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.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Short Answer

Step by step solution

Determine the electric field in y-direction:

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Geometrical and Physical Optics

Energy Physics

Particle Model of Matter

Further Mechanics and Thermal Physics

Conservation of Energy and Momentum

Rotational Dynamics

Study anywhere. Anytime. Across all devices.