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We know the electric field of an infinite plane is $\frac{N}{2e_0}$, where N is the charge density, but for conducting objects its $\frac{N}{e_0}$?

This is a contradiction. Or because the plane is infinite, the conductor formula fails to work? Or do conductors need volume

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A conductor must consist of a union of surfaces and bulk. Therefore, a conducting sheet must have two sides separated by a neutral bulk(However thin that might be). Therefore, within a given volume of an insulating sheet and a conducting sheet, the latter has twice the amount of charge and produces twice the electric field twice.

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