---
title: "Single-mode range, air-filled 2×1 cm"
subject: "Electromagnetism"
chapter: 6
key: "semester_3/electromagnetism:waveguide-2x1cm-single-mode-range"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/waveguide-2x1cm-single-mode-range
site: BCT Engineering
---

# Single-mode range, air-filled 2×1 cm

Electromagnetism · Chapter 6: Wave guides

- Asked in: 2076 Asoj, 3 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/waveguide-2x1cm-single-mode-range
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-6

## Question

An air-filled rectangular waveguide has dimensions a = 2 cm and b = 1 cm. Determine the range of frequencies over which the guide will operate in single mode (TE10).

## Short answer

TE_10 cut-off: f_c = c/(2a) = 3×10⁸/(2×0.02) = 7.5 GHz. Next higher modes (TE_20, TE_01) start at 15 GHz. Single-mode operation range: 7.5 GHz < f < 15 GHz.

## Answer

Given: a = 2 cm, b = 1 cm, air-filled.

```text
f_c(TE₁₀) = c/(2a) = 3×10⁸ / (2 × 0.02) = 7.5 GHz
f_c(TE₂₀) = c/a    = 15 GHz
f_c(TE₀₁) = c/(2b) = 3×10⁸ / (2 × 0.01) = 15 GHz
f_c(TE₁₁, TM₁₁) = (c/2)√[(1/0.02)² + (1/0.01)²] = 1.5×10⁸ × √12500 = 16.77 GHz
```

The next higher modes after TE₁₀ start at 15 GHz (TE₂₀ and TE₀₁, equal because a = 2b).

Only the dominant TE₁₀ mode propagates for

**7.5 GHz < f < 15 GHz**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the cut-off of TE10 for a = 2 cm, b = 1 cm (air)?
   A. 15 GHz
   B. 7.5 GHz
   C. 3.75 GHz
   D. 16.77 GHz
   Answer: B. c/(2a) = 3×10^8/0.04 = 7.5 GHz. (Key point: TE10 cut-off)

2. Why do TE20 and TE01 have the same cut-off here?
   A. Both are TM modes
   B. Both have m = n
   C. Because a = 2b, so c/a = c/(2b) = 15 GHz
   D. Because the guide is square
   Answer: C. f_c(20) = c/a and f_c(01) = c/(2b); with a = 2b these are both 15 GHz. (Key point: Next mode cut-off)

3. What is the single-mode (TE10 only) frequency range?
   A. 7.5 GHz < f < 16.77 GHz
   B. 15 GHz < f < 16.77 GHz
   C. 0 < f < 7.5 GHz
   D. 7.5 GHz < f < 15 GHz
   Answer: D. Operation must be above 7.5 GHz and below the next cut-off, 15 GHz. (Key point: Single-mode band)

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