I Charge movement relative to magnetic field frame dependence/invariance

[FusionJim]

The Faraday disc generates current if spinning in a homogeneous magnetic field, it also generates current if there is a non symmetric magnetic field but then the magnet producing that field has to be kept stationary with respect to the disc.
In fact back in Faraday's time he had first a field like that and this worked like an eddy current brake. Current under magnet pole face goes radially one way and where the magnetic flux loops back the current goes radially in the opposite direction forming closed loops within the disc.

Yet when such a asymetric magnet is placed on the disc and co-rotates rigidly on the disc no current loops exist within the disc.
I have measured this myself by spinning such a disc with a block magnet attached to it and the disc temperature doesn't rise at all indicating there is no current generated.

So this got me thinking. Any charged particle that is moving with respect to a magnetic field should experience a Lorentz force that deflects the particle.

A hall effect sensor has this situation, a non symmetric magnetic field where a flat conductor is atop it and as current passes through the electrons get deflected in the transverse (to current) direction setting up a charge imbalance.
Once the charge imbalance is set up there is now a transverse E field that opposes any additional charge deflection.
So because the conductor has limited space in the transverse direction therefore the charges have nowhere to move and they "pile up" and set up an E field that stops further deflection.


Is this the same reason why a non-symmetric magnet rigidly co-rotating with a Faraday disc where the magnetic field doesn't cover the whole disc surface cannot create eddy currents within the disc similarly to how they arise if the magnet is kept stationary?

The co-rotating magnet initially causes radial deflection currents but because the field only covers a limited area the charges cannot move further and they "pile up" creating an opposing E field (E field is azimuthal) that cancels any further charge deflection and therefore no current can loop and flow within the disc?



In other words what I am asking is this. Does a non symmetric co-rotating magnetic field can still cause the same Lorentz force on the moving charges as a symmetrical frame invariant magnetic field does? Only with the asymetric co-rotating field the force cannot result in sustained current?

Another example that has caused me some confusion why I ask this is this - imagine an electron beam in a straight path, then take any rectangular bar magnet and make it move just under the beam with the same exact speed as the electrons , what happens?
Do the electrons feel anything ? If they do , does the same thing happen as in the disc with the co-rotating bar magnet, respectively, the electrons get deflected but only to the point where the field extends and then pile up causing a E field that stops any further deflection?

 

[TSny]

If a magnet is at rest, it produces a magnetic field but no electric field. If the magnet is in motion, it produces both a magnetic field and an electric field. This follows from the transformation laws for E and B fields when switching frames of reference.

Suppose the disk is spinning in the lab frame of reference, and the magnet is at rest.

The free electron shown inside the disk is moving toward the right in the upward magnetic field of the magnet. The electron of charge $-e$ experiences a magnetic force $evB$ toward the center of the disk, or into the page in the picture below.

Next, suppose the disk is not spinning but the magnet is moving toward the right with speed $v$. In the lab frame of reference, the moving magnet produces an electric field into the page at the location of the electron. The magnetic field of the magnet does not produce a force on the electron when the electron is at rest.

The magnitude of the electric field is related to the magnitude of the magnetic field and the speed of the magnet. Assuming $v$ is much smaller than the speed of light, it can be shown that the magnitude of this electric field is $vB$. So, the electron experiences an electric force out of the page of magnitude $evB$. This force tends to drive the electron out toward the rim of the disk.

Finally, suppose the disk is spinning and the magnet is moving with the disk.

In the lab frame of reference, the electron experiences a magnetic force $evB$ into the page and an electric force $evB$ out of the page. The two forces cancel, so the net force on the electron is zero.

In a frame of reference moving with the electron and magnet, the electron and magnet are at rest. In this "comoving" frame, there is no electric force on the electron and no magnetic force on the electron. So, observers in the lab and comoving frames agree that the net force on the electron is zero.

 

[FusionJim]

@TSny thank you for the reply.
I realized long ago that magnetic field lines are just a man made concept, so we only truly have field strength and geometry as the only physical quantities.

If this is the case then it seems weird that a non axis symmetric magnet co-rotates with the disk and no EMF is generated but when a symmetric magnet rotates an EMF is generated.
I suppose this is because of the geometry, because a symmetric magnet rotating around axis of symmetry its field doesn't change in space and time but when such a magnet is rotated in a non symmetric way (where the movement doesn't correspond with the axis of field symmetry) then the "added" E field cancels the contribution that would otherwise arise from the B field?


Taking into account your explanation @TSny it seems to me that everytime a conductor is stationary but the magnet is moving through space/time induction happens because of the effect of the E field created by the moving B field. Meanwhile everytime the magnet is stationary but the conductor is moving instead, induction happens because of the V x B force aka Lorentz force on the moving electrons.
And finally in order for there to be EMF generated within a conductor that is moving together with a magnet the magnet can only "move" or rather rotate around axis of symmetry otherwise the magnet's B field due to movement through space creates an E field that cancels the effect of the B field on the moving electrons?

If this is so, then would it be correct to say that a dynamo generator generates current differently than a modern alternator? Both generate current because of induction but in one the coil is physically moving through a B field at rest but in the other the B field is physically moving through space and passes through/by a coil at rest. Because according to what you said, the dynamo has a static B field but instead the coil is moving , so the magnet doesn't move so there is no E field from it?

[Post edited by a Mentor at the request of the poster]

 

[TSny]

@TSny thank you for the reply.
I realized long ago that magnetic field lines are just a man made concept, so we only truly have field strength and geometry as the only physical quantities.

I agree. The basic laws of classical electromagnetism are Maxwell's equations, the Lorentz force law, and the constitutive relations for material media. These are expressed in terms of the E and B vector fields. The application of these laws does not require the concept of field lines. Of course, the field lines are very helpful for visualization.

If this is the case then it seems weird that a non axis symmetric magnet co-rotates with the disk and no EMF is generated but when a symmetric magnet rotates an EMF is generated.
I suppose this is because of the geometry, because a symmetric magnet rotating around axis of symmetry its field doesn't change in space and time but when such a magnet is rotated in a non symmetric way (where the movement doesn't correspond with the axis of field symmetry) then the "added" E field cancels the contribution that would otherwise arise from the B field?

For the moving non-axisymmetric magnet, we could find the E and B fields in the lab frame by first considering the fields in the magnet's instantaneous comoving frame. This idea was used in post #2.

But, I'm unsure how to use this idea for the axis-symmetric magnet spinning about its axis. You would need to go to a rotating frame of reference to see the magnet at rest. I'm not comfortable with transforming electromagnetic relations between rotating and nonrotating frames. So, I can't be of much help here.

Taking into account your explanation @TSny it seems to me that everytime a conductor is stationary but the magnet is moving through space/time induction happens because of the effect of the E field created by the moving B field.

I would not say the E field is "created" by the moving B field. This might just be semantics. The source of the electric field of the moving magnet can be thought of in the following way. In the magnet's rest frame, the material of the magnet carries a certain magnetization but no electric polarization. So, in its rest frame, the magnet produces only a magnetic field (no electric field). In a reference frame where the magnet is in translational motion, it can be shown using special relativity that the material will generally have electric polarization in addition to magnetization. The electric polarization can be considered the source of the electric field associated with the moving magnet.

Does this mean a spinning magnet will acquire electric polarization and therefore produce an electric field? I'm not sure. I haven't yet found a good reference on the E and B fields of rotating magnets.

Meanwhile everytime the magnet is stationary but the conductor is moving instead, induction happens because of the V x B force aka Lorentz force on the moving electrons.

Yes

And finally in order for there to be EMF generated within a conductor that is moving together with a magnet the magnet can only "move" or rather rotate around axis of symmetry otherwise the magnet's B field due to movement through space creates an E field that cancels the effect of the B field on the moving electrons?

If this is so, then would it be correct to say that a dynamo generator generates current differently than a modern alternator? Both generate current because of induction but in one the coil is physically moving through a B field at rest but in the other the B field is physically moving through space and passes through/by a coil at rest. Because according to what you said, the dynamo has a static B field but instead the coil is moving , so the magnet doesn't move so there is no E field from it?

I don't know how to answer these questions. I generally don't think of E fields or B fields as "physically moving through space". Anyway, it's a very interesting topic.

 

[FusionJim]

In a reference frame where the magnet is in translational motion, it can be shown using special relativity that the material will generally have electric polarization in addition to magnetization. The electric polarization can be considered the source of the electric field associated with the moving magnet.

I suppose one thing is certain. The E field that will result from a magnet being moved/rotated in any other way than around it's field symmetry axis is a non conservative E field? Meaning that the field lines do not end on surfaces perpendicularly to them like they would from an electrostatic field?
Because an electrostatic field cannot move charges (apart from charges rearranging to cancel the field) but a non conservative E field can cause transverse motion of charges within a conductor correct?

Does this mean a spinning magnet will acquire electric polarization and therefore produce an electric field? I'm not sure. I haven't yet found a good reference on the E and B fields of rotating magnets.

Well, if the magnet is highly conductive like the protection layer made from copper and nickel applied to most NdFeb type magnets then I would suspect atleast in the case of rotary symmetry around field axis there to be a polarization going on. But in the non symmetrycal case I don't know.
Technically if that non symmetrycal magnet being rotated on a co-rotating disc cannot result in any charge movement on the disc, then in theory it also should not result in any effect on the charges in the magnet? Because both would experience the same effect

 

[TSny]

I suppose one thing is certain. The E field that will result from a magnet being moved/rotated in any other way than around it's field symmetry axis is a non conservative E field?

Yes. A nonconservative electric field requires the existence of a time-dependent magnetic field.

A symmetric magnet that rotates at a constant angular speed about its axis of symmetry will not produce a time-dependent magnetic field. So, any E field produced by the spinning magnet would be conservative.

If a magnet moves such that it produces a time-varying B field in a frame of reference, then there will be a nonconservative electric field in this frame.

Meaning that the field lines do not end on surfaces perpendicularly to them like they would from an electrostatic field?
Because an electrostatic field cannot move charges (apart from charges rearranging to cancel the field) but a non conservative E field can cause transverse motion of charges within a conductor correct?

I'm not sure I'm understanding what you're saying here (my fault). It might help if you could provide a specific example that illustrates your question.