Electric field outside a conductor and its surface charge

[Happiness]

Imagine a surface charge $\sigma$. The boundary condition on $E$ is

$E_{above}-E_{below}=\frac{\sigma}{\epsilon_0}\hat{n}$, where $\hat{n}$ points outwards perpendicularly to the surface.

Because the field inside a conductor is zero, it requires that the field immediately outside is $E=\frac{\sigma}{\epsilon_0}\hat{n}$.

But if the conductor is a large, flat plane with a uniform surface charge $\sigma$, shouldn't the field be $E=\frac{\sigma}{2\epsilon_0}\hat{n}$?

 

[nasu]

You are mixing two different things.
The field of a plane of charge (fixed charge) is not the same as the field of a charged sheet of conducting material.
This issue was discussed previously on the forum. For example, here:
https://www.physicsforums.com/threads/electric-field-of-an-infinite-plane-and-a-conductor.241108/