Question

Show that the mode $\mathit{T}{\mathit{E}}_{\mathbf{00}}$ cannot occur in a rectangular wave guide. [Hint: In this case role="math" localid="1657512848808" $\frac{\mathbf{\omega }}{\mathbf{c}}\mathbf{=}\mathit{k}$, so Eqs. 9.180 are indeterminate, and you must go back to Eq. 9.179. Show thatrole="math" localid="1657512928835" ${\mathit{B}}_{\mathbf{z}}$ is a constant, and hence—applying Faraday’s law in integral form to a cross section—thatrole="math" localid="1657513040288" ${\mathit{B}}_{\mathbf{z}}\mathbf{=}\mathbf{0}$ , so this would be a TEM mode.]

Short answer

It is proved that the$T{E}_{00}$mode cannot occur in a rectangular waveguide.

Step-by-step solution

Determine the electric field in y-direction:

• First equation:

Write the expression for Maxwell’s equation.

$\frac{\mathbf{\partial }{\mathbf{E}}_{\mathbf{z}}}{\mathbf{\partial }\mathbf{y}}\mathbf{-}\mathit{i}\mathit{K}{\mathit{E}}_{\mathbf{y}}\mathbf{=}\mathit{i}\mathit{\omega }{\mathit{B}}_{\mathbf{x}}$ …… (1)

For a rectangular waveguide, as $\frac{\omega }{c}=k\text{and}{E}_z=0$then, equation (1) becomes,

localid="1657514438209" $$\begin{gathered} \frac{\partial \left(0\right)}{\partial y}-i\frac{\omega }{c}{E}_y=i\omega B_x \\ {E}_y=-c{B}_x \end{gathered}$$

• Second Maxwell’s equation:

Write the expression for Maxwell’s equation

$\mathit{i}\mathit{k}{\mathit{E}}_{\mathbf{x}}\mathbf{-}\frac{\mathbf{\partial }{\mathbf{E}}_{\mathbf{z}}}{\mathbf{\partial }\mathbf{x}}\mathbf{=}\mathit{i}\mathit{\omega }{\mathit{B}}_{\mathbf{y}}$ …… (2)

• Third Maxwell’s equation:

Write the expression for Maxwell’s equation.

$\frac{\mathbf{\partial }{\mathbf{B}}_{\mathbf{z}}}{\mathbf{\partial }\mathbf{y}}\mathbf{-}\mathit{i}\mathit{k}{\mathit{B}}_{\mathbf{y}}\mathbf{=}\mathbf{-}\mathit{i}\frac{\mathbf{\omega }}{{\mathbf{c}}^{\mathbf{2}}}{\mathit{E}}_{\mathbf{x}}$ …… (3)

• Fourth Maxwell’s equation

Write the expression for Maxwell’s equation.
$\mathit{i}\mathit{k}{\mathit{B}}_{\mathbf{x}}\mathbf{-}\frac{\mathbf{\partial }{\mathbf{B}}_{\mathbf{z}}}{\mathbf{\partial }\mathbf{x}}\mathbf{=}\mathbf{-}\frac{\mathbf{i}\mathbf{\omega }}{{\mathbf{c}}^{\mathbf{2}}}{\mathit{E}}_{\mathbf{y}}$ …… (4)

Show that the mode TE00cannot occur in a rectangular guide wave.

Substitute $k=\frac{\omega }{c}\text{and}\frac{\partial E_z}{\partial x}=0$in the equation (2).

role="math" localid="1657513845935" $$\begin{gathered} i\frac{\omega }{C}{E}_x-0=i\omega B_y \\ {E}_x=c{B}_y \end{gathered}$$

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

$$\begin{gathered} \frac{\partial B_z}{\partial y}-ik{B}_y=-i\frac{\omega }{c^2}\left(c{B}_y\right) \\ \frac{\partial B_z}{\partial y}=ik{B}_y-ik{B}_y\mid \\ \frac{\partial B_z}{\partial y}=0 \end{gathered}$$

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

$ik{B}_x-\frac{\partial B_z}{\partial x}=-\frac{i\omega }{c^2}\left(-c{B}_x\right)$

$ik{B}_x-\frac{\partial B_z}{\partial x}=ik{B}_x$

$\frac{\partial B_z}{\partial x}=0$

Hence,

$\frac{\partial B_z}{\partial x}=\frac{\partial B_z}{\partial y}=0$

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 $T{E}_{00}$mode cannot occur in a rectangular waveguide.

Therefore, the role="math" localid="1657514122509" $T{E}_{00}$mode cannot occur in a rectangular waveguide.