Chapter 9: Q28P (page 430)

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

Short Answer

Expert verified

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.

$\frac{\partial E_{z}}{\partial y} - i K E_{y} = i ω B_{x}$ …… (1)

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

localid="1657514438209" $\frac{\partial (0)}{\partial y} - i \frac{ω}{c} E_{y} = i ω B_{x} E_{y} = - c B_{x}$

Second Maxwell’s equation:

Write the expression for Maxwell’s equation

$i k E_{x} - \frac{\partial E_{z}}{\partial x} = i ω B_{y}$ …… (2)

Third Maxwell’s equation:

Write the expression for Maxwell’s equation.

$\frac{\partial B_{z}}{\partial y} - i k B_{y} = - i \frac{ω}{c^{2}} E_{x}$ …… (3)

Fourth Maxwell’s equation

Write the expression for Maxwell’s equation.

i k B_{x} - \frac{\partial B_{z}}{\partial x} = - \frac{i ω}{c^{2}} E_{y}

…… (4)

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

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

role="math" localid="1657513845935" $i \frac{ω}{C} E_{x} - 0 = i ω B_{y} E_{x} = c B_{y}$

Substitute $E_{x} = c B_{y}$ in the equation (3).

$\frac{\partial B_{z}}{\partial y} - i k B_{y} = - i \frac{ω}{c^{2}} (c B_{y}) \frac{\partial B_{z}}{\partial y} = i k B_{y} - i k B_{y} ∣ \frac{\partial B_{z}}{\partial y} = 0$

Substitute $E_{y} = c B_{x}$ in the equation (4).

$i k B_{x} - \frac{\partial B_{z}}{\partial x} = - \frac{i ω}{c^{2}} (- c B_{x})$

$i k B_{x} - \frac{\partial B_{z}}{\partial x} = i k B_{x}$

$\frac{\partial B_{z}}{\partial x} = 0$

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.

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Short Answer

Step by step solution

Determine the electric field in y-direction:

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

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