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Suppose we have two charged shells with a distance between the two centers that they cannot be considered as point-like charges in the mutual interaction, for example three times the radius. Can we still calculate the force between the two spheres by thinking of them as charges concentrated in their center? and why?

Qmechanic
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Andrew Carrol
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    Does this answer your question? What's the exact gravitational force between spherically symmetric masses? – Vincent Thacker Jun 02 '21 at 12:19
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    or this – mikuszefski Jun 02 '21 at 12:21
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    Does this answer your question? Electric field due to two charged conducting spheres – Bhavay Jun 02 '21 at 12:26
  • @VincentThacker mmm not so much.. i know you can use Gauss’ theorem to demonstrate the electric field outside of a shell is the same as a point charge but i can’t understand how you can use it to imply both bodies are treated like 2 point charges. – Andrew Carrol Jun 02 '21 at 13:47
  • @mikuszefski well they’re using the distance approximation so the force can be seen as uniformly distributed on the second body so.. i don’t think it’s the same thing – Andrew Carrol Jun 02 '21 at 13:56
  • @AndrewCarrol The answer depends on properties of these shells (dielectric or metallic). In general case there is electrostatic induction so that the force is not like between 2 point charges. The theory of this force is very complicated but nevertheless there are several papers about it like this one https://royalsocietypublishing.org/doi/10.1098/rspa.2012.0133 – Alex Trounev Jun 03 '21 at 19:04
  • Sorry, should have read the linked answer more carefully. If the charge interaction changes the distribution this is an entirely different game, of coarse. I personally think about this immediately ( only because I worked in this field for so long ) in terms of multipole moments ( assuming that the charges stay spherical symmetric) I know that the series expansion of the interaction in terms multipole moments converges if the distance is larger than the charges radii. Each shell has only a monopole moment. So that is all there is. I know, a slightly strange approach, but works for me. – mikuszefski Jun 04 '21 at 06:20
  • "i know you can use Gauss’ theorem to demonstrate the electric field outside of a shell is the same as a point charge but i can’t understand how you can use it to imply both bodies are treated like 2 point charges." You just need to note that the force on A due to P is the opposite of the force on P due to A, as in my Answer. – Keith McClary Jun 07 '21 at 17:07

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Suppose $S$ and $A$ are spherically symmetric distributions. Then the force on $A$ due to the field of $S$ is the same as with $S$ replaced by a point charge $P$. This is opposite to the force on $P$ due to the field of $A$, which is the same as with $A$ replaced by a point charge.

Keith McClary
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