #176]
$$\vec{\nabla} \times \vec{E}=-\partial_t \vec{B} \; \Rightarrow \; \int_{\partial A} \mathrm{d} \vec{r} \cdot \vec{E}=\mathcal{E}=-\mathrm{d}_t \int_A \mathrm{d}^2 \vec{f} \cdot \vec{B}.$$
See also the answer by "Ricky Tensor" in the stackexchange discussion.

 

[weirdoguy]

They have physical meaning

No they don't as has been discussed on PF dosens of times. If all physicists tell you they don't, and you don't have the proper knowledge and a bigger picture of electrodynamics, just stick to what they say instead of arguing.

 

[Charles Link]

If all physicists tell you they don't

I think this one is far from unanimous. See post 172 and 174. This was discussed earlier in this thread as well. IMO post 107 also has some calculations that would lend some validity to the idea that a Coulombic field gets established in the conductor coil , and thereby will appear external to the coil as well. Meanwhile the fellow on the Physics Stack Exchange seems to make a good case for it as well. There is too much other good physics out there to get the feathers ruffled over one item such as this one, but this one generates correct answers in any case, and I do prefer it over the other explanation (Faraday's law for the whole circuit) where the mathematics works, but the physics IMO is missing a piece or two.

 

[SredniVashtar]

I think the problem here is that some people are viewing the problem in the limited setting of the quasistatic approximation , while others are trying to see it in the most general setting possible.

Yes, the repartition of the electomagnetic field into electric and magnetic fields depends on the frame of reference, but in the problem of a coil fixed in the reference frame of the observer, you do not have any ambiguity. Likewise, when the frequency - or the rate of change of the fields - is so low that there is no appreciable retardation effect or radiation, you can safely use the Coulomb gauge to simplify the problem.

In these settings the decomposition of Etot into Ecoul and Eind makes perfect physical sense to me. One is the irrotational electrostatic field generated by the seemingly fixed interface and surface charge at a given instant in time, the other is the solenoidal induced electric field associated with the changing magnetic flux. They happen to be equal in magnitude and opposite in sign inside the perfect conductor , and they different greatly outside of it. One has sources and sinks, the other has closed field lines.

To say that this is unphysical is in my eyes the same as insisting that we should use relativistic mechanics to describe the motion of billiard balls.(Sorry I wanted to post this several pages ago, but in the last month or so I have been forced to face the worst aspects of real life :-( )

 

[Charles Link]

Thank you @SredniVashtar . I look forward to the feedback from your post. Perhaps others might also agree with your assessment.

 

[vanhees71]

I think the problem here is that some people are viewing the problem in the limited setting of the quasistatic approximation , while others are trying to see it in the most general setting possible.

Yes, the repartition of the electomagnetic field into electric and magnetic fields depends on the frame of reference, but in the problem of a coil fixed in the reference frame of the observer, you do not have any ambiguity. Likewise, when the frequency - or the rate of change of the fields - is so low that there is no appreciable retardation effect or radiation, you can safely use the Coulomb gauge to simplify the problem.

In these settings the decomposition of Etot into Ecoul and Eind makes perfect physical sense to me. One is the irrotational electrostatic field generated by the seemingly fixed interface and surface charge at a given instant in time, the other is the solenoidal induced electric field associated with the changing magnetic flux. They happen to be equal in magnitude and opposite in sign inside the perfect conductor , and they different greatly outside of it. One has sources and sinks, the other has closed field lines.

To say that this is unphysical is in my eyes the same as insisting that we should use relativistic mechanics to describe the motion of billiard balls.(Sorry I wanted to post this several pages ago, but in the last month or so I have been forced to face the worst aspects of real life :-( )

It is not so much the relativistic covariance, which is of importance here, and there's also no problem with the quasistatic approximations used in AC circuit theory, which of coarse breaks Lorentz covariance, but it's about gauge dependence. No gauge dependent quantity can have direct physical significance. Physical properties must be expressed in terms of gauge-independent quantities. In classical electrodynamics that means you can express everything in terms of the complete fields $\vec{E}$ and $\vec{B}$. Also you can split these fields in parts, based on the potential in an arbitrary gauge, but you cannot interpret these gauge-dependent parts physically.

 

[SredniVashtar]

I'm sorry but I don't follow you. It's classical electrodynamics we are talking about, right?
So, once we have determined the total E and B fields (in the coil's static frame of reference), what prevents you from applying Helmoltz's theorem to the total E field, in order to uniquely decompose it into its irrotational and solenoidal parts?

To my knowledge, for well behaved fields (associated with charge and current density distribution that vanish at infinity) the Helmoltz decomposition is unique so you will find one Ecoul component that is conservative, and one Eind component that is solenoidal. And these have clear physical interpretazions (although one might argue about their observability).

What is not unique are the potentials. What makes the solenoidal and irrotational components of the electric field not unique? Can you give a concrete example?

Also, in a previous message you mentioned that you consider voltage to be a potential difference; but if the only field you are using is the total electric field, how can you find a potential function for E=Etot?

 

[Charles Link]

And these have clear physical interpretazions (although one might argue about their observability).

Yes, and for $E_{induced}$ the clear physical interpretation is that it goes hand-in-hand with the changing magnetic field inside the inductor. Otherwise, if we just take a Faraday's law approach over the whole circuit, how does the $E_{induced}$ find its way into the other part of the circuit outside the coil?

The Coulombic response to the $E_{induced}$ inside the coil seems to be a very good explanation for how this occurs.

Edit: and note the $E_{induced}$ from the changing magnetic field occurs over a broad region, including outside the inductor, but it doesn't all of the sudden pick up a factor of $N$ for the $N$ turns anywhere. This factor of $N$ is something the Coulombic response picks up though in its path integral along (through) the coil, (and this path integral has the same value between the same two points outside the coil). The Coulombic response is IMO the reason behind why Faraday's law for the whole circuit loop works so well.

 

[vanhees71]

I'm sorry but I don't follow you. It's classical electrodynamics we are talking about, right?
So, once we have determined the total E and B fields (in the coil's static frame of reference), what prevents you from applying Helmoltz's theorem to the total E field, in order to uniquely decompose it into its irrotational and solenoidal parts?

To my knowledge, for well behaved fields (associated with charge and current density distribution that vanish at infinity) the Helmoltz decomposition is unique so you will find one Ecoul component that is conservative, and one Eind component that is solenoidal. And these have clear physical interpretazions (although one might argue about their observability).

What is not unique are the potentials. What makes the solenoidal and irrotational components of the electric field not unique? Can you give a concrete example?

Also, in a previous message you mentioned that you consider voltage to be a potential difference; but if the only field you are using is the total electric field, how can you find a potential function for E=Etot?

That's the point. If $\vec{\nabla} \times \vec{E}=-\partial_t \vec{B} \neq 0$ there is no potential field, and induction as in a transformer doesn't work in the static case, where the electric field has a potential.

Of course you can decompose any vector field in a potential and a solenoidal part, but as you argue yourself the corresponding scalar and vector potentials are unique only up to a gauge transformation. So this decomposition is to some extent arbitrary.

 

[Charles Link]

That's the point. If ∇→×E→=−∂tB→≠0 there is no potential field, and induction as in a transformer doesn't work in the static case, where the electric field has a potential.

I believe it was meant to read "[as] in the static case."

If the changing magnetic field can be considered to be contained in the inductor or transformer, it then does have a fairly well-defined scalar potential. We all ( @SredniVashtar , @vanhees71 , and myself) discussed this back in posts 71 and 72. There seems to always be the perhaps small problem for the EE that he can pick up or lose one turn of changing flux depending on how he strings the wires of his voltmeter w.r.t. the transformer core. Feynman even discusses the voltage from an inductor following his equation (22.3), but he omits the fine detail of this plus or minus extra turn of changing flux, which is really the whole idea behind what Professor Lewin presents as a paradox.
See https://www.feynmanlectures.caltech.edu/II_22.html

One reason we got to talking about this topic again in this thread is that there was (just a couple days ago) considerable disagreement on the explanation that was needed for an introductory homework problem for the voltage from an inductor with a changing current. We decided to bring the discussion back to this thread, rather than to have a debate on the homework thread of something that is far beyond the introductory level. We have had a couple of the participants from that discussion view this thread, and one or two have also given their inputs.

 

[SredniVashtar]

Of course you can decompose any vector field in a potential and a solenoidal part, but as you argue yourself the corresponding scalar and vector potentials are unique only up to a gauge transformation. So this decomposition is to some extent arbitrary.

Let's start from this last paragraph, first:

I am not sure we can have it both ways. Either the decomposition is unique, or it is arbitrary. And since there is a mathematical theorem to prove it, I would say it's settled: it is unique.
The potentials, on the other hand, are not unique. But this is not a problem and does not reflect neither on the uniqueness of Etot and Btot (I am considering a fixed frame of reference), nor on the uniqueness of the decomposition of Etot in Ecoul and Eind.

Now the first paragraph:
----------
That's the point. If $\vec{\nabla} \times \vec{E}=-\partial_t \vec{B} \neq 0$ there is no potential field, and induction as in a transformer doesn't work in the static case, where the electric field has a potential.
----------

I don't see how this could be the point in regard to the arbitrariness/uniqueness of the decomposition Ecoul, Eind. When dB/dt is nonzero, the Eind contribution that comes into being is perfectly determined in the given frame of reference (as a matter of fact, one can compute it withouth having to consider the coil at all). Due to this addition/interaction, the resulting electric field no longer admits an 'overall' potential function and therefore voltage (defined as the line integral of the total electric field) becomes path-depedent. The path independent part of voltage that admits a potential function is the line integral of the (uniquely defined, in the given frame of reference) partial component Ecoul.

So, how can you say (post 73) that for you "voltage is a potential difference" if you also think that the split of Etot in Ecoul and Eind is artificial and there is no physics in it (post 75)? What electric field do you use in your definition of voltage? Or do you just surrender the concept of voltage when dB/dt in nonzero?
In that post you also said that the Ecoul, Eind decomposition is gauge dependent. Care to reconsider, in light of Helmoltz's decomposition theorem?

 

[vanhees71]

I don't understand your problem with simply calculating $\vec{E}$ and $\vec{B}$ as it is done in the standard textbooks. You don't need artificial splits or at least you can't use arbitrary splits in gauge-dependent parts to interpret them physically. Of course, you use the potentials (in this case of quasistationary AC circuit theory the Coulomb gauge) to derive gauge-independent properties, e.g., the inductance matrix of a system of circuits, leading to the magnetic fluxes through the corresponding loops of the circuit,
$$\phi_j=\sum_k L_{jk} i_k,$$
which only involves gauge-independent quantities and thus is physically interpretable. The relation to the electromotive forces along these loops then is
$$\mathcal{E}_j=-\dot{\phi}_j=-\sum_k L_{jk} \dot{i}_k.$$

 

[Charles Link]

The textbooks seem to be somewhat deficient on this topic. Feynman did address it when he discussed the inductor, and did say in his derivation that the electric field is approximately zero in the conductor coil, but he left much of the finer detail to the reader. I applaud @SredniVashtar for doing such a fine job of defending some physics that has received much undue criticism.

 

[vanhees71]

A very detailed derivation can be found in Sommerfeld, Lectures on theoretical physics vol. 3.

 

[SredniVashtar]

I don't understand your problem with simply calculating $\vec{E}$ and $\vec{B}$ as it is done in the standard textbooks.

Assuming you are referring to me, I do not have a problem with computing E and B. What I am trying to ascertain is the reason why you say that the split of E (which I tend to call Etot) into Ecoul and Eind is arbitrary and gauge dependent when there is a mathematical theorem (Helmoltz's) that says such decomposition is unique (in the given frame of reference and under very general conditions met by the given setup).

Once we have determined E, which you seem to agree is the 'real' field, what breaks Helmoltz's theorem?

And why do you keep saying that the split (in the given fixed frame of reference) is not physically interpretable?
Eind is the electric field that is present in the space occupied and around the variable flux region due to its time dependency (without any coil around), while Ecoul is the field that would be generated by the configuration of displaced surface and interface charges in the material. The field E is the superposition of these two fields. Is this not a physical interpretation?
Maybe I am experiencing another language issue and we have different concepts of what is physically interpretable?

 

[Charles Link]

When dB/dt is nonzero, the Eind contribution that comes into being is perfectly determined in the given frame of reference (as a matter of fact, one can compute it withouth having to consider the coil at all).

This is from post 185, and I think this is a very important part of this whole concept. You can do the calculation for a transformer with just a primary coil, before adding the secondary coil, and the presence of the secondary coil does not appreciably affect the $E_{induced}$ that was computed without it.

The other part then comes in because the total electric field in the secondary coil is very nearly zero, so that there is necessarily a $E_c=-E_{induced}$ in the secondary coil. Since $\nabla \times E_c=0$, this electrostatic component has that $\int E_c \cdot dl$ is the same whether you run the integral through the coil or over the same two points external to it. IMO introducing the two components of the electric field is a very useful way of doing this computation, and it gives much insight to the underlying physics.

So far, we are not getting any rebuttal to post 189 , and I do think @SredniVashtar has presented a very solid case for what I think should be accepted as some very sound physics.

 

[SredniVashtar]

...and the presence of the secondary coil does not appreciably affect the $E_{induced}$ that was computed without it.

Yes, I keep thinking in terms of the Lewin ring problem where the magnetic field associated with the current in the ring is negligible. In any case Eind is the field associated with the resultant (changing) magnetic flux, and Ecoul is the field associated with the charge distribution.

But my point was not to defend a way to compute the fields, but the interpretation of the split.

 

[Charles Link]

It may be worth posting a "link" to the introductory physics homework problem where this topic recently presented itself.

See https://www.physicsforums.com/threads/simple-inductance-problem.1059240/

I really think the "split" electric fields concept merits a much better reception. I am grateful that you @SredniVashtar have done such a good job in defending it.

The standard Faraday's law EMF approach over the whole loop is mathematically correct, (as I have said before), and does get the correct answer, but IMO it is missing a piece or two, in that it doesn't explain how the electric field $E_{induced}$ all of the sudden emerges from the conductor coil and moves into another part of the circuit.