The Faraday disk in the context of the theory of relativity

[Ivan Nikiforov]

Good afternoon! I would like to understand how the Faraday disk works and get answers to two questions. The working conditions are as follows: 1) we rotate the external circuit with a voltmeter relative to a stationary disk; 2) the external circuit is not in a magnetic field. Questions: 1) will an electric current flow in such a system? 2) what forces will act on the external circuit with the voltmeter? I would like to consider these processes precisely from the point of view of the theory of relativity, since the Faraday disk is based on the principle of relativity of simultaneity. Thank you in advance!

 

[anuttarasammyak]

On external circuit which does not include the disk, Lorentz force of v X B appears and makes electrons move where v is r$\omega$.

 

[TSny]

View attachment 356917
Good afternoon! I would like to understand how the Faraday disk works and get answers to two questions. The working conditions are as follows: 1) we rotate the external circuit with a voltmeter relative to a stationary disk; 2) the external circuit is not in a magnetic field.

Here is an elaboration of @anuttarasammyak's comment.

By the "external circuit," I think you mean this part

This part of the circuit is in a magnetic field. The field of the magnet extends to this part.

Questions: 1) will an electric current flow in such a system?

Yes.

2) what forces will act on the external circuit with the voltmeter?

As the external circuit rotates in the direction shown in the figure, the free charge carriers in the external circuit move through the B field. So, a carrier with charge $q$ and velocity $\mathbf v$ experiences a magnetic force $\mathbf F = q \mathbf v \times \mathbf B$. For simplicity, imagine the B field to be uniform:

In the horizontal parts $ab$ and $cd$, the magnetic force on electrons is toward the axle: from $b$ toward $a$ and from $c$ toward $d$. See this video for a discussion of the motional electromotive force (emf) induced in a rod rotating in a B field. Since $ab$ is longer than $cd$, the emf in $ab$ is greater than in $cd$. Thus, the net emf in the external circuit drives electrons in the external circuit in the counterclockwise direction: $d$ to $c$ to $b$ to $a$. Electrons arriving at $a$ flow down the axle and back to $d$ through the stationary disk.

I would like to consider these processes precisely from the point of view of the theory of relativity, since the Faraday disk is based on the principle of relativity of simultaneity. Thank you in advance!

I'm not clear on what you want here.

 

[Ivan Nikiforov]

On external circuit which does not include the disk, Lorentz force of v X B appears and makes electrons move where v is r$\omega$.

Thanks for the comment. By the condition of the problem, the external circuit is not in a magnetic field, therefore the Lorentz force cannot appear.

 

[Ivan Nikiforov]

I'm not clear on what you want here.

Thanks for the comment. Yes, you understand correctly which part of the system rotates. I would like to understand how this system will work if the external circuit is not in a magnetic field. Let's say that due to the design, we will make it so that the magnetic field bypasses the external circuit.
The following is stated in the literature: "An observer moving at a velocity u, relative to a medium with magnetization M, detects an electric moment P equal to:

At the same time, it is not claimed that the moving observer must be in a magnetic field. Based on this statement, we can give a slightly abstract example. A magnet with a conductive surface (a metal magnet) lies motionless on the table. A copper wire moves relative to the magnet, at a great distance from it and perpendicular to the magnetic field lines. An observer located in a copper wire detects an electric field on the sides of the magnet. If a copper wire is connected by brushes to the sides of a magnet, current will flow in a closed circuit. At the same time, the copper wire is not in a magnetic field. Is this interpretation correct?

A magnetic field can be closed in addition to an external circuit, for example, like this.

 

[anuttarasammyak]

Thanks for the comment. By the condition of the problem, the external circuit is not in a magnetic field, therefore the Lorentz force cannot appear.

Then apply B=0 to my comment, please. No B No emf.

 

[Ivan Nikiforov]

Then apply B=0 to my comment, please. No B No emf.

I generally agree with you. At the same time, it turns out that everything is not so obvious. If in this model (when the external circuit is not in a magnetic field) we start rotating the disk relative to the external circuit again, an electric current is likely to flow. According to the principle of relativity, the rotation of the disk and the rotation of the external circuit are equivalent. Indeed, in empty space we cannot understand whether the disk is rotating or the external circuit is rotating. We can only say that there is relative motion between the disk and the external circuit. Therefore, we come to the conclusion that the rotation of the circuit, which is not in a magnetic field, relative to the disk, leads to a current. Is this interpretation correct?

 

[anuttarasammyak]

Rotation of the circuit where B=0 and rotation of the disk where B$\neq$ 0 should be investigated independently. You don't have to think of the correlation between.
There are three things; maget, disk and circuit. What kind of motion involving the three are you interested in ?

 

[Ivan Nikiforov]

In the literature, the occurrence of electric polarization is explained as follows. It is argued that the magnetic field is a set of elementary circuits with an electric current. This refers to the rotation of an electron around an atom. It is argued that an elementary circuit with an electric current at rest in the reference frame S0 acquires an electric moment in the moving reference frame S. In a simplified way, this is explained as follows. For a moving observer, the time of movement of an electron in the direction of motion differs from the time of movement of an electron in the opposite direction. At the same time, it is not claimed that the moving reference frame S should be in a magnetic field.

 

[Ivan Nikiforov]

Rotation of the circuit where B=0 and rotation of the disk where B$\neq$ 0 should be investigated independently. You don't have to think of the correlation between.
There are three things; maget, disk and circuit. What kind of motion involving the three are you interested in ?

Thank you. An interesting explanation. I'll think about it.
I am interested in the motion of the external circuit relative to the magnet and the disk (the magnet and the disk are one).

 

[anuttarasammyak]

Then as said no B no emf.

 

[Ivan Nikiforov]

Then as said no B no emf.

Then it turns out that the rotation of the disk and the rotation of the external circuit are not equivalent. Why? How can this be explained from the point of view of the theory of relativity?

For example, such a model is given in the literature. The picture shows a metal band that moves inside an inductor, that is, in a magnetic field. At the edges of the tape, sliding contacts of an external circuit are connected, which is not in a magnetic field. In this model, the metal band and the external chain have a rectilinear uniform relative motion, that is, they represent inertial reference frames. So, in the general case, we cannot understand whether the tape is moving or the external circuit is moving. There is only a relative motion at which current flows. Therefore, if we move only the outer circuit, we will get the same result. Is this statement true?

 

[Ivan Nikiforov]

In fact, the first question of the topic can be formulated as follows. An electric current circuit is fixed in the laboratory (reference frame S0). Relative to this circuit with an electric current and at a great distance from it, a material point (reference frame S) is moving. The material point is not located in a magnetic field. An observer located in the frame of reference S of a material point discovers that a circuit with an electric current has an electric moment:

Is this interpretation correct?
I would like to get an answer to this question precisely in the context of the theory of relativity.

 

[Nugatory]

According to the principle of relativity, the rotation of the disk and the rotation of the external circuit are equivalent. Indeed, in empty space we cannot understand whether the disk is rotating or the external circuit is rotating. We can only say that there is relative motion between the disk and the external circuit.

The principle of relativity says that inertial frames are equivalent, but a rotating frame is not inertial. Even in otherwise empty space, we can determine whether the disk is rotating; an accelerometer attached to the disk will show non-zero proper acceleration if the disk is rotating.

Then it turns out that the rotation of the disk and the rotation of the external circuit are not equivalent. Why? How can this be explained from the point of view of the theory of relativity?

Because the theory never claimed that they were equivalent.

 

[Ibix]

Then it turns out that the rotation of the disk and the rotation of the external circuit are not equivalent

Of course. Attach a mass on a spring to each component - if one is spinning the spring will extend.

How can this be explained from the point of view of the theory of relativity?

It's only inertial frames that are all equivalent. Acceleration is absolute - that is, detectable in an arbitrarily small closed box with something like my mass on a spring.

Whether you are sliding in a straight line past a straight rail or the rail is sliding past you is unanswerable. Moving along a curved rail has an answer - one or more of you will be accelerating.

 

[Ivan Nikiforov]

The principle of relativity says that inertial frames are equivalent, but a rotating frame is not inertial. Even in otherwise empty space, we can determine whether the disk is rotating; an accelerometer attached to the disk will show non-zero proper acceleration if the disk is rotating.
Because the theory never claimed that they were equivalent.

Thanks for the comment. I mean equivalent rotation in terms of electromagnetism, not mechanics. In comment No. 12, I specifically indicated a model in which the metal band and the external chain move evenly and in a straight line. This model is essentially a Faraday disk, if you make a geometric sweep of it. Is it possible to apply your arguments to the model mentioned in comment No. 12?

 

[Ivan Nikiforov]

Whether you are sliding in a straight line past a straight rail or the rail is sliding past you is unanswerable.

Thanks for the comment. It also seems to me that with uniform rectilinear motion, it doesn't matter if the disc is moving or the external circuit is moving or if they are moving simultaneously towards each other. However, the fact is that the electromagnetic processes occurring during the rectilinear motion of the disk and the external circuit are completely identical to the electromagnetic processes occurring during the rotation of the disk and the external circuit. This has been proven experimentally.

 

[anuttarasammyak]

Then it turns out that the rotation of the disk and the rotation of the external circuit are not equivalent. Why?

Why do you think equivalent ?

I am interested in the motion of the external circuit relative to the magnet and the disk (the magnet and the disk are one).

From your figure I interpret the motions here are rotation. Spinning magnet is an advanced matter so I would set Magnet is at rest with no spin in a IFR. In this IFR,
Disk : spinning with angular velocity $\Omega$ where $B\neq0$
Circuit : spinning with angular velocity $\omega$ where $B=0$

$\Omega$ contribute to emf with integrated contribution of $r \Omega B$ where r is radius of a part of Disk.
$\omega$ has nothing to do with emf due to $B=0$.
"Relative rotation" angular velocity $\Omega-\omega$ does not show as you might have expected.

[EDIT]
Now we know $\omega$ is irrelevant. Let’s think of relativity on $\Omega$. On Disk inherent FR which is not a IFR but a rotational FR, magnet is spinning with $-\Omega$. emf is observed as it is in the IFR.

The next configuration : in a IFR Disk is at rest and Magnet is spinning with $-\Omega$. Is emf observed? I do not think so because v=0 but it is worth doing an experiment.

 

[Ivan Nikiforov]

Why do you think equivalent ?

The model is shown in comment No. 12. If the disk is expanded (geometrically transformed) into a straight part, then we get the same thing. If in the model of comment No. 12, the movement of the metal band and the external circuit are equivalent, then the rotation of the disk and the external circuit are also equivalent. In this case, as I understand it, acceleration is not taken into account, since it does not affect electromagnetic processes (electric and magnetic fields).

It has been experimentally proved that the electromagnetic processes for a rotating disk and an external circuit are completely identical to the electromagnetic processes occurring during the rectilinear motion of a magnet and an external circuit. If you roll up the magnet in this picture into a ring, we get a disk.

 

[anuttarasammyak]

You start from rotation but the figure shows linear motion. I do not think it is a good way to visit different topics at a time if you really want to study.

 

[Ivan Nikiforov]

You start from rotation but the figure shows linear motion. I do not think it is a good way to visit different topics at a time if you really want to study.

Excuse me, but in the literature, when considering the electromagnetic processes occurring in a Faraday disk, it is argued that this is the same thing - the rotation of the disk and the translational motion of the magnetic bar. That's not my opinion. I am only referring to sources and based on them I am trying to build a correct interpretation in relation to the conditions of the problem.

 

[Ibix]

The model is shown in comment No. 12. If the disk is expanded (geometrically transformed) into a straight part, then we get the same thing.

If you make your disc infinitely large you make its angular velocity zero. What results would you expect from a Faraday disc with zero angular velocity?

 

[Ivan Nikiforov]

If you make your disc infinitely large you make its angular velocity zero. What results would you expect from a Faraday disc with zero angular velocity?

Please forgive me, these are probably translation features. I was referring to a geometric transformation in which a disk turns into a rectangle (a magnetic bar, a metal ribbon). This explains the transition from rotational motion to rectilinear motion, and vice versa. Ultimately, this shows that the models in comments No. 12 and No. 19 are equivalently converted to a disk. In the literature, these models are used to explain the electromagnetic processes occurring in the Faraday disk.

 

[Ivan Nikiforov]

In this book, in chapter 22, the question of how applicable to rotation are the results obtained for rectilinear motion is discussed in detail. It is stated that for rotational and rectilinear motion are the same: 1) all the results concerning the possible variants of the relative motion of the magnetic field, the metal bar and the external circuit; 2) the electromagnetic process of production and the magnitude of the electromotive force. Thus, for my problem, rotation and rectilinear motion are equivalent.

In this book, in chapter 21, paragraph 178a, it is stated that the detection of an electric field when moving relative to a magnetic field, as well as the detection of a magnetic field when moving relative to an electric field, is a first-order relativistic effect that depends on u/c, and is a consequence of the relativity of simultaneity.

 

[anuttarasammyak]

Chapter 22 of my Panofsky-Phillips second edition is "Radiation, scattering and dispersion" which seems not our topics. What is the chapter title of your book ?

Referring to these books, what is your own conjecture in OP setting for which you have posed only questions ?

 

[Ivan Nikiforov]

Chapter 22 of my Panofsky-Phillips second edition is "Radiation, scattering and dispersion" which seems not our topics. What is the chapter title of your book ?

Referring to these books, what is your own conjecture in OP setting for which you have posed only questions ?

My book was translated into Russian in 1963. It is possible that the chapter numbers do not match the original. In my book, chapter 22 is called "The covariant form of field equations for material media and conservation laws." This chapter describes how the electrical polarization of a circuit with an electric current occurs, which is detected by a moving observer.
This chapter also discusses the extent to which the results obtained for rectilinear motion are applicable to rotation.
The chapter titled "Radiation, Scattering, and Dispersion" in my book has the number 21.
As I understand it, according to the rules of the forum, I can only ask questions. If I state my own understanding, it may be a violation. Therefore, at the end of the text, I write the phrase: "Is this true?"
Based on literary sources, I came to the following conclusion. When the external circuit moves relative to the disk, an electric current must flow in the system. That is, the answer to the first question of the topic is in the affirmative. On the second question of the topic, about what forces will act on the external circuit in this case, I do not yet have an accurate understanding. Just speculation. Therefore, I wanted to address these issues on the forum and understand: 1) am I drawing the right conclusions on the first question; 2) get opinions on the second question.

 

[anuttarasammyak]

Based on literary sources, I came to the following conclusion. When the external circuit moves relative to the disk, an electric current must flow in the system

Yes, as stated in post my #2 and @TSny ‘s #3 and also Panofsky Philips the figure of which you quoted in #19 with B there , but you said no B in #4. By which condition shall we go ?

 

[anuttarasammyak]

This chapter also discusses the extent to which the results obtained for rectilinear motion are applicable to rotation.

I do not find such a discussion in my second edition. Maybe I am a careless reader. I should appreciate it if someone who has the book check it also.

 

[Ivan Nikiforov]

Yes, as stated in post my #2 and @TSny ‘s #3 and also Panofsky Philips the figure of which you quoted in #19 with B there , but you said no B in #4. By which condition shall we go ?

In comment No. 19, it is said that, in relation to the conditions of the problem, rotation and rectilinear motion are equivalent. The model in comment No. 19 is given solely to demonstrate that the electromagnetic processes in the disk are completely identical to the electromagnetic processes in the magnetic bar. The very formulation of the first question of the topic remains unchanged: will current flow if the moving external circuit is not in a magnetic field? I came to the conclusion that the current will flow.

 

[anuttarasammyak]

will current flow if the moving external circuit is not in a magnetic field?

As I answered in #11, no.