Question $2$: $[30$ pts$]$

The standard rectangular cross$-$section metallic waveguide WR $15$ has dimensions of the longer and shorter sides as $3\mathrm{.}7592$ mm and $1\mathrm{.}8796$ mm $.$ As is well known, the waveguide's use for different applications is based on the first two lowest cut$-$offs of the waveguide. Find the frequency range of operation for this waveguide based on the lowest two order modes in the waveguide.

Hint: Note that frequency range of usage for any waveguide is between the lowest order's cutoff frequency and cutoff frequency of the mode which is one order higher. Use the cutoff frequency for the first two modes to define the operational frequency range.

Question $3$: $[30$ pts$]$

Find the incidence angle $\backslash (\backslash$theta$\_\{$i$\}\backslash )$ at which the reflection of a parallely polarized plane wave will be zero from the soil in the following scenario. Calculate above angle of incidence for two scenarios based on the wetness of the soil, which changes the dielectric constant soil:

$1)\backslash (\backslash$epsilon$\_\{$r$\}=3\backslash )$

$2)\backslash (\backslash$epsilon$\_\{$r$\}=15\backslash )$

How will the responses change when the wave has a perpendicularly polarized E$-$field?