I’ve often heard it said that Earnshaw’s theorem rules out the possibility of levitating anything with static magnets. Is that correct? I’m uncertain because as I understand it the theorem talks about stabilizing *point* particles, but if I take a bunch of magnets and glue them to different bits of a rigid structure, then it’s no longer a point particle I’m trying to stabilize. For example, in the geometry in the linked diagram, along which axis would the levitating ‘top’ be unstable? Nested magnet diagram The diagram shows magnets with polarity represented by color and this is a 2D cut-away (ie the structure is rotationally symmetric).
Comments
I think that only applies to electric charges, not magnetism
There are plenty of desktop decorations using magnetic levitation. Static magnets, arranged in a circle, in both base and floater, with floater having a smaller circle
Well, the obvious issue I see is stability. Currently, our inner and outer circumferential magnets are aligned. I will assume that the weight is perfectly offset by the face magnets on top at this exact point.
If the outer shell is lifted above this point, then all of the circumferential magnets are now lifting the weight. If the weight was perfectly offset before this point, then it is now overcome, and the shell continues to lift until the force of the circumferential magnets has dropped off enough with distance that our shell may settle.
At this point, they have inherently moved up a distance much greater than the original gap between the two cylinders, and so their main force component is vertical. They are now no longer stabilizing horizontally, and our top cylinder is free to tilt to one side and settle physically pressed against the bottom one.
It’s likely it would be unstable against tilting. The big magnets at the top tend to behave that way.
There are several versions of Earnshaw’s Theorem. The one that applies to static electric fields from point charges is easy to prove and usually seen in undergrad physics texts. But another version applies to static magnetic dipoles. A partial proof can be found on Wikipedia but the full proof is apparently really nasty.
It’s also fundamentally 3-d, so 2-d cutaways like your diagram are misleading.
In your diagram, I think the system is unstable to tilting: if the outer shell kicks up on one side and down on the other, all the magnets get farther from each other.