Problem #$5(20$ pts$)$:

An infinitely thick slab of dielectric material with ${}_r=9$ is excited with a $30$ GHz plane wave at normal incidence in air. A sheet with a dielectric constant of ${}_{r1}=4$ and a thickness $d_1=2\mathrm{.}5$ mm is placed on top of the slab as in Fig. $5$a$.$ Use the speed of light in free space as $c=3{10}^8\frac{m}{s}.$ Determine:

a$)$ The equivalent transmission line model of the structure

b$)$ The impedance looking into the structure.

Now, a second sheet of dielectric material with a relative permittivity of ${}_{r2}$ is placed on top of the first sheet as in Fig. $5b.$ If I wish for the impedance looking into this structure to be $0\mathrm{.}5{}_0$ :

a$)$ How should I choose the relative permittivity ${}_{r2}$ and the thickness $d_2$ of the second sheet?

b$)$ If the incident electric field is given by vec$(E)(z)=((hat(x))E_{0x}+(hat(y))E_{0y})e^{-jkz},$ find the phasor expression for the total electric field $($incident and reflected$)$ in the air region. Leave your answer in terms of $E_{0x},E_{0y},$ and $z($and only $E_{0x},E_{0y},$ and z$).$

Fig. $5$b