91Ó°ÊÓ

Write the expression for Maxwell’s equation.

$\frac{∂E_z}{∂y}-iK{E}_y=i\mathrm{Ó\neg }{B}_x$ …… (1)

For a rectangular waveguide, as$\frac{\mathrm{Ó\neg }}{c}=k$ and $E_z=0$then, equation (1) becomes,

$\begin{array}{l}\frac{∂\left(0\right)}{∂y}−i\frac{\mathrm{Ó\neg }}{c}{E}_y=i\mathrm{Ó\neg }{B}_x\\ E_y=−c{B}_x\end{array}$

Write the expression for Maxwell’s equation.

$ik{E}_x-\frac{∂E_z}{∂x}=i\mathrm{Ó\neg }{B}_y$ …… (2)

Write the expression for Maxwell’s equation.

$\frac{∂B_z}{∂y}-ik{B}_y=-i\frac{\mathrm{Ó\neg }}{c^2}{E}_x$ …… (3)

Write the expression for Maxwell’s equation.

$ik{B}_x-\frac{∂B_z}{∂x}=-\frac{i\mathrm{Ó\neg }}{c^2}{E}_y$ …… (4)

Substitute$k=\frac{\mathrm{Ó\neg }}{c}$and $\frac{∂E_z}{∂x}=0$in the equation (2).

$\begin{array}{c}i\frac{\mathrm{Ó\neg }}{c}{E}_x−0=i\mathrm{Ó\neg }{B}_y\\ E_x=c{B}_y\end{array}$

Substitute $E_x=c{B}_y$in the equation (3).

$\begin{array}{c}\frac{∂B_z}{∂y}−ik{B}_y=−i\frac{\mathrm{Ó\neg }}{c^2}\left(c{B}_y\right)\\ \frac{∂B_z}{∂y}=ik{B}_y−ik{B}_y\\ \frac{∂B_z}{∂y}=0\end{array}$

Substitute$E_y=c{B}_x$in the equation (4).

$\begin{array}{c}ik{B}_x−\frac{∂B_z}{∂x}=−\frac{i\mathrm{Ó\neg }}{c^2}(−c{B}_x)\\ ik{B}_x−\frac{∂B_z}{∂x}=ik{B}_x\\ \frac{∂B_z}{∂x}=0\end{array}$

Hence,

$\frac{∂B_z}{∂x}=\frac{∂B_z}{∂y}=0$

If the boundary is just inside the metal, the value of E will be zero. So, the value of B will also be zero.

Hence, this is a TEM mode, and ${\text{TE}}_{00}$mode cannot occur in a rectangular waveguide.

Therefore, the ${\text{TE}}_{00}$mode cannot occur in a rectangular waveguide.