You Can't Even Hope to Contain Him

I was dumping some packing peanuts into a trash can and they stopped pouring in — enough charge had built up that the additional peanuts were repelled by the ones in the can. Here’s a couple of attempts to throw peanuts in.

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You might be tempted to think you could confine a peanut (or any charged particle) this way — all of the charges repel, so you form a potential well which traps the extra charge. Unfortunately it doesn’t work out. The electric field you get will counteract gravity, but there’s no field directed radially inward, at all points, and no way to get one. Electric fields only converge on charges. The best you could get would be a field that was “leaky” — inward at one point, but outward somewhere else. All of this is shown mathematically in Earnshaw’s theorem. It works for magnetic fields, too, with the loophole that it doesn’t apply to diamagnets. (but Earnshaw didn’t know about those)