So I understand that it means you cannot levitate objects by using static permanent magnets and that’s bc there isn’t a state of equilibrium for the object. I would like to know why that is the case.
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“Equilibrium” means the thing we’re levitating will stay in position. In particular, since it’s levitating, that means that if it moves in any direction away from the stable point it will get pushed back towards the stable point by the magnetic field. If that weren’t true, there’d be some direction that it *wasn’t* getting pushed back towards the stable point and it would eventually drift off in that direction.
So that implies that there’s at least one spot somewhere in the magnetic field where all the forces point inwards towards that spot (or at least converge, like spiraling inwards). We measure how much the forces come together/apart in a field like that by something called “divergence”, a vector calculus thing we don’t need to get into here, but think of it as “how much new stuff is appearing or disappearing in this spot”. If all the forces are coming together at a spot, we must have “stuff” disappearing in that spot. Similarly, if the forces are spiraling inwards we’ve got a tendency for the forces to spin around that spot (measured by a different vector calculus thing called “curl”).
So…to levitate, we need at least one spot where the divergence or the curl isn’t zero…a spot where all the forces generally point inwards towards the spot we want to levitate.
Except you can’t have that…a different chunk of physics (Gauss’s law) shows that a stable electric or magnetic field in “free space” (i.e. air, vacuum, whatever) has a curl and divergence of zero…there’s no spot *anywhere* that does what we want. So there’s no spot we can levitate the object. The proof of Gauss’s law is probably way beyond our scope, but it falls out of the equations that model how electric and magnetic fields behave.
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