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Given six strong magnets, what if we fix three of them to a table in a simple, horizontal, equilateral triangle-like formation (with their north poles sticking up), and the other tree to the points of a rigid wooden (same size) equilateral triangle, with their north poles facing down? We can overlay these two triangles in a perfect "Star of David" formation, and statically levitate the latter triangle above the table by the repulsion of the fields. No moving charges or fields required, just some wood and six strong permanent magnets. The weight of the triangle & magnets will prevent "flipping", while the repulsive magnetic fields will prevent "spinning" or lateral movement as well.

Does this violate Earnshaw's theorem somewhere?

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    $\begingroup$ You cannot violate Earnshaw's theorem with static magnets. For the possible loopholes, see en.wikipedia.org/wiki/Earnshaw%27s_theorem#Loopholes $\endgroup$ – Peter Diehr Dec 29 '18 at 2:26
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    $\begingroup$ @ManRow: Sorry, but your device is unstable without a tether. You can test this by touching it lightly, from any direction. Try the experiment: build it, and video the results! $\endgroup$ – Peter Diehr Dec 29 '18 at 2:43
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    $\begingroup$ @ManRow: just look at the link provided earlier for the conditions. The mathematical proof is in Earnshaw's theorem. If you study the proof carefully, you should be able to see that any such configuration is unstable. The proof is generally done as homework in a junior/senior level field theory class. $\endgroup$ – Peter Diehr Dec 29 '18 at 12:51
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    $\begingroup$ Here's an empirical reason why this violates Earnshaw's theorem: can you buy one? Because such a thing would be a seriously cool toy, at least. $\endgroup$ – tfb Jan 21 at 20:10
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    $\begingroup$ Indeed not: Earnshaw's theorem says they are required (or at least diamagnetic materials). My 'empirical' comment was merely meant to back the theorem up: not only do we theoretically know such things don't work, we don't observe them in practice. I'm not going to respond further as this is not going anywhere. $\endgroup$ – tfb Jan 21 at 20:35
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So, essentially my idea was to create a setup where the individual "saddle paths" of the levitated platform's magnets and/or charges are oriented in such a way that one part of the levitated platform moving towards lower potential would require the rest to move into higher potential (referring to the total sum of gravity & electrostatic potential energies), in such a way so as to create an (gravity + electrostatic) potential energy well for the "overall structure" as a whole at the levitation position.

While there can be numerous different arrangements of opposing individual saddle paths for individual charges or magnets fixed to a levitating platform above some collection of charges or magnets fixed to the ground, there doesn't seem to be an obvious arrangement of everything that would necessarily lead to a (gravitational + electrostatic + ferromagnetic) potential energy well for the levitated structure as a whole.

I have not fully analyzed all the full equipotential surface(s) of my setup specifically, but looking at some simpler similar examples it doesn't look like these kind of potential energy wells can exist, even if the specific saddle path(s) of my full setup can be difficult or nontrivial to find or visualize (comprising some specific complicated or unforseen combination of rolling, spinning, spiraling, etc... movement(s))

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Earnshaw's theorem only applies to electrostatic systems, which consist of static point charges. It does not have anything to say either way about the stability of your system, since that is a system composed of magnetic components.

The origin of Earnshaw's theorem can be traced to Gauss' law, which essentially states that the electric potential $\phi$ obeys Poisson's equation $$-\nabla^2 \phi = \rho$$ where $\rho$ is the charge density. You may recall from an analysis class that solutions to Laplace's equation have the property that over some region of space $R$ they have no local extrema (unless they are constant). Thus, there is no point where the electric force (the derivative of $\phi$) vanishes and is attractive in all directions.

If the system is not static or involves magnetic fields, then the electric potential $\phi$ is not a complete description of the dynamics and Earnshaw's theorem won't apply. Additionally, since magnetic forces don't originate from a Gauss' law-like form (recall there are no magnetic charges), Earnshaw's theorem can't be applied.

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    $\begingroup$ There is more than one version of Earnshaw's theorem. There are versions that deal with magnetism, as described in the WP article: en.wikipedia.org/wiki/Earnshaw's_theorem $\endgroup$ – Ben Crowell Jan 20 at 22:20
  • $\begingroup$ Earnshaw's theorem does apply to magnets whose field does not depend on applied fields I think. $\endgroup$ – tfb Jan 21 at 19:18
  • $\begingroup$ Interesting, my mistake. In this case, judging from the "loopholes" section of the linked Wikipedia page, it seems like constrained motion may provide another way to circumvent the instability. $\endgroup$ – Jonathan Curtis Jan 23 at 3:02
  • $\begingroup$ Yup, essentially the idea is to create a setup where the individual "saddle paths" of the levitated magnets and/or charges are oriented in such a way that one part of the levitated platform moving towards lower potential would require the rest to move into higher potential (referring to the total sum of gravity & electrostatic potential energies), in such a way to create an overall (gravity + electrostatic) potential energy well for the "overall structure" at the levitation position. $\endgroup$ – ManRow Jan 24 at 6:27

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