Inducing EMF Through a Coil: Understanding Flux

[vanhees71]

$B=H$ holds in vacuo, where $M=0$ (in Gaussian or Heaviside-Lorentz units). What else should be $M$ without matter present?

The magnetic-pole model works, because it's equivalent to assuming the current density to be $\propto \vec{\nabla} \times \vec{M}$, including possible surface-current densities like in my example of the homogeneously magnetized sphere, which is the most simple example due to symmetry.

In SI units the constitutive equation is $\vec{B}=\mu_0 (\vec{H}+\vec{M})$. I also mess up where the $\epsilon_0$ and $\mu_0$ should go. The mnemonics is that the material sources like magnetization belong to $\vec{H}$ and then you need $\mu_0$ to get $\vec{B}$ dimensionally correct ;-). By definition of the SI units the macroscopic Maxwell equations read
$$\vec{\nabla} \cdot \vec{B}=0, \quad \vec{\nabla} \times \vec{E}+\partial_t \vec{B}=0$$
and
$$\vec{\nabla} \times \vec{H} -\partial_t \vec{D}=\vec{j}, \quad \vec{\nabla} \cdot \vec{D}=\rho,$$
with $\rho$ and $\vec{j}$ the "free charges and currents", including polarization charge densities $\rho_{\text{pol}}=\vec{\nabla} \cdot \vec{P}$ and magnetization currents $\vec{j}_{\text{mag}}=\vec{\nabla} \times \vec{M}$ with possible surface-charge densities and surface-current densities at boundaries between different media (or a medium and vacuum).

 

[Charles Link]

Note in the above post 131, that the current loop running around the brick is a more accurate description of what is going on with the building block of the magnetic moment than a plus magnetic charge at one face and a minus at the other. Thereby, at least in this classical description, we need to compute what $B$ gives in the material using this other building block, and compare it to what the plus and minus block gives. The result we find is that $B=H+4 \pi M$,(cgs units) , with the $4 \pi M$ being the correction term.

I'm repeating some of what I previously said, but this new part is something I didn't put in the original explanation. Even the current loop is a classical description=it may not be the perfect description, but it is deemed to be closer to the complete description than a (fictitious) plus and minus magnetic charge.

The plus and minus charge pole model says that $B=H$ everywhere, but the current loop model says that in the material $B=H + 4 \pi M$, so we add the $4 \pi M$ to the $H$ and consider that to be a description that is much more mathematically accurate. It still is an approximation, as @vanhees71 mentions in post 139, but over the years it has been accepted as being a fairly good one.

 

[vanhees71]

The $4 \pi M$ is not "a correction term" but a source term due to the presence of a medium. It's describing the response of the medium to the external electromagnetic fields, i.e., the rearrangement of the charge and magnetic-moments distribution of the particles making up the medium.

I don't understand why you think that $\vec{B}=\vec{H}$ everywhere in the magnetic-charge-dipole model. Of course, what you call $\vec{H}$ is indeed model-dependent, i.e., in which arbitrary way you split the sources into "free/external" and "internal" parts. You can arbitrarily shuffle sources to fields and vice versa. The physical results are the same, particularly $\vec{E}$ and $\vec{B}$, which are the observable fields.

 

[Charles Link]

The 4πM is not "a correction term" but a source term due to the presence of a medium

What I am using as a source term is the surface current per unit length term $K_m= c \, M \times \hat{n}$ that even Griffiths has in his computation of the vector potential $A$ in section 6 of his book. There is no $M$ source term. (The other source term is a $J_m=c \, \nabla \times M$). It is surprising that the surface current term calculation gives $4 \pi M$ as the correction, so it looks like a source term of $4 \pi M$, but it comes from a Biot-Savart or amperes law computation of $B$ from the surface current per unit length $K_m= c \, M \times \hat{n}$ on a long cylinder, where we get $B=4 \pi M$.

If the cylinder is chosen to be of finite length, then surprisingly enough, the result of the Biot-Savart computation of the surface currents of the magnetized cylinder is precisely $B=H + 4 \pi M$.

If you look this one over very carefully, I think you might come to agree with me on this.

Edit:

The $4 \pi M$ then gets added into the pole model description as a source term for $B$, where we have discovered that the pole model result that $B=H$ everywhere isn't completely accurate, by using our improved description of the current loop as a building block. Thereby the pole model then uses the $4 \pi M$ as a source term, but the pole model itself, at least as far as I can tell, would give no information on what this correction term, if any, might need to be.

If we stick with our fictitious plus and minus charge model as the building block, we would conclude that $B=H$ everywhere, (where $H$ is computed from the poles, i.e. $\rho_m=-\nabla \cdot M$ and $\sigma_m=M \cdot \hat{n}$). and note the $\sigma_m=M \cdot \hat{n}$ is a magnetic surface charge=on the endfaces=it is not a volume source term. When we compute $B$ for a long magnetized cylinder, using the pole model, the $\sigma_m$ on the endfaces are assumed to have little effect, so that $H =0$ for the long cylinder, even though it has $\sigma_m=\pm M$ on the endfaces. We then throw in our correction term (that we discovered from our surface current calculations) with the result that $B=4 \pi M$ for the long cylinder.
(and note, although I may be stating the obvious, the endfaces for a long cylinder are not infinite sheets of charge in which case we would have $H=-4 \pi M$. Instead their contribution is minimal, and considered to be zero).

Yes, the pole model does use $4 \pi M$ as a source term for $B$, but I believe it comes as the result of a more complete study of the surface current/current loop description. The pole model, with its plus and minus magnetic charges, by itself would have no information on what the correction term needed to be. The pole model uses $H$ and $4 \pi M$ as sources, but only the $H$ itself comes from the magnetic charge description. Had the fictitious plus and minus magnetic charge model been deemed to be completely accurate, then there would be no $4 \pi M$ correction term in the calculation of the magnetic field $B$.

 

[vanhees71]

Ok course the surface current is a source term. What else should it be? I've shown explicitly the equivalence of both descriptions in this example of the homogeneously magnetized sphere. It's of course valid for magnets of any shape and any polarization.

 

[Charles Link]

@vanhees71 Please read the last three paragraphs in the above post 144 that I just added (=below Edit). I think that might help tie it all together.

The pole model with $H$ from the poles (the plus and minus magnetic charges) was thought to represent the magnetic field at one time. Then the surface current calculation (with the microscopic current loops as building blocks) showed that the $B$ in the material was not just $H$, but was found to be $B=H+4 \pi M$, but that $B=H$ was indeed correct outside the material.

Note that using plus and minus charges as building blocks does give the correct $E$ in the material when we consider the $P$ in dielectric materials, (note that we do not ever need an additional $4 \pi P$ to get the fundamental physical field $E$), but the $H$ that is computed from magnetic charges does not represent the $B$ that is calculated from current loops in the material.

This so far does not include currents in conductors. Finally $H$ is then redefined, (as is necessary to keep the formula $B=H+4 \pi M$ intact), to include currents in conductors as sources of $H$ using Biot-Savart and basically adding it to both sides of the formula, so that $H$ is now at the level of a mathematical construction.

With the introduction of $B=H+4 \pi M$ in the material, $H$ was already determined to not represent the actual field $B$ in all cases. Had the $H$ not been redefined to include the currents in conductors, the formula $B=H+4 \pi M$ would not be valid when currents in conductors are present.

With these modifications to the pole model, the surface current method and the pole model give identical results for the computed magnetic field $B$ for virtually any magnetization function $M$ and any distribution of currents in conductors. (Note that $B=H+4 \pi M$ is basically a pole model formula. The surface current method does not recognize poles as sources. For the pole model, you first compute $H$ and then you get compute $B$ as $B=H +4 \pi M$ ).

@alan123hk I would also welcome your feedback on this. I think you might find it interesting, starting around post 130.

 

[alan123hk]

@alan123hk I would also welcome your feedback on this. I think you might find it interesting, starting around post 130.

I am really interested in this topic, but obviously my ability in this area is not enough, so what I can do at this stage is to slowly learn and then understand.

 

[Charles Link]

I am really interested in this topic, but obviously my ability in this area is not enough, so what I can do at this stage is to slowly learn and then understand.

@alan123hk Perhaps a good place to start would be the following thread that I started. I tried to keep things simple in this thread:

See https://www.physicsforums.com/threads/a-magnetostatics-problem-of-interest-2.971045/

 

[Charles Link]

I would like to point out one item that might help to emphasize it=when the magnetization current runs around the outside of the rectangular building block, the result for bricks stacked together is that the current between the bricks cancels, but we get a surface current flowing (e.g. counterclockwise with $M$ in the $+\hat{z}$ direction=note $K_m=c \, M \times \hat{n}$ where $\hat{n}$ is the outer pointing unit normal vector to the surface) around the outer surface of the bricks. We then use Biot-Savart or ampere's law to compute the magnetic field $B$ both in the material and outside the bricks. If we make the bricks into the shape of a long cylinder, we find $B=4 \pi M$ is the result we get for $B$ inside the material in cgs units, with surface current per unit length $\vec{K}_m=c \, \vec{M} \times \hat{n}$.

This surface current is a well-known result that Griffiths comes up with in section 6 of his E&M text in the derivation of the vector potential $A$. There is no charge transport in these surface currents, so one can argue whether they are completely real or the result of the mathematics of the magnetic moment.

Edit: Note also that for the plus and minus magnetic charges on each block as building blocks, the charges butted up against each other from adjacent bricks in the z direction cancel each other, and we are left with a magnetic surface charge density $\sigma_m=+M$ on the +z end, and a magnetic surface charge density of $\sigma_m=-M$ on the -z end.

When we do the Coulomb's law (magnetic form ) for these bricks, we get the same result as the current loop bricks (using Biot-Savart) external to the material, (i.e. $B=H$), but inside the material we find $B=H+4 \pi M$ (cgs units), where $H$ is the Coulomb's law result, and $B$ is the Biot-Savart result.

 

[Charles Link]

The question arises, why do we even use an $H$ anymore if it is found to not represent the magnetic field $B$? The answer is that it is one of the more useful mathematical constructions. It comes in very handy with a transformer where we have a version of Maxwell's equations $\nabla \times H =J_{conductors}+\dot{D}$, which in the steady state with Stokes' theorem becomes $\oint H \cdot dl=NI$. (MKS units).

It also comes in very handy for computing the magnetic flux when a transformer has an air gap, where we get magnetic poles on the two faces at the air gap. $H$ is then assumed to take on two different values=one in the material and another in the air gap, and the magnetic material is assumed to be linear. We then have with $B$ being continuous that $B=\mu H_{material}=\mu_o H_{gap}$ and magnetic flux $\Phi=BA$. With the integral around the complete inside path of the transformer we have $\oint H \cdot dl=NI$, so that we have two equations to solve for the unknowns $H_{material}$ and $H_{gap}$. Feynman must have considered this one to be important as well, because he did a write-up on it. See https://www.feynmanlectures.caltech.edu/II_36.html right around equation (36.26).The $H$ is something that comes out of the pole model of magnetization in materials. It really doesn't show up in a surface current presentation. It is also used in hysteresis curves of $M$ vs. $H$, where the $H$ typically comes from the current in a conductor of a solenoid around a long sample of cylindrical shape. Note that as mentioned in post 146, besides coming from contributions from the poles, which are insignificant in the case of a long cylinder, $H$ is defined to also include the currents in conductors as sources, using Biot-Savart.

For a Physics Forums thread of a transformer with an air gap see https://www.physicsforums.com/threads/absolute-value-of-magnetization.915111/ also referenced in post 111 of this thread.

 

[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.