Capacitance of a Parallel Plate Capacitor Without a Dielectric

Start with the bare parallel-plate arrangement, no dielectric present: when charge $+Q$ is placed on the isolated plate, an equal and opposite $-Q$ is induced on the INNER face of the earthed plate (facing the charged plate), while an equal $+Q$ is simultaneously induced on that earthed plate's OUTER face -- but since that outer face is directly connected to earth, this induced $+Q$ is genuinely free to (and does) drain away into the ground immediately, leaving only the bound $-Q$ on the inner face.

Consider the field in three distinct regions. In the OUTER regions -- beyond either plate, on the far sides of the pair -- the individual fields due to the two oppositely-charged plates point in OPPOSITE directions and cancel EXACTLY, giving zero net field outside the capacitor altogether: $E=\dfrac{\sigma}{2\epsilon_0}-\dfrac{\sigma}{2\epsilon_0}=0$. In the INNER region -- the gap directly between the two plates -- the two plates' individual fields instead point in the SAME direction and simply ADD: $E=\dfrac{\sigma}{2\epsilon_0}+\dfrac{\sigma}{2\epsilon_0}=\dfrac{\sigma}{\epsilon_0}=\dfrac{Q}{\epsilon_0A}$, directed from the positive plate toward the negative one -- exactly TWICE the single-sheet result of section 8.2.3, because here there are two charged sheets, not one, both contributing in the same sense inside the gap. …