What do you mean exactly by Gauss's law?
For a point charge $q$, we have $E=\frac{q}{4\pi\epsilon_0 r^2}$.
More generally, Gauss's law says that the surface integral of $E$ is equal to the enclosed charge $Q$ divided by $\epsilon_0$:
$$\bigcirc\!\!\!\!\!\!\!\!\iint E\cdot dS = \iiint \frac\rho{\epsilon_0} dV= \frac Q{\epsilon_0}$$
I may be going on a rampage, but suppose we look at just one plate with charge $Q$ in a cross section of area $A$.
And suppose we define a cylinder with the plate in the middle, a cross section with area $A$, and a distance $d$ to each end of the cylinder.
Then due to symmetry, we have that the surface integral is $2\cdot E \cdot A$.
Gauss's law says that this is equal to $\frac {Q}{\epsilon_0}$.
So:
$$2EA = \frac {Q}{\epsilon_0} \quad\Rightarrow\quad E = \frac{Q}{2\epsilon_0 A}$$
Are you still with me?
Or am I going on a rampage? (Wondering)
