Faraday's Paradox: Induced Current Explained

[carrz]

disk and magnet spin together -> induced currentI ask can anyone properly explain why there is induced current in the setup?

 

[jartsa]

lorentz force pushes electrons in the wires

 

[carrz]

Why, how? To properly explain it is not sufficient to just say Lorentz force did it. There is always Lorentz force between the magnet's magnetic field and electron's magnetic fields in the conducting disk. But to actually strip atoms of their electrons in the disk and move them to the rim or the center, this Lorentz force must therefore be somehow different, or stronger, than when both the magnet and the disk are stationary. Proper explanation must address how and why Lorentz force changes the way it does for each particular setup.I did not find any other paper or article says that, when both the magnet and the disk are spinning together, current is actually induced in external circuit and not the disk.

 

[Dale]

This thread is reopened with some editing to focus the topic, the OP will post some further clarification.

Note, the scenario in the previously linked Wikipedia page references a uniform magnetic field and an axisymmetric magnet so the analysis is not applicable and references to it were removed.

 

[vanhees71]

Ok, let's first describe the depicted setup. You should always label such figures properly. So I can only guess by (my personal ;-)) common sense, what's depicted. The horse-shoe shaped thing is the magnet, and the disk is a conductor. The voltage ("measured" by the light bulb) is between a point at the center of the disk to its rim.

I cannot explain this quantitatively, because I'm not aware of a full expression for the magnetic field of the horse-shoe magnet. But qualitatively it's easy to discuss the three setups, that where described originally. Let's start with the one left in the edited thread:

(a) Magnet and disk are rotating together (i.e., the magnet is fixed on the disk). There is a time-dependent magnetic field, giving rise to a electric field and thus a Lorentz force $\vec{F}=-e(\vec{E}+\vec{v} \times \vec{B}/c)$ is acting on each electron in the disk. Here you can assume $\vec{v}=\vec{\omega}\times \vec{x}$ for the velocity of the electrons, because the velocity according to the drift due to conductivity can be neglected for such usual household setups. So there will be a current flowing through the light bulb. If it's large enougth, the light bulb will be "on".

(b) Only the disk is rotating and the magnet is fixed. Then the magnetic field is stationary and no electric field is induced by it. Nevertheless there is a (now purely magnetic) Lorentz force acting on each electron in the rim $\vec{F}=-e \vec{v} \times \vec{B}/c$. This leads to a drift of charges and the buildup of a charge density at the rim of the disk. Thus you have a voltage between the center and the ring of the disk, giving rise to a current through the light bulb, which is shining again, if this current is large enough.

(c) Only the magnet is rotating around the disk. Again an electric field is induced due to the time varying magnetic field, and the same qualitative explanation as under (a) applies.

I hope that settles the issue. There is no paradox. It is only important that such setups are to be solved by using the complete Maxwell+mechanical (relativistic!) laws, including the Lorentz force on the conduction electrons in the metallic pieces of the setup. There is also no contradiction with Faraday's Law in integral form, which has to be written in its complete form, including the time variation of the surface boundary due to its motion if such a surface is used to calculate the flux.

 

[carrz]

(a) Magnet and disk are rotating together (i.e., the magnet is fixed on the disk). There is a time-dependent magnetic field, giving rise to a electric field and thus a Lorentz force $\vec{F}=-e(\vec{E}+\vec{v} \times \vec{B}/c)$ is acting on each electron in the disk. Here you can assume $\vec{v}=\vec{\omega}\times \vec{x}$ for the velocity of the electrons, because the velocity according to the drift due to conductivity can be neglected for such usual household setups. So there will be a current flowing through the light bulb. If it's large enougth, the light bulb will be "on".

Where is time-dependent magnetic field, why is it? How and why is that time-dependent magnetic field giving rise to Lorenz force? What is the velocity in that equation of, and what is it relative to?

 

[Dale]

disk and magnet spin together -> induced current

Personally, I am not certain that there is an induced current in this setup.

The Lorentz force has two terms, one due to E fields and one due to the cross product of the v and B field. The second term is clearly non-zero, but because a time-varying B field induces an E field the first term is also non-zero.

If this were linear motion then it would be completely clear that these two terms cancel out exactly, but since it is rotational motion there may be some slight non-cancellation. I would have to see or do the math to be certain either way.

 

[vanhees71]

Well, to do the math is pretty complicated here. Unfortunately I've not the equipment to do an experiment. Perhaps we should look in the literature for this particular setup ;-)).

 

[Dale]

Agreed, I don't have a dedicated numerical solver for Maxwell's and I would not want to try this with anything less (or an experiment). So I just don't know the outcome.

The one thing that I do know is that there will be eddy currents induced, since without the lightbulb and associated connections this is the design of an eddy current brake.

 

[Nugatory]

The one thing that I do know is that there will be eddy currents induced, since without the lightbulb and associated connections this is the design of an eddy current brake.

The eddy current brakes that I am familiar with are all of the fixed magnet and moving rotor variety?

 

[carrz]

Personally, I am not certain that there is an induced current in this setup.

It's the first kind of homopolar generator Faraday used and with which the paradox most likely originated. I think disk shaped magnets covering the whole of conducting disk only came into existence later on.

In all the papers and articles about either the paradox or homopolar generators I did not see anyone suggests in any way there are any differences, but both designs seem to be equally represented. Although, unfortunately, explanations do vary, but they do describe those same effects.

The Lorentz force has two terms, one due to E fields and one due to the cross product of the v and B field. The second term is clearly non-zero, but because a time-varying B field induces an E field the first term is also non-zero.

Where do you see "time-varying B field" when the magnet is fixed to the disk? Does magnetic field magnitude not stay the same for any point on the disk itself, just like if they were completely stationary? Why would B field be any stronger when they are moving than when they are static? And where is it this B field grows stronger, around electrons in the magnet, around electrons in the conducting disk, or both?

If this were linear motion then it would be completely clear that these two terms cancel out exactly, but since it is rotational motion there may be some slight non-cancellation. I would have to see or do the math to be certain either way.

Are you saying rotating magnet creates "time-varying" B field, but a magnet that moves in a straight line does not create "time-varying" B field?

 

[Drakkith]

Carrz, i doubt that Faraday was using this setup when he discovered the paradox, as it doesn't work the same as the descriptions say. In this example rotating the magnet will result in a flow of current. Usually the setup used to explain the paradox considers the magnetic field to be homogenous throughout the disk, as if you used a big cylindrical magnet on each side instead of a small horseshoe one. This can easily be seen in different examples by looking at how the magnetic field is setup. In the picture the wiki article uses, they don't even bother to show a magnet, they just show the direction of the magnetic field (which is probably detrimental to explaining the paradox and contributes to the confusion here).

 

[Dale]

unfortunately, explanations do vary

I am not sure why you consider that unfortunate. I personally enjoy having multiple correct explanations. Often different people will prefer different explanations, so having multiple ones helps.

Where do you see "time-varying B field" when the magnet is fixed to the disk?

The magnet is the source of the field so as its position changes over time then the field at any fixed point necessarily changes over time.

Does magnetic field magnitude not stay the same for any point on the disk itself, just like if they were completely stationary?

That is true, but not relevant. The criterion for determining if the field is time varying is whether or not the fields stay the same (magnitude and direction) for every fixed point in a given inertial frame. The disk is not fixed in any inertial frame.

I believe that we have discussed this before.

Are you saying rotating magnet creates "time-varying" B field, but a magnet that moves in a straight line does not create "time-varying" B field?

No, they both create time varying fields. The difference is that for straight line motion the resulting E field clearly cancels out the force from the B field.

 

[vanhees71]

I also think that it confuses the issue only even more if you try to explain things in non-inertial frames, although this is possible in principle of course. You come, however, close to the complexity of general relativity (Einstein-Maxwell equations), which is not necessary to treat the homopolar generator.

I'm looking through the literature a bit in order to start writing an FAQ article about this issue. I'm really surprised, how confusing the explanations are, hundred years after Einstein's famous paper "On the electrodynamics of moving bodies" of 1905 (translation of the title mine). There are even wrong statements in the American Journal of physics. One can emphasize only Feynman's statement:

If in doubt, go back to the fundamental equations, which are Maxwell's equations in local (differential) form and the Lorentz-force Law $\vec{f}=\rho (\vec{e} + \vec{v} \times \vec{B}/c)$ for charge distributions $\rho$ and the assiciated current distributions $\rho \vec{v}$.

Also one should keep the relativistic expressions everywhere, including the constitutive relations for macroscopic electrodynamics a la Minkowski. Then usually apparent paradoxes like the various Faraday disks as well as others like Feynman's disk, all issues with the socalled "hidden momentum" (solved already in 1911 by von Laue in his textbook on special relativity), etc.

 

[carrz]

Carrz, i doubt that Faraday was using this setup when he discovered the paradox, as it doesn't work the same as the descriptions say.

I'm pretty sure in dozens of papers and articles I went through someone would have mention something about it, but instead what I saw is that the same effects are described for either type of homopolar generator.

In this example rotating the magnet will result in a flow of current.

Magnet rotating alone when the disk is stationary will not produce current, that's one paradox. Magnet rotating together with the disk will produce current, that's the second paradox. And as far as I know both types of homopolar generator produce these same two effects.

Usually the setup used to explain the paradox considers the magnetic field to be homogenous throughout the disk, as if you used a big cylindrical magnet on each side instead of a small horseshoe one. This can easily be seen in different examples by looking at how the magnetic field is setup. In the picture the wiki article uses, they don't even bother to show a magnet, they just show the direction of the magnetic field (which is probably detrimental to explaining the paradox and contributes to the confusion here).

I don't see how to settle this. Either you show me some reference that explains how the two types of generator produce different effects, or I show you some reference that explains those same "disk magnets" effects with the horseshoe type of homopolar generator. I'll search for it later on.

 

[Drakkith]

I'm pretty sure in dozens of papers and articles I went through someone would have mention something about it, but instead what I saw is that the same effects are described for either type of homopolar generator.

Then I'm sure you wouldn't mind looking through them once more and seeing if they describe the actual setup used. I took a look at the wiki page again, and it specifically uses a cylindrical magnet in its example.

Magnet rotating alone when the disk is stationary will not produce current, that's one paradox. Magnet rotating together with the disk will produce current, that's the second paradox. And as far as I know both types of homopolar generator produce these same two effects.

That is not correct, as has been explained already. With a horseshoe magnet the disk is subjected to a time varying magnetic field when the magnet is moved around the disk since the field is concentrated around the magnet. With a cylindrical magnet the field is not varying in time since it is the same everywhere throughout the disk.

I don't see how to settle this. Either you show me some reference that explains how the two types of generator produce different effects, or I show you some reference that explains those same "disk magnets" effects with the horseshoe type of homopolar generator. I'll search for it later on.

Feel free.

 

[carrz]

I am not sure why you consider that unfortunate. I personally enjoy having multiple correct explanations. Often different people will prefer different explanations, so having multiple ones helps.

There can be only one correct explanation. For the 3rd scenario when both the disk and the magnet are spinning together, some say it's because magnetic field stay static, some say it's because current is induced in connecting wires, and as far as I know only you say it is because of "time-varying" B field.

The magnet is the source of the field so as its position changes over time then the field at any fixed point necessarily changes over time.

Change over time in what property, what location, relative to what, caused by what? Both electrons in the magnet and electrons in the disk are sources of magnetic fields, and protons contribute their spin magnetic moments which we can ignore I suppose. In any case we are not talking about any absolute fixed points, we are talking only about relation between magnetic fields in the magnet and magnetic fields in the disk.

That is true, but not relevant. The criterion for determining if the field is time varying is whether or not the fields stay the same (magnitude and direction) for every fixed point in a given inertial frame. The disk is not fixed in any inertial frame.

Yes, and both the disk and the magnet are in the same inertial frame when they are spinning together. So why exactly do you think magnitude and direction, of either magnetic fields of the magnet or magnetic fields of the disk, would change for any fixed point in their own inertial frame?

No, they both create time varying fields. The difference is that for straight line motion the resulting E field clearly cancels out the force from the B field.

How does that work?

 

[jartsa]

Why, how? To properly explain it is not sufficient to just say Lorentz force did it. There is always Lorentz force between the magnet's magnetic field and electron's magnetic fields in the conducting disk. But to actually strip atoms of their electrons in the disk and move them to the rim or the center, this Lorentz force must therefore be somehow different, or stronger, than when both the magnet and the disk are stationary. Proper explanation must address how and why Lorentz force changes the way it does for each particular setup.

Here's a quick course about magnetism of electrons:

Let us consider electrons in a circle formation, held in place somehow. Is the circle magnetic? No.

The circle starts rotating. Is the circle magnetic now? Yes. That was the course.

Let us consider two identically rotating circles of electrons. A sideview picture: ||

What is the direction of the magnetic force in this case? The circles attract each other. Like this: -> <-

Now let's make one of the circles to rotate faster. What happens? Force increases. Like this: --> <--

Now let's make the slower circle smaller. What happens? Direction of force changes. Electrons are pulled outwards in the smaller circle, inwards in the larger circle.

And now we cut out a large sector of the large circle. We have now the homopolar generator pictured in post #1. And we see that there is a force pushing electrons radially in that kind of generator. I planned to show the opposite. :grumpy:

(The large circle is the magnet. 50% of the rotation of the circle is there to make the circle magnetic. The other 50% is there to make it a rotating magnet)

 

[Drakkith]

Thread locked for the moment, pending possible moderation.

 

[Dale]

There can be only one correct explanation.

That is not true, but it is also off topic, so I won't argue the point. However, if you want only one explanation then asking on an internet forum is not going to accomplish your objective.

For the 3rd scenario when both the disk and the magnet are spinning together, some say it's because magnetic field stay static, some say it's because current is induced in connecting wires, and as far as I know only you say it is because of "time-varying" B field.

What is "it". What do I say is because of the time-varying B field? I believe that all I said is that I don't know if there is a current in this scenario and would need to see the math (or an experiment).

That there is a time-varying B field is a fact. Because of that fact the analysis of the scenario is not obvious to me. That is all I have claimed. I have not attempted to offer an explanation for the behavior of the system because I am not even sure what that behavior is.

Change over time in what property, what location, relative to what, caused by what?

Change over time in the B field at any fixed location relative to the inertial frame where the center of the disk is at rest which is caused by the rotation of the magnet and disk about that center.

In any case we are not talking about any absolute fixed points, we are talking only about relation between magnetic fields in the magnet and magnetic fields in the disk.

That is not how Maxwell's equations work.

Yes, and both the disk and the magnet are in the same inertial frame when they are spinning together. So why exactly do you think magnitude and direction, of either magnetic fields of the magnet or magnetic fields of the disk, would change for any fixed point in their own inertial frame?

Nonsense. They don't have an inertial frame. They are rotating, so they are non-inertial by definition.

How does that work?

Consider a charge, q, at rest in a magnetic field, B. The Lorentz force is:
$$f=q(E+v \times B) = q(0+0\times B) = 0$$

In a frame where the charge is moving with velocity v the fields are:
$E'=-\gamma v \times B$
$B'=\gamma B - v(\gamma-1)(B.v)/(v.v)$

and the force is:
$$f'=q(E'+v\times B') = q(-\gamma v \times B + v \times (\gamma B - v (\gamma-1)(B.v)/(v.v)) = q(-\gamma v \times B + v \times \gamma B) = 0$$

 

[Drakkith]

Thread re-opened for the time being.

 

[Voltage Drop]

I stumbled across this thread and would like to offer what I think is a simple explanation to the original poster's question.

"I ask can anyone properly explain why there is induced current in the setup?"If the magnet and the disk are attached and rotate together and I am sitting on the magnet, what do I see? I see flux lines extending up from my feet.

If the magnet and disk are rotating and I look over my head, what do I see? I see wires connecting the load cutting these flux lines, once per revolution.

The voltage is generated in the wires connected to the load! If no load were connected there would be no voltage generated.

Now let's make the magnet and disk remain stationary in the room and put the load on a platform which may be rotated about the disk. When the load is rotated about the disk there will be a voltage generated in the load even though the magnet and disk are stationary relative to the room as a frame of reference.

I've never understood why this is a paradox when both types of Faraday machine are similar depending on the frame of reference from which they are viewed.

There is one difference.

In the conventional Homopolar machine with the magnet stationary and the disk rotating, there will be eddy currents generated in the disk.

With the Faraday Homopolar machine, there are no eddy currents in the disk.