Question

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

Short answer

It is proved that the ${\text{TE}}_{00}$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.

$\frac{\partial E_z}{\partial y}-iK{E}_y=i\omega B_x$ …… (1)

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

$\begin{array}{l}\frac{\partial \left(0\right)}{\partial y}-i\frac{\omega }{c}{E}_y=i\omega B_x\\ E_y=-c{B}_x\end{array}$

• Second Maxwell’s equation:

Write the expression for Maxwell’s equation.

$ik{E}_x-\frac{\partial E_z}{\partial x}=i\omega B_y$ …… (2)

• Third Maxwell’s equation:

Write the expression for Maxwell’s equation.

$\frac{\partial B_z}{\partial y}-ik{B}_y=-i\frac{\omega }{c^2}{E}_x$ …… (3)

• Fourth Maxwell’s equation:

Write the expression for Maxwell’s equation.

$ik{B}_x-\frac{\partial B_z}{\partial x}=-\frac{i\omega }{c^2}{E}_y$ …… (4)

Show that the mode cannot occur in a rectangular guide wave.

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

$\begin{array}{c}i\frac{\omega }{c}{E}_x-0=i\omega B_y\\ E_x=c{B}_y\end{array}$

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

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

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

$\begin{array}{c}ik{B}_x-\frac{\partial B_z}{\partial x}=-\frac{i\omega }{c^2}(-c{B}_x)\\ ik{B}_x-\frac{\partial B_z}{\partial x}=ik{B}_x\\ \frac{\partial B_z}{\partial x}=0\end{array}$

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

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