$(18$ pts$)$ Two infinite conducting sheets located at $z=0m$ and $z=2m$ carry charge density ${}_s=+-lo{n}_0$ as shown in the figure below. The electrostatic potential $V(z)$ in between the plates is given below

$V(z)=\{\begin{array}{cc}\frac{z}{3},& 0zh\\ z-\frac{1}{3},& hz2\end{array}$

a$)$ Which plate will have positive charge. Justify your answer.

b$)$ Find the electric field $\frac{?}{\text{b}\text{a}\text{r}}(E)$ between the plates for both regions.

c$)$ Find the permittivities $lo{n}_1$ and $lo{n}_2$

${?}_{\text{s}\text{h}\text{e}\text{e}\text{t}}^{\frac{?}{\text{b}\text{a}\text{r}}(E)}=\frac{P_s}{2{}_0}$hat$(R)$

d$)$ Find the thickness $h$ of the bottom region.