Could Iron Filings Form Rings Around a Magnetized Sphere in Space?

[James William Hall]

I saw a physics problem on Craig’s List (of all places) that piqued my interest. I’ll paraphrase it:

An astronaut on the ISS placed a large magnetized sphere outside the station far enough where the station had no effect on the magnet then threw a large pail of iron filings at the sphere. He was wondering if some of the filings would form rings around the sphere. What did he see? What if the sphere was rotating?

 

[berkeman]

What do you think would happen?

 

[James William Hall]

The attraction follows the inverse square law so if the iron filings cloud were large enough most of the filings would stick to the sphere and the rest would form a scatter pattern in the surrounding space. Some filings would drift away but some would form in rings or in a pattern rather. If there were a slow rotation the pattern would persist. As the rotation increased I don’t know what would happen when centrifugal force took effect except on the filings stuck to the sphere which would move away from the sphere. The foregoing is mostly intuition based on the forces in effect which are magnetic, gravity (short term nil effect), and centrifugal. What do you think would happen Berkeman?

 

[berkeman]

What do you think would happen Berkeman?

For the non-rotating sphere, I think the filings will all be attracted to the sphere and will make their way to it and stick to it. The key is that the field is very non-uniform, which exerts an attractive pull on the metal filings:

https://www.mezzacotta.net/100proofs/images/008-Dipole.png

In the case of a rotating magnetized sphere, it depends on whether the rotation is about the magnetic axis or perpendicular to it. If it is rotating about its magnetic axis (about the vertical axis in the diagram above), then I don't think it's much different from the non-rotating case. If it is rotating about a perpendicular axis so that the magnetic poles are spinning around, that will modulate the strength and direction of the B-field in space, and will cause an erratic but still attractive force on the filings.

In either spinning case they should still all be attracted and make it into stick to the magnet, unless as you say the rotation is so fast that they gain enough kinetic energy to fly back off.

It would be interesting to see if in the case of a rapidly rotating sphere whether the filings make it back down to the sphere and regain enough KE from the rotation to fly back off again. I suppose some semi-stable pattern of filings could exist around the sphere in that case...

 

[James William Hall]

Thank you, Berkeman.

 

[Haborix]

This is an interesting thought experiment. Thinking of the iron filing and the magnetized sphere as two dipoles, I don't think the force between them follows an inverse-square law. There is a fun little theorem called Bertrand's theorem which tells us which kind of central potentials have closed, bound orbits. But I'm not sure the theorem applies since the force between dipoles depend on their orientation.

This is all to say that even you didn't toss the filings directly at the sphere, there may not be an initial velocity that allows a ring to form (a ring being indicative of a closed and bound orbit). Anyway, I'll stop handwaving now.

 

[Keith_McClary]

I think the filings will all be attracted to the sphere and will make their way to it and stick to it.

If it was an inverse square force, they would go into elliptical orbits. I can't find any reference for dipoles.

 

[Hornbein]

If it was an inverse square force, they would go into elliptical orbits. I can't find any reference for dipoles.

There is no simple law for dipoles. Isaac Newton found it was approximately inverse cube. It might be possible to cook up an arrangement of magnets that supported orbiting filings. Then again maybe not.

 

[mfb]

For two dipoles the general scaling is 1/r⁴ multiplied by a term that depends on the relative alignment. It's growing too fast with decreasing distance, so there won't be any stable orbits. Everything escapes or sticks to the magnet.

 

[cmb]

Do you mean like an orbit ring? AFAIK the maths for orbits confined by one centrally directed force, only, requires a 1/r^2 centripetal force function, which a permanent magnet field doesn't have. Stuff will be unstable as the field from a permanent magnet does not follow that function, so will just head straight for, or spiral into, the magnet.

Something that is set in motion in an orbit at just the right speed for a given orbit radius is at an unstable point and will either spiral in or head off, it won't be stable.

 

[Haborix]

I share the intuition of many of those who have posted about the behavior for central potentials. However, as a nitpick, I'm not entirely convinced that just because something isn't 1/r or r^2 for potential there can't be closed orbits. Bertrand's theorem is the mathematical statement of what many have posted thus far. However, because the potential for two dipoles interacting depends not only on the radial distance but also the direction, I'm not convinced Bertrand's theorem applies. In other words, I'm not convinced there isn't some configuration of initial dipole moments and radial distances for which there might be a closed orbit. Anyway, I think I repeat myself.

 

[cmb]

I share the intuition of many of those who have posted about the behavior for central potentials. However, as a nitpick, I'm not entirely convinced that just because something isn't 1/r or r^2 for potential there can't be closed orbits. Bertrand's theorem is the mathematical statement of what many have posted thus far. However, because the potential for two dipoles interacting depends not only on the radial distance but also the direction, I'm not convinced Bertrand's theorem applies. In other words, I'm not convinced there isn't some configuration of initial dipole moments and radial distances for which there might be a closed orbit. Anyway, I think I repeat myself.

I don't think it has to be 1/r^2, but I think it has to tend to 1/r^2 from a field with a characteristic 1/r^(less than 2). By that means the gradient of the potential will mean something can drift out and find its 'natural' solution, if there is one.

My prose explanation (sans equations) for this would be;

For a permanent magnet it may tend to 1/r^2 at some point but it tends from 1/r^3 in its nearfield. This would mean the energy gained by some small perturbation inwards will be greater than a omega^2 function (as angular momentum must be preserved, even if the orbiting particle gains radial kinetic energy).

So the result is that if the field is stronger than 1/r^2 then an orbiting part will have to gain inward radial kinetic energy for an inward perturbation (or vice versa lose outward kinetic energy) if it is to preserve angular momentum (i.e. heads for the magnet or spins off).

 

[Haborix]

I appreciate the explanation but I think it still ignores the issue which I believe complicates the analysis. That is, the dependence of the force/potential on the relative orientation of various vector quantities in the problem.

 

[mfb]

This is not Bertrand's theorem. We don't need closed orbits.
Of course there is a solution with a closed orbit. Both magnets rotating bound to each other with a perfect circular orbit. It's just an unstable solution.

If you transfer the system to a co-rotating frame you get an effective 1/r² potential from centrifugal force that is repelling objects. Combine it with a -1/r potential and you get a potential well in the radial direction - a stable configuration, the particle will stay in a given radius range. If you combine it with any 1/r² potential, however, you don't have that stability.

 

[sophiecentaur]

So why wouldn’t all the filings end up stuck to the sphere? My argument would be that magnetic ‘lines of force’ are regarded as the path that an isolated North Pole would follow.
I suggest that, as with electrostatic induction, there would be an equivalent magnetic polarization and the field near the nearer pole would (net) attract the filing. So you will get the same pattern of filings as you get on the ground only more symmetrical. (No G effect). But every random shake would cause them all to end up on the surface. Magnetic forces are much stronger than g.

 

[hutchphd]

Magnetic forces are much stronger than g.

My intuition says that, given the opportunity, the filings would form head to toe along the field lines in 3D like they do in 2D on a piece of paper. The energetics would space them apart (have you ever sketched field lines by hand?) and you would have a round satellite carrying a "drawing" of its magnetic field. Obviously at some spin rates and orientations this may not work
Can we get a bay on the ISS to test this out? I'll bet iron filings are not their favorite cargo inside. I know most folks with vacuum systems blanch at the sight of steel wool.

 

[sophiecentaur]

My intuition says that, given the opportunity, the filings would form head to toe along the field lines in 3D like they do in 2D on a piece of paper.

.. . . only the lines of filings could be much longer and more impressive.

This is it in a nutshell and it seems reasonable. g is very low, even down here and you get sagging chains of filings between poles. So perhaps the ISS experiment wouldn't be necessary close up to the magnet. If the filings were replaced by frictionless balls - small ball bearings - wouldn't the divergence of the field around the poles cause the same effect as you get with electrostatic attraction. It's probably down to this induced quadrapole field round the filings being different from the induced dipole field on the dust/ paper pieces.

 

[hutchphd]

Is there a way to get ferromagnetic "filings" to be neutrally buoyant in a transparent fluid? Perhaps you could coat them and differentially float them to select their individual density.
So I did extensive research. Apparently "ferrofluids" were invented by NASA in the 1963! Interesting Wikipedia:https://en.wikipedia.org/wiki/Ferrofluid#Optics

 

[sophiecentaur]

Is there a way to get ferromagnetic "filings" to be neutrally buoyant

It wouldn't need to be too accurately neutral. A slow fall or rise would be good enough for an experiment. If you can get an oil, heavy enough to support them, you can add light oil to get what you want. (Something for the long winter evenings)

I remember 'tuning' my ancient Lava Lamp by adding small amounts of salt to the water. It works a treat these days and the mix is quite critical.

 

[Keith_McClary]

A slow fall or rise would be good enough for an experiment.

You could watch this at 0.25 speed:

 

[hutchphd]

I liked the one with sand even better:

 

[hutchphd]

I see that the ferrofluids are actually very very small ferro particles colloidally suspended. Much fun.

 

[mfb]

So why wouldn’t all the filings end up stuck to the sphere?

They would all end up at the magnet or escape. On Earth other forces can keep them elsewhere, floating in space there is nothing that stops them from going to the magnet.

 

[sophiecentaur]

They would all end up at the magnet or escape. On Earth other forces can keep them elsewhere, floating in space there is nothing that stops them from going to the magnet.

I was thinking in terms of friction (electric forces) between the filings, rather than gravity. Whilst I was thinking along the same lines of your post, gravity is less and less relevant, even on Earth in small scale interactions.

For filings with less than an escape velocity they would 'follow the lines' but, if they hit an existing column of filings, why not just stick to it?

I imagine thermal vibrations could cause eventual collapse.

 

[James William Hall]

Thank you all for your opinions on this thought experiment. If I understand properly I can visualize the experiment as follows:

The magnetic sphere is placed outside the ISS at some distance not spinning. A limited quantity of iron filings is gently wafted in a cloud at a distance of 40 times the radius of the ball. At first the cloud hangs together moving slowly in straight lines [A]. Then the filings closest to the ball begin to veer conically toward the ball accelerating first to the closest pole growing spikes quickly then as the height of the fill reaches a certain point the remaining filings form spikes steadily back toward the other pole along the field lines until there are no more filings left for symmetry and so we are left with a tear-shaped spiky ball [C]. Had there been lots of filings there would eventually be a symmetrical distribution. Or would there be a symmetric distribution no matter the quantity of filings. Meanwhile, the outer most filings waft by the ball in a straight line headed to far space. The filings just a bit nearer the ball bend a bit, pass the ball, and continue into outer space at an angle . As an aside, if we suddenly demagnetized the sphere the shape would hold as there is no force to accelerate the organized filings.

Now, as I think what Berkeman said earlier, in the case of a rotating sphere about the magnetic axis or even perpendicular to it, the image of the final steady state [C] would not significantly change.