Electric field outside a conductor and its surface charge

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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}##?
 

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