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Can anyone please help explain why the charge layers on the inner of a conductor, must be equal and opposite to the charge layers on the neighbouring conductors.

I originally thought this would be because the electric field inside a conductor has to be 0 but I am unsure how I would approach this from purely the laws of electrostatics, can someone please help explain this concept I really would like to understand from an electrostatic perspective.

Thank you, please feel free to edit the tags of this question if any are missing.

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The fact that there is 0 electric field inside any conductor can directly be derived using the following assumptions:

  1. Existance of electric charges.
  2. Existance of an electric field $\vec E$.
  3. An electric field $\vec E$ exerts a force $\vec F = q \vec E$ on a charge $q$.

I would say these are all laws of electrostatics. If you disagree, feel free to point out which are not and why. Additionally, I will use the definition of a conductor, namely that it contains free electric charges in any small subvolume of it, since it is impossible to say anything about a conductor without defining what that is.

Now let's assume there is a conductor with volume $V$ in an equilibrium state and there is an electric field $\vec E$ with $\exists \vec r \in V: \vec E(\vec r) \neq 0$. According to the definition of a conductor there is a free charge $dq$ in the small subvolume $dV$ around $\vec r$, which is accelerated by the force $d\vec F = dq ~\vec E(\vec r)$, thus the conducor is not in the equilibrium. As a consequence, the initial assumption is wrong and for any conductor in an equilibrium state it has to hold $\forall \vec r \in V: \vec E(\vec r) = 0$.

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