Consider a parallel plate capacitor. This is a setup of two very large parallel plates, each a conductor and each with area $A$, and one having positive charge $Q$ and the other negative charge $-Q$.
My question is about the value of the electric field between these conductors.
Now, if these parallel plates were non conducting charged planes then we could compute the electric field due to each plate and use superposition.
By Gauss' Law using a pillbox as a Gaussian surface for one of the plates we have
$$E\cdot 2A = \frac{\sigma A}{\epsilon_0}$$
$$E=\frac{\sigma}{2\epsilon_0}$$
where $\sigma$ is the surface charge density on the plate.
When we use superposition we obtain a field of $\frac{\sigma}{\epsilon_0}$ between the plates and zero elsewhere.
But what we really have are conductors, and conductors have no electric field inside of them. Thus, when we use Gauss' Law on one of the plates we have
$$EA=\frac{\sigma A}{\epsilon_0}$$
$$E=\frac{\sigma}{\epsilon_0}$$
Now it seems to me that this calculation is the same whether there is a second plate or not. After all, the second plate generates a field that contributes zero flux on the pillbox we're using as a Gaussian surface.
On the other hand, why can't we use superposition to write the electric field at a point between the plates as
$$E=E_1+E_2=2\frac{\sigma}{\epsilon_0}$$
A quick google search for the electric field between capacitor plates gives me this link, which shows a calculation that considers the plates as infinitely charged planes. So another way to frame my question, perhaps, is to ask why such calculations don't take into account the fact that we have conductors and the electric field inside of these conductors is zero?
