91Ó°ÊÓ

Write the expression for the components of electric and magnetic fields along the z-axis in a rectangular wave.

$\begin{array}{c}[\frac{{∂}^2}{∂x^2}+\frac{{∂}^2}{∂y^2}+{\left(\frac{\mathrm{Ó\neg }}{c}\right)}^2-k^2]E_z=0\\ [\frac{{∂}^2}{∂x^2}+\frac{{∂}^2}{∂y^2}+{\left(\frac{\mathrm{Ó\neg }}{c}\right)}^2-k^2]B_z=0\end{array}$

Here, $E_z$is the longitudinal component of electric field and$B_z$ is the longitudinal component of a magnetic field,$\mathrm{Ó\neg }$ is the frequency of a wave, c is the speed of light, and k is the wavenumber.

For the TM wave, the value of the longitudinal component of the magnetic field is zero.

Write the boundary conditions at the wall.

$\begin{array}{c}{E}^{âˆ\yen }=0\\ B^{âŠ\yen }=0\end{array}$

Let, $E_z(x,y)=X\left(x\right)Y\left(y\right)$.

Here, .$X\left(x\right)=A\sin \left(k_{x}x\right)+B\cos \left(K_{x}x\right)$

At walls $E_z=0$, then at$x=0$and $x=a$, the value of X and B will be,

$\begin{array}{c}X=0\\ B=0\end{array}$

Hence, it is known that:

$\begin{array}{c}{k}_x=\frac{m\mathrm{Ï€}}{a};m=1,2,\mathrm{3....}\\ k_y=\frac{n\mathrm{Ï€}}{a};n=1,2,\mathrm{3....}\end{array}$

So, the longitudinal electric field will be,

$E_z=E_0\sin \left(\frac{m\mathrm{Ï€}x}{a}\right)\sin \left(\frac{n\mathrm{Ï€}y}{a}\right)$

${\mathrm{Ó\neg }}_{mn}=c\mathrm{Ï€}\sqrt{{\left(\frac{m}{a}\right)}^2+{\left(\frac{n}{b}\right)}^2}$Write the expression for wave number.

Hence, the cut off frequency will be,

$k=\sqrt{{\left(\frac{\mathrm{Ó\neg }}{c}\right)}^2−{\mathrm{Ï€}}^2[{\left(\frac{m}{a}\right)}^2+{\left(\frac{n}{b}\right)}^2]}$

Write the expression for the wave velocity,

$v=\frac{\mathrm{Ó\neg }}{k}$ …… (1)

Write the expression for the lowest cut-off frequency $\left({\mathrm{Ó\neg }}_{11}\right)$for mode${\text{TM}}_{11}$ .

${\mathrm{Ó\neg }}_{11}=c\mathrm{Ï€}\sqrt{{\left(\frac{1}{a}\right)}^2+{\left(\frac{1}{b}\right)}^2}$ …… (2)

Hence, the wavenumber in terms of the cut-off frequency will be,

$k=\frac{1}{c}\sqrt{{\mathrm{Ó\neg }}^2−{\mathrm{Ó\neg }}_{mn}^2}$