# Waveguides

 Question 1
A waveguide consists of two infinite parallel plates (perfect conductors) at a separation of $10^{-4}$ cm, with air as the dielectric. Assume the speed of light in air to be $3 \times 10^{8}$ m/s. The frequency/frequencies of TM waves which can propagate in this waveguide is/are _______.
 A $6 \times 10^{15}Hz$ B $0.5 \times 10^{12}Hz$ C $8 \times 10^{14}Hz$ D $1 \times 10^{13}Hz$
GATE EC 2022   Electromagnetics
Question 1 Explanation:
Cut-off frequency.
\begin{aligned} f_c&=\frac{c}{2a}\;\;\;(m=1) &=\frac{3 \times 10^8}{2 \times 10^{-4} \times 10^{-2}}\\ &=1.5 \times 10^{14}Hz \end{aligned}
$f \gt f_c$ will only propagate
A and C will propage.
 Question 2
A standard air-filled rectangular waveguide with dimensions $\text{a=8 cm, b=4 cm}$, operates at $\text{3.4 GHz}$. For the dominant mode of wave propagation, the phase velocity of the signal in $v_{p}$. The value (rounded off to two decimal places) of $v_{p}/c$, where c denotes the velocity of light, is _____
 A 2.24 B 3.82 C 1.2 D 4.6
GATE EC 2021   Electromagnetics
Question 2 Explanation:
$f_{c 10}=\frac{c}{2 a}=\frac{3 \times 10^{8}}{2\left(8 \times 10^{-2}\right)}=1.875 \mathrm{GHz}$
Guide phase velocity,$V_{p}=\frac{C}{\sqrt{1-\left(\frac{f_{C 10}}{f}\right)^{2}}}$
$\frac{V_{p}}{C}=\frac{1}{\sqrt{1-\left(\frac{f_{C 10}}{f}\right)^{2}}}=\frac{1}{\sqrt{1-\left(\frac{1.875}{3.4}\right)^{2}}}=1.198$

 Question 3
The refractive indices of the core and cladding of an optical fiber are 1.50 and 1.48, respectively. The critical propagation angle, which is defined as the maximum angle that the light beam makes with the axis of the optical fiber to achieve the total internal reflection, (rounded off to two decimal places) is _____ degree.
 A 18.3 B 3.45 C 9.37 D 5.82
GATE EC 2021   Electromagnetics
Question 3 Explanation:
Given that
Refractive index of core $\eta_{1}=1.50$
Refractive index of clad $\eta_{2}=1.48$
Critical propagation angle $\left(\theta_{P}\right)$
$\theta_{P}=\sin ^{-1}\left[ \frac{ \sqrt{\eta_{1}^{2}-\eta_{2}^{2}}}{\eta_{1}} \right]=\sin ^{-1} \left[ \frac{ \sqrt{1.5^{2}-1.48^{2}} }{1.5} \right] =9.37$
 Question 4
Consider a rectangular coordinate system (x,y,z) with unit vectors $a_{x}\:a_{y}$ and $a_{z}$. A plane wave traveling in the region $z\geq 0$ with electric field vector $E=10\cos\left ( 2\times 10^{8}t\:+\:\beta z\right )a_{y}$ is incident normally on the plane at $z=0$, where $\beta$ is the phase constant. The region $z\geq 0$ is in free space and the region $z \lt 0$ is filled with a lossless medium (permittivity $\varepsilon \:=\:\varepsilon _{0}$, permeability $\mu \:=\:4\mu _{0}$ , where $\varepsilon _{0}\:=\:8.85\times 10^{-12}\:\text{F/m}$ and $\mu _{0}\:=\:4\pi \times 10^{-7}\:\text{H/m})$. The value of the reflection coefficient is
 A $\frac{1}{3}$ B $\frac{3}{5}$ C $\frac{2}{5}$ D $\frac{2}{3}$
GATE EC 2021   Electromagnetics
Question 4 Explanation:
Given $\vec{E}=10 \cos \left(2 \pi \times 10^{8} t+\beta z\right) \hat{a}_{y}$ for $z \geq 0$ having free space. For $z \lt 0$ medium has
$\epsilon_{r 2}=1 ; \mu_{r 2}=4$

$\Gamma=\frac{\eta_{2}-\eta_{1}}{\eta_{2}+\eta_{1}}=\frac{120 \pi(2)-120 \pi}{120 \pi(2)+120 \pi}=\frac{1}{3}$
 Question 5
A rectangular waveguide of width w and height h has cut-off frequencies for $TE_{10} \; and \;TE_{11}$ modes in the ratio 1:2. The aspect ratio w/h, rounded off to two decimal places, is ___________
 A 0.85 B 1.25 C 1.73 D 1.92
GATE EC 2019   Electromagnetics
Question 5 Explanation:
$f_{c m n}=\frac{c}{2} \sqrt{\left(\frac{m}{a}\right)^{2}+\left(\frac{n}{b}\right)^{2}}$
For TE mode
$f_{c10}=\frac{c}{2 w} \quad\ldots(i)$
and For $TE_{11}$ mode.
\begin{aligned} f_{c11} &=\frac{c}{2} \sqrt{\frac{1}{w}^{2}+\left(\frac{1}{h}\right)^{2}} &\ldots(i)\\ &=\frac{c}{2 w} \sqrt{1+\left(\frac{w}{h}\right)^{2}} &\ldots(ii)\\ \text{given}\quad\frac{f_{c10}}{f_{11}} &=\frac{1}{2} &\ldots(iii) \end{aligned}
put (i). (in) in (ii)
$\Rightarrow \frac{\frac{c}{2 w}}{\frac{c}{2 w} \sqrt{1+\left(\frac{w}{h}\right)^{2}}}=\frac{1}{2} \Rightarrow \sqrt{1+\left(\frac{w}{h}\right)^{2}}=2$
On solving above equation, we get,
$\frac{w}{h}=\sqrt{3}=1.732$

There are 5 questions to complete.