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(forgive me any grammatical flaws, english is not my native language)

hello,

i know that according to Earnshaw's theorem a stable levitation cannot exist by only using permanent magnets.

Similar questions has been already answered (e.g) Can I levitate an object without using an electromagnet? but they don't state what actually would happen and why.

...with a setup like that for exampe.

enter image description here

We asume that we have a cup and a sphere. Since a perfect equilibrium cannot exists, what happens actually then?

  1. no levitation (no repulsive force) at all due to negation
  2. "instable levitation" (sphere jitters and spins)
  3. the sphere either sticks to the cup or pops out of it depending on it's weight
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The sphere will not levitate due to issue 1. The magnetic field of the sphere as depicted in your picture will not be like a magnetic monopole with field lines radiating outward. The magnetic field outside the sphere will be nearly zero. In fact, such a field (like an electric point charge) is impossible to create since magnetic monopoles don't exist. The result of this is that the sphere falls to the bottom of the cup since it feels no magnetic force.

Beyond that, let's consider an ideal case: a sphere and cup that are both electrically charged with the same sign charge (electric charges interacting with each other would behave the same as magnetic charges interacting with each other). This will result in situation 3. If the ball is heavy enough, it will sink below the lip of the cup. From that point, the vertical electric field gets weaker as you approach the bottom, since more an more of the cup is above the ball, pushing it down. So the ball would fall all the way to the bottom. If the ball is too light, it floats upward from the electric force. Above the cup, the electric fields spread outward, and if the ball is even slightly off-center, it would be pushed away and out of the cup. An exactly-right weight ball would not be stable, as any vertical disturbance would cause the ball to fall into or out of the cup as explained earlier.

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