# Why are charge layers on the inner of a conductor equal and opposite to the charge layers on the neighbouring conductors?

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.

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 $$q$$.
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 $$\forall \vec r \in V: \vec E(\vec r) = 0$$.