I have a quick question:
In deriving the displacement current term for Ampere's Law, my book has the line:
$$\Phi_E= \int_S \mathbb{E} \cdot \hat{n} da= \int_S \frac{\sigma}{\epsilon_0} da = \frac{Q}{A \epsilon_0} \int_S da= \frac{Q}{\epsilon_0}$$
My question is:
Here the electric field is the electric field between two conducting plates (capacitor) neglecting edge effects. $\sigma$ is the charge density of a plate given by: $\frac{Q}{A}$ where A is the capacitor plate area. My book substitues $\sigma= \frac{Q}{A}$ and arrives at: $$\frac{Q}{A \epsilon_0} \int_S da = \frac{Q}{A \epsilon_0} A = \frac{Q}{\epsilon_0}.$$ Why do the "A"'s cancel? One $A$ is the area of the capacitor plate and the other area is the area of the gaussian surface. In general, these area's will not be equal. The image to go along with the derivation is similar to the one shown below. 
Best Answer
The flux is only has a nonzero part where there is a nonzero electric field. Next if we assume the $\sigma$ is the charge density on the plate, then when you replace the $\bf E$ integral with a $\sigma$ integral then the area there is just over the area of the capacitor $C$ because that is where the electric field is nonzero, and equals $\sigma/\epsilon_0.$
$$\Phi_E= \int_S \mathbf E \cdot \hat{\mathbf n}\,\mathrm da= \int_C \frac{\sigma}{\epsilon_0} \mathrm da$$