A Chord of Given Triangle Area

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has eccentricity $\dfrac{\sqrt 2}2$ and major axis of length $4$.

1. Find the equation of $C$. 2. A line $l$ through the point $(0,-2)$ meets $C$ at two points $A$ and $B$; let $O$ be the origin. If the area of $\triangle OAB$ is $\sqrt 2$, find $|AB|$.