I am very confused about plates (conductors/insulators) and applying Gauss law. It seems like gauss law for the isolator is very often derived wrongly.
In short: if you have an infinite plate (insulators) then you can place a cylinder perpendicular to the surface and often the following derivation follows:
Gauss law:
$$ \oint_S EdA = \frac{Q_{\text{enclosed}}}{\epsilon_0} $$
$$ E 2(\pi r^2)= \frac{\sigma (\pi r^2)}{\epsilon_0} $$
$$ E = \frac{\sigma}{2\epsilon_0}$$
However even for a very thin infinite plate you enclose two surfaces, so you then you find for an insulator: $$ E 2(\pi r^2)= \frac{\sigma 2(\pi r^2)}{\epsilon_0} $$ $$ E = \frac{\sigma}{\epsilon_0}$$
The confusing thing is it is usually followed by a derivation with a conductor, giving the last equation.
So my specific question is, is the usual derivation for the insulating plate wrong? The proper derivation you have to use charge density per volume instead of surface density
$$ E 2(\pi r^2)= \frac{\rho d (\pi r^2)}{\epsilon_0} $$ $$ E = \frac{\rho d }{2\epsilon_0}, \rho d = \sigma $$ $$ E = \frac{\sigma}{2\epsilon_0}$$
The first derivation can be done for a conductor I think, because all charge is on the surface.
