Inducing EMF Through a Coil: Understanding Flux

[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.

 

[Charles Link]

Even though this thread is an old one, and also controversial, I do think we introduced some very good physics here, and I find it much easier to understand what is going on in an inductor/conductive coil when I take the approach that the $E_{induced}$ gets balanced by the conservative $E_{electrostatic}$.

 

[Charles Link]

and a follow-on to the above: With a long solenoidal type inductor, even with an iron core, we know very well that the magnetic field is very localized. We even know by symmetry what $E_{induced}$ is, so why not use that extra information and say that we know what $E_{induced}$ is inside the coil? and that if we have a twisted pair coming off the coil, we don't need to sit there in ignorance with the Kirchhoff's law that is in a much more watered-down form, that says we don't know where the $E_{induced}$ or EMF is located in the circuit loop. We know almost exactly where the $E_{induced}$ is, (it is in the coils), and we also know that in the conductor there will be a conservative $E_{electrostatic}$ to balance the $E_{induced}$, and we also know that this $E_{electrostatic }$, by its conservative nature, will also display itself away from the conductive coil in any circuit loop that we form.