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So, I've been told by Earnshaws Theorem that it's impossible to trap a charged particle using an purely electrostatic configuration. However, this doesn't seem to make intuitive sense. If I were to create a sphere whose surface consisted purely of positive charges, evenly distributed over the surface and then placed a single positive charge in the middle, would that not be trapping a particle in an electrostatic configuration? There would even be a restoration force as if the particle was "nudged" slightly, the net force would be to push the particle back into the center. I'm mostly looking for an intuitive answer. Have I missed something? Is this not defined as electrostatic?

A mathematical explanation would also be appreciated alongside an intuitive one.

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  • $\begingroup$ The thing is the positive charges in the surface will repel themselves and destroy the configuration unless there are other interactions involved. $\endgroup$
    – secavara
    Apr 28, 2018 at 16:18

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What you have missed is the fact that, inside a uniformly charged spherical shell, there is no electric field. The electric potential is uniform inside the shell, so there is no restorative force; there is no force at all.

However if instead of a spherical shell you have a spherical wire grid, the story is a little different. It's still true that inside the sphere there would be no electric field, but outside the sphere there would be an electric field. So, if the sphere were positively charged, it would be able to confine a negatively charged particle. Any negatively charged particle that departed from the interior of the sphere would be shoved back as soon as it entered the region outside the sphere. Of course, the grid would intercept a significant fraction of the particles; and any particle with more kinetic energy than its potential energy difference (between the surface of the sphere and infinity) could escape.

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