$)[10$ Marks$]$ In a rectangular waveguide with dimensions of $a=1\mathrm{.}5cm$ and $b=0\mathrm{.}8cm$ and filled with a aterial characterized by $={}_0$ and $=4{}_0$ the z$-$component of the electric field is given as

termine:

$E_z=2sin(\frac{x}{a})sin(\frac{3y}{b})cos({10}^{11}t-z),\frac{V}{m}$

a$)$ The mode of operation $($mode order$)$:

b$)$ The cutoff frequency: $(1$ marks$)$

c$)$ The phase constant $$ : $(1$ mark

$q,(1$ mark$)$

d$)$ The propagation constant $$ : $q,(1$ mark$)$

e$)$ The intrinsic wave impedance: $q,(0\mathrm{.}5$ marks

f$)$ The group velocity: $q,(0\mathrm{.}5$ marks

planation $($Steps to find the answers$)$: $(5$ marks$)$