$7-9.$ A common commercial size of rectangular hollow$-$conductor waveguide has $($inner$)$ dimensions $a=2\mathrm{.}286$ by $b=1\mathrm{.}016$ cm $.$

$($a$)$ What is the single$-$mode bandwidth of this size waveguide$-$that is$,$ what is the frequency range $($in gigahertz$)$ within which the $T{E}_{10}$ mode propagates but the next higher mode does not? $[$Most practical waveguide systems are operated under this single$-$mode condition.$]$

$($b$)$ The maximum field strength that air can support without breakdown is about $30,000\frac{V}{c}m($at standard atmospheric pressure$).$ What is the maximum power that can be transmitted by this size waveguide at frequencies within the single$-$mode band?