Can a Rotating Conducting Disk Generate a Magnetic Field?

[khemist]

In a recent homework problem for my physics class, there was a question regarding a non conducting disk with a charge, and this rotating disk (axis or rotation perpendicular to the surface, through the center) will produce a B field because of the movement of charges.

Now suppose I have a conducting disk, rather than a non conductor. When rotated, will this also produce a magnetic field? My guess is that the resistivity of the material will come into play, but earlier in the course we mention that charges on conductors are able to move, and if the disk is rotating, the charges could theoretically stay in one spot, relative to an outside observer. However, the resistivity of the material could force the charges to move, unless this disk is actually a super conductor.

 

[Curl]

If the conducting disk has some net charge on it, yes. But the charges will move in the radial direction until some equilibrium is reached, so they gave you a non-conductive disk to make it simpler for you.

 

[khemist]

What do you mean that an equilibrium will be reached?

And yea I know that's why they made it easier, so I am asking here about the harder version ;)

 

[Cleonis]

In a recent homework problem for my physics class, there was a question regarding a non conducting disk with a charge, and this rotating disk (axis or rotation perpendicular to the surface, through the center) will produce a B field because of the movement of charges.

Now suppose I have a conducting disk, rather than a non conductor. When rotated, will this also produce a magnetic field?

You haven't specified whether this conducting disk is charged.conducting disk, negatively charged, rotating
The surplus electrons give rise to electrostatic repulsion. So rather than than spreading out evenly over the volume of the disk the surplus electrons will tend to move to the perimeter. That is, most of the surplus electrons will be located near the perimeter.
Compared to a non-conducting charged disk the magnetic field arising from the rotation won't be much different, since the total amount of surplus electrons is the same.conducting disk, not charged, rotating
The electron population in a conductor behaves somewhat like a gas. So when the conducting disk is spinning the electrons will be subject to a centrifugal effect.

Near the hub there will be a slight depletion of electrons, near the perimeter there will be a corresponding slight surplus of electrons. If I hazard a guess I think the two magnetic fields (from near-the-hub and from near-the-perimeter) will not cancel entirely.

So on theoretical grounds I do expect a magnetic field surrounding an uncharged, fast spinning conducting disk. I don't know how difficult it would be to try and get experimental confirmation.

 

[khemist]

Yes, the test disk is charged.

So although the charges are free to move, they are still unable to "slip" completely while the disk is rotating so there are still the same amount of moving charges, if the non conductor and conductor have the same amount of charge on them. (and I am guessing it is due to the resistivity of the material. If the disk was a super conductor, there would be no moving charges?)

That is quite interesting about the not charged disk. I would think that there is in fact no surplus of electrons, and the "space" vacated by the electrons due to the centripetal force would cancel the space that is occupied. How would one go about solving a problem like this?

 

[Vanadium 50]

The charged conducting disk version of the problem is never given, because it requires giving up a lot of the idealizations. In the simplest view, the conductor moves and the additional electrons stay in place, because there's no field in the conductor to move them. But if you think about it more, the electrons are all in the skin, and the skin of a physical conductor is also typically an oxide layer, so you have some resistance. Now you have a field, so the electrons can respond to it.

You'll quickly find yourself in a situation where you have many different places where the idealization breaks down, and the answer depends on details like the resistivity and thickness of the "skin". So there's really no pedagogical reason to introduce this kind of problem.

 

[Cleonis]

[...] if the disk is rotating, the charges could theoretically stay in one spot, relative to an outside observer. However, the resistivity of the material could force the charges to move, unless this disk is actually a super conductor.

My apologies, I had not read your question properly.

I do know of experiments where a ring-shaped tube was filled with Helium, and then cooled to a temperature where the Helium becomes a superfluid liquid. Then they spun up that conduit, and sure enough the Helium did not follow that motion. A superfluid flows without friction. Without friction there is no way for the conduit to drag the fluid in it along.

By analogy, if you have a disk that is a superconductor, and you spin it up then I expect that the free-moving electrons in that material will not come along in that spinning up.

If the disk is a normal conductor rather than a superconductor then I think the resistivity of the material will offer sufficient friction to pull the free electrons to co-rotating motion. There may be a bit of a lag, but I expect that to be minimal. The inertial mass of electrons is very small; electrons have about 1/1800th the mass of protons.

 

[khemist]

You guys confirmed my initial thought. Thanks for the informative responses.

 

[Curl]

You forgot that if a magnetic field is generated, the free electrons in the conductor will be moving through a B field which exerts the Lorrentz force on them.

That's why I told you in the beginning this is more complicated than you'd think.

 

[Creator]

By analogy, if you have a disk that is a superconductor, and you spin it up then I expect that the free-moving electrons in that material will not come along in that spinning up.

.

That's true, Cleonis; but I hope you are not implying that therefore a magnetic field does not develop...because contrary to expectation it is because of the very reason you gave that a dipole magnetic field does actually develop in any rotating superconductor (in the lab frame).
In fact monitoring that London magnetic moment (as it is called) in rotating SC niobium was the working principle in the Gravity Probe-B experiment.


Creator

 

[Cleonis]

By analogy, if you have a disk that is a superconductor, and you spin it up then I expect that the free-moving electrons in that material will not come along in that spinning up.

[...] a dipole magnetic field does actually develop in rotating a superconducting disk (in the lab frame).

Yeah.
What I intended to point out is what you are pointing out in your post.

Comparing 4 cases:

- The case of a charged disk, rotating:
There is a net charge on the move, giving rise to a corresponding magnetic field.

- The case of a non-charged disk, rotating. I like to interpret that situation in terms of the superposition principle of electrodynamics: both positive charge and negative charge is on the move. The contributions of each are a perfect match for the other. The contrbutions cancel each other, therefore no magnetic field will be detected.

- The case of a non-charged, normal conducting disk, during phase of spinning up:
Theoretically the electrons will lag behind slightly during spin-up, but that effect will be very transient.

- The case of a superconducting disk, during spinning up and subsequent uniform rotation:
The positive charges of the disk are brought in circumnavigating motion, but the free electrons in the superconductor don't get to participate in that circumnavigating.
In that situation I expect that there will be a magnetic field, corresponding to the motion of the positive charge carriers.

 

[tiny-tim]

The charged conducting disk version of the problem is never given, because …

in a charged conductor (not near another charged object) doesn't most of the charge go to the places of smallest radius of curvature (on the surface), which in this case would be the edge of the disc, even if it's not rotating?

so would the rotation make any appreciable difference?

 

[khemist]

You forgot that if a magnetic field is generated, the free electrons in the conductor will be moving through a B field which exerts the Lorrentz force on them.

That's why I told you in the beginning this is more complicated than you'd think.

I am not quite sure what was stopping people from explaining this. Thank you for pointing that out.

On a side note, I never actually thought about the direction of the magnetic field for any of these situations. Would the field be directed perpendicular to the area of the disk? Is this field in the same direction in each situation? My guess is that it would depend on the direction of the rotation and the sign of the charge that is moving. We can assume that each disk is rotated in the same direction, but can we assume that the sign of the charge is the same, if say, the charged conductor has a positive net charge on it?

 

[Creator]

Now suppose I have a conducting disk, rather than a non conductor. When rotated, will this also produce a magnetic field? My guess is that the resistivity of the material will come into play,...

khemist; in a related issue...
You ought also to realize that even without net surface charge, rotating a disk can create a magnetization in the material via the Barnett Effect, in which a net magetizaton develops porportional to the angular velocity and also the magnetic susceptibility and inversely proportional to the gyromagnetic ratio of the material:
see here: http://en.wikipedia.org/wiki/Barnett_effect

Also as I mentioned to Cleonis , a rotating superconductor will always generate a dipole B field proportional to angular velocity,w, (aligned with the rotation axis) and it will be quantized in units of 2m/q...where m and q are the mass and charge of the electron, respectively. This was predicted by Fritz London and is referred to as the "London Magnetic moment". In effect, rotation reverses the Meissner effect and allows a net magnetic field to extend throughout the SC material as well as into the space around the SC.

Creator

 

[khemist]

khemist; in a related issue...
You ought also to realize that even without net surface charge, rotating a disk can create a magnetization in the material via the Barnett Effect, in which a net magetizaton develops porportional to the angular velocity and also the magnetic susceptibility and inversely proportional to the gyromagnetic ratio of the material:
see here: http://en.wikipedia.org/wiki/Barnett_effect

Also as I mentioned to Cleonis , a rotating superconductor will always gererate a dipole B field proportional to angular velocity,w, and it will be quantized in units of 2m/q...where m and q are the mass and charge of the electron, respectively. This was predicted by Fritz London and is referred to as the "London Magnetic moment". In effect, rotation reverses the Meissner effect and allows a net magnetic field to extend throughout the SC material as well as into the space around the SC.

Creator

Interesting, thanks.