Some reasoning about Alfvén’s frozen-in flux theorem

[ZX.Liang]

TL;DR Summary

There seems to be a problem with Alfvén’s frozen-in flux theorem.

Alfvén’s theorem is very famous in plasma physics. It is also often used in astrophysics.
The link in Wiki: https://en.wikipedia.org/wiki/Alfvén's_theorem
However, after a series of continuous reasoning, it seems that this theorem has problem.
What errors can be hidden in the reasoning processes below?

0. The ideal conductor (or fluid) cannot cut the magnetic field line, which is a conclusion of Alfvén’s frozen-in flux theorem. The reasoning processes:
1. If the ideal conductor cannot cut the magnetic field line, its internal magnetic flux will not change;
2. If the magnetic flux does not change, there will be no eddy electric field inside it;
3. If there is no eddy electric field, there will be no eddy current;
4. If there is no current, the physics process will be independent of the resistivity; (This step of reasoning is very important)
5. If the physical process is independent of resistivity, it is unnecessary to distinguish between ideal conductor, non-ideal conductor and insulator;
6. The result of the successive reasoning is that if the ideal conductor cannot cut the magnetic field line, its performance in the magnetic field is the same as that of the non-ideal conductor and insulator.
Did you see it? If the frozen-in theorem is hold, it will lead to an absurd conclusion: in the magnetic field, the ideal conductor and insulator will have the same performance!
Conversely, if one is convinced that the ideal conductor and insulator have different performances, he must deny the frozen-in theorem. That is, we must admit that ideal conductors can cut magnetic field lines. This proves that the frozen-in theorem is wrong.

What errors can be hidden in above reasoning processes?

For more information about this problem, see the following link:
http://www.kpt-planet.com/Alfven's theorem/Alfven's theorem.PDF

 

[Baluncore]

What errors can be hidden in above reasoning processes?

You are switching the conductivity during the reasoning.
Analyse the situation independently for each scenario.

 

[Ken G]

The error is in step 3. You imagine that if the electric field is zero, the current is also zero. This is a classic error in mathematics called "multiplication by zero." The current is the conductivity times the field, but if the conductivity is infinite and the field is zero, the current is not zero, it is indeterminate. To determine what it is, you must back away from the impossible limit of perfect conductivity, and then you have no problem. If your only point here is "perfect conductivity is impossible," we must all say yes, obviously, it is merely a convenient limit for what is actually very high conductivity. But that limit is not convenient if you do not apply it correctly.

 

[ZX.Liang]

You are switching the conductivity during the reasoning.
Analyse the situation independently for each scenario.

Thank you for your reply.

Conductivity is meaningful only when current exists.
If the prerequisite is that the current does not exist at all, the conductivity is meaningless.

If the frozen-in theorem holds, the induced electric field and current have been prohibited. So, the conductivity is meaningless and there is no difference between an ideal conductor and an insulator.

 

[ZX.Liang]

The error is in step 3. You imagine that if the electric field is zero, the current is also zero. This is a classic error in mathematics called "multiplication by zero." The current is the conductivity times the field, but if the conductivity is infinite and the field is zero, the current is not zero, it is indeterminate. To determine what it is, you must back away from the impossible limit of perfect conductivity, and then you have no problem. If your only point here is "perfect conductivity is impossible," we must all say yes, obviously, it is merely a convenient limit for what is actually very high conductivity. But that limit is not convenient if you do not apply it correctly.

Thank you for your reply.

The mathematical relationship you said is right.

The problem we are discussing is not just a pure mathematical problem but a physical problem. In this physical problem, there is a causal relationship between the induced electric field and the induced current, that is, there is no current without the electric field.

If there is no electric field but a current, what factors determine the magnitude and direction of the current? This brings new problem.

This issue is discussed in section 2 of our linked document. The conclusion is that the magnitude and direction of the current are determined by Lenz's law.

 

[Ken G]

Thank you for your reply.

The mathematical relationship you said is right.

The problem we are discussing is not just a pure mathematical problem but a physical problem. In this physical problem, there is a causal relationship between the induced electric field and the induced current, that is, there is no current without the electric field.

Yes.

If there is no electric field but a current, what factors determine the magnitude and direction of the current? This brings new problem.

The issue before was the ratio of the current to the field, which is called "conductivity." In a very good conductor, that ratio is large. So try your logic again using a large value for that ratio, but not an infinite one. If you use infinity, your logic falls apart because you cannot claim the current is zero when the field is zero. If you don't use infinity, then it is not necessary that the field be zero in the first place. Either way, you see the flaw in your logic. Is that clear now?

This issue is discussed in section 2 of our linked document. The conclusion is that the magnitude and direction of the current are determined by Lenz's law.

Is there another question here?

 

[Baluncore]

If the prerequisite is that the current does not exist at all, the conductivity is meaningless.

Many things do exist, but are meaningless or irrelevant this time. Don't dismiss them out of hand, or they may come back to bite you.

Different situations will be applicable at different times, in different places. You seem to assume that the system is static, has always been and will always be. You must consider the evolutionary path to the present situation, as waves of change migrated through the system.

The frozen-in flux theorem may be the limit at t=∞. It may be an impossible bound. But that does not mean it cannot help us understand or explain what we see.

 

[ZX.Liang]

The issue before was the ratio of the current to the field, which is called "conductivity." In a very good conductor, that ratio is large. So try your logic again using a large value for that ratio, but not an infinite one. If you use infinity, your logic falls apart because you cannot claim the current is zero when the field is zero. If you don't use infinity, then it is not necessary that the field be zero in the first place. Either way, you see the flaw in your logic. Is that clear now?

I said that there is no current without electric field, which is based on causality, not on the ratio of current to electric field (conductivity), because the causal relationship of "no electric field, must no current" is more fundamental than Ohm's law.
In the relationship between current and electric field, in addition to the ratio (conductivity), there are inductance (inductive reactance) and capacitance (capacitive reactance). Because the magnetic freezing equation and frozen-in theorem study a time-varying system rather than a stable system, inductance and capacitance must be considered to accurately describe the relationship between current and electric field.
In a steady-state system, the relationship between current and electric field is determined by resistance (Ohm's law). In a time-varying system, the relationship between current and electric field is determined by complex impedance. If the resistance and capacitance in the time-varying system can be ignored, the relationship between current and electric field is determined by Lenz's law. This is the key to the problem. The linked document discussed this relationship.

 

[ZX.Liang]

Different situations will be applicable at different times, in different places. You seem to assume that the system is static, has always been and will always be. You must consider the evolutionary path to the present situation, as waves of change migrated through the system.

The frozen-in flux theorem may be the limit at t=∞. It may be an impossible bound. But that does not mean it cannot help us understand or explain what we see.

In the linked document, we introduce inductance, capacitance, and lenz's law into the time-varying system. If you look carefully, you may have a new view.

 

[Ken G]

I said that there is no current without electric field, which is based on causality, not on the ratio of current to electric field (conductivity), because the causal relationship of "no electric field, must no current" is more fundamental than Ohm's law.
In the relationship between current and electric field, in addition to the ratio (conductivity), there are inductance (inductive reactance) and capacitance (capacitive reactance). Because the magnetic freezing equation and frozen-in theorem study a time-varying system rather than a stable system, inductance and capacitance must be considered to accurately describe the relationship between current and electric field.
In a steady-state system, the relationship between current and electric field is determined by resistance (Ohm's law). In a time-varying system, the relationship between current and electric field is determined by complex impedance. If the resistance and capacitance in the time-varying system can be ignored, the relationship between current and electric field is determined by Lenz's law. This is the key to the problem. The linked document discussed this relationship.

I thought you were asking about what was wrong with the wrong argument you gave above. I told you what was wrong, you incorrectly reasoned that an infinite conductance implied a zero current. That's simply not correct, I'm not sure any more if you really want to know why.

 

[ZX.Liang]

I thought you were asking about what was wrong with the wrong argument you gave above. I told you what was wrong, you incorrectly reasoned that an infinite conductance implied a zero current. That's simply not correct, I'm not sure any more if you really want to know why.

Maybe what I said is not clear enough.

I never thought that infinite conductance would cause zero current. I think that zero electric field must cause zero current.

In the linked document, we want to prove that a non-zero electric field and an infinite conductivity will not produce an infinite current (So, it is unnecessary that Alfven is worried about the infinite current), because an ideal conductor in a time-varying system, Lenz's law determines the magnitude and direction of the current.

 

[Ken G]

Maybe what I said is not clear enough.

I never thought that infinite conductance would cause zero current. I think that zero electric field must cause zero current.

OK, one final time:
You said infinite conductance forces there to be zero field, which forces there to be zero current. Ergo, by the most basic of logic, called a syllogism (https://en.wikipedia.org/wiki/Syllogism), you claimed that infinite conductance requires there to be zero current. So you did think that infinite conductance would cause zero current, this was precisely your logic and there's not much point in denying it now. I told you why that is a wrong thing to think. If you don't care that you were thinking wrongly there, there just isn't a whole lot more I can do.

In the linked document, we want to prove that a non-zero electric field and an infinite conductivity will not produce an infinite current (So, it is unnecessary that Alfven is worried about the infinite current), because an ideal conductor in a time-varying system, Lenz's law determines the magnitude and direction of the current.

It is obvious that nothing will produce an infinite current, so there's not much point having some long discussion to prove something that is already obvious.

 

[ZX.Liang]

You said infinite conductance forces there to be zero field, which forces there to be zero current.

Maybe my expression is not clear enough, which makes you misunderstand my view.
"Infinite conductivity force there to be zero field, which force there to be zero current" This is not my view. This is an absurd deduction from the conclusion of the magnetic freezing theorem. All my discussions are aimed at proving that this conclusion is wrong. The root of the error is "The idea conductor (or fluid) cannot cut the magnetic field line". It is the starting point of reasoning that is wrong, not the process of reasoning.
Please look at the beginning post again:

0. The ideal conductor (or fluid) cannot cut the magnetic field line, which is a conclusion of Alfvén’s frozen-in flux theorem. The reasoning processes:
1. If the ideal conductor cannot cut the magnetic field line, its internal magnetic flux will not change;
2. If the magnetic flux does not change, there will be no eddy electric field inside it;
3. If there is no eddy electric field, there will be no eddy current;
4. ……

Step 0 here is not my opinion, but the conclusion of the frozen-in theorem. The error is at this starting point. Starting from the conclusion of the frozen-in theorem, we can deduce an absurd conclusion to prove that the frozen-in theorem is wrong. This is our logic.

Why is the magnetic freezing theorem wrong? The answer is given in the linked document.

It is obvious that nothing will produce an infinite current, so there's not much point having some long discussion to prove something that is already obvious.

You, I and Alfvén all think that there can be no infinite current. However, why can't there be infinite current? Alfvén 's answer is that an ideal conductor cannot cut magnetic field lines. We gave a completely different answer in the linked document:
An ideal conductor can cut magnetic field lines and generate electric field, but it cannot generate infinite current.
1. In three special cases, the induced electric field is irrotational, so eddy current cannot be generated.
2. In general, the amplitude of induced current is limited by Lenz's theorem.
Therefore, the ideal conductor cutting the magnetic field line will not cause infinite current.

Our understanding will bring many benefits. for example:
According to the frozen-in flux theorem, when an external force pushes an ideal conductor in the magnetic field, there will be a magnetic resistance due to dragging the frozen magnetic lines. But what factors are related to the magnetic resistance? There is no clear answer. The common saying is that the stronger the magnetic field, the greater the resistance.

With our theory, the magnitude of magnetic resistance is proportional to the change of exogenous magnetic flux and the magnetic gradient.
Especially, the magnitude of magnetic resistance has nothing to do with the magnetic field strength, only with the magnetic field gradient. We also proved this with experiments.

I hope you can take a closer look at whether the view in our document is tenable.

 

[Ken G]

Maybe my expression is not clear enough, which makes you misunderstand my view.
"Infinite conductivity force there to be zero field, which force there to be zero current" This is not my view.

Quotes from you:

2. If the magnetic flux does not change, there will be no eddy electric field inside it;
3. If there is no eddy electric field, there will be no eddy current;

Now you are saying this was not your view, it was just something wrong that you decided to post. Nevertheless, you asked in your initial question why it was wrong, and I told you why it was wrong. You then argued it was wrong for other reasons. The problem with your logic is that when something is wrong for one reason, it can also be wrong for a thousand reasons, that is just the nature of wrong things.

This is an absurd deduction from the conclusion of the magnetic freezing theorem. All my discussions are aimed at proving that this conclusion is wrong.

So you say wrong things, then prove they are wrong. That sounds like a complete waste of time.

 

[ZX.Liang]

Quotes from you:

2. If the magnetic flux does not change, there will be no eddy electric field inside it;
3. If there is no eddy electric field, there will be no eddy current;

Now you are saying this was not your view, it was just something wrong that you decided to post. Nevertheless, you asked in your initial question why it was wrong, and I told you why it was wrong. You then argued it was wrong for other reasons. The problem with your logic is that when something is wrong for one reason, it can also be wrong for a thousand reasons, that is just the nature of wrong things.So you say wrong things, then prove they are wrong. That sounds like a complete waste of time.

It seems that you still don't understand what I asked at the beginning.
My question is: “What errors can be hidden in the reasoning processes?”

The "reasoning processes" here refers to the step 1 to step 6, excluding the step 0, because the step 0 is the precondition of our reasoning and is not my reasoning.

If you can tell me directly which step is wrong from step 1 to step 6, I will know where I am wrong.

If all the reasonings from step 1 to step 6 are correct, but the absurd conclusion is obtained, it will prove that the first half sentence of step 1 (equivalent to the step 0) is wrong, that is, the frozen-in theorem is wrong.

Can you tell me directly which step is wrong from step 1 to step 6?

 

[berkeman]

It seems that you still don't understand what I asked at the beginning.
My question is: “What errors can be hidden in the reasoning processes?”

Wouldn't such errors on your part have been caught in the peer-review process for these published papers of yours? They are peer-reviewed by credible journals, correct?

 

[ZX.Liang]

Wouldn't such errors on your part have been caught in the peer-review process for these published papers of yours? They are peer-reviewed by credible journals, correct?

Our two papers were approved by two editors and four reviewers. Professor Bellan of Caltech was also one of the reviewers. He was recommended by us as a reviewer, and he was contacted later. He wrote a book called "fundamentals of plasma physics".

Of course, they may all be wrong. So, hope you can tell me directly which step is wrong in my reasoning from Step 1 to Step 6.

 

[Ken G]

If all the reasonings from step 1 to step 6 are correct, but the absurd conclusion is obtained, it will prove that the first half sentence of step 1 (equivalent to the step 0) is wrong, that is, the frozen-in theorem is wrong.

Can you tell me directly which step is wrong from step 1 to step 6?

I can only repeat what I already said. There is an error going from step 2 to step 3, it is the familiar mathematics error of multiplying infinity by zero and claiming the result must be zero, which is not correct. To wit:

1. If the ideal conductor cannot cut the magnetic field line, its internal magnetic flux will not change;
(you are saying that if the conductivity is infinite, which is the meaning of an "ideal conductor", the internal magnetic flux inside a fluid parcel cannot change, i.e., the field is "frozen in". The reason it cannot change is because infinite conductivity cannot support a finite electric field.)

2. If the magnetic flux does not change, there will be no eddy electric field inside it;
(Yes, that is why the contained flux is preserved.)

3. If there is no eddy electric field, there will be no eddy current;
(This is the wrong step. When the conductivity is infinite, a current can appear even when the electric field is zero. The current is the infinite conductivity times the zero field. Of course that is indeterminate, so one must be more careful, and use a very very large conductivity and a very very small electric field. But the current is normal. The action of the current is to make the magnetic field strength change so as to maintain the same magnetic flux crossing any fluid parcel, however the parcel moves and contracts/expands.

 

[berkeman]

After a Mentor discussion, this thread is done. @ZX.Liang -- please do not try to use PF to peer review your "peer reviewed" publications.