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A rectangular wave guide of cross section $5 \mathrm{~cm} X 2 \mathrm{~cm}$ is used to propagate $T M_{11}$ mode at $9 \mathrm{G} \mathrm{Hz}$. Determine the cutoff wavelength and wave impedance.
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Solution:

$\lambda_{\mathrm{c}}=\frac{2 a b}{\sqrt{m 2 b 2+n 2 a 2}}$

Here, $a=2 \times 10^{-2} \mathrm{~m} ; \quad b=5 \times 10^{-2} \mathrm{~m}$

Thus, Cutoff Wave length $\lambda_{\mathrm{c}}=9.285 \mathrm{~cm}$ and $\mathrm{f}_{\mathrm{c}}=\frac{c}{\lambda c}=3.23 \mathrm{GHz} \\$. Wave impedance for TM mode $=\eta \sqrt{1-\left(\frac{f c}{f}\right)^2} \$

$=377 \sqrt{1-\left(\frac{3.23}{9}\right)^2} \\$ $=351.8$ ohms