A slab of conductor (region 1) with a thickness $a$ exists with a covering of dielectric (region 2) as shown in the accompanying figure. The slab and the dielectric are very large in the $\hat{y}$ and $\hat {z}$ directions.

The conductor extends over the range $0 < x < a$ in the $\hat{x}$ direction, and the dielectric extends over the range $b < x < b$ in the $\hat{x}$ direction. The conductor has an uniform surface charge density $\sigma_o$, and the dielectric has a polarization given by $\vec{P}(x)=C_o x \, \hat{x}$, where $C_o$ is a constant.

What is the value of the electric field $E$ in region 2 ($a < x < b$)?

What is the value of the electric field $E$ in the region ($r > b$)?