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I have 2 questions about Newton's Shell Theorem.

  1. Is it valid when mass is distributed arbitrarily in 3D space? not like sphere?

  2. When it comes to electrostatic force, can we define "Center of Charge" like "Center of Mass" used in Gravitation's shell theorem? As if all of those charges distributed along the space are concentrated in one point inside the space.

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For the first question: no, it only applies to spherically-symmetric distributions. There are generalisations to some other distributions I think, but it is certainly not true for arbitrary distributions.

For the second, yes, you can do that: if you're dealing with a spherically-symmetric distribution of charge you can treat it as a point charge at the centre of the distribution.

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