---
title: "Define TM mode; TM11 cut-off 5×3 cm"
subject: "Electromagnetism"
chapter: 6
key: "semester_3/electromagnetism:tm-mode-define-cutoff-5x3-guide"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/tm-mode-define-cutoff-5x3-guide
site: BCT Engineering
---

# Define TM mode; TM11 cut-off 5×3 cm

Electromagnetism · Chapter 6: Wave guides

- Asked in: 2067 Mangsir, 6 marks
- Video: [Fields for T.E.  and T.M.  Mode | Field Visualization](https://www.youtube.com/watch?v=ivYNjkraG8c) — RTU Wallah
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/tm-mode-define-cutoff-5x3-guide
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-6

## Question

Define transverse magnetic mode. A rectangular waveguide has dimensions a = 5 cm and b = 3 cm. The medium within the waveguide has ε_r = 1, μ_r = 1, σ = 0 and conducting walls of the waveguide are perfect conductors. Determine the cutoff frequency for TM11 mode.

## Short answer

Transverse magnetic (TM) mode has no longitudinal magnetic field (H_z=0). Using f_c=(c/2)√((m/a)²+(n/b)²) with m=n=1, a=5 cm, b=3 cm gives f_c≈5.83 GHz.

## Answer

#### Transverse magnetic (TM) mode

A **TM mode** is a wave mode in which the magnetic field has no component along the direction of propagation (H_z = 0), while the electric field has a longitudinal component E_z ≠ 0. In a rectangular guide

E_z = E₀ sin(mπx/a) sin(nπy/b) e^{−jβz}

with m ≥ 1 and n ≥ 1 (if either is zero all the field components vanish). The lowest TM mode is TM₁₁.

#### Cut-off of TM₁₁ (a = 5 cm, b = 3 cm, air-filled, perfect walls)

```text
f_c = (c/2) √[(m/a)² + (n/b)²]

(1/a)² = (1/0.05)² = 400
(1/b)² = (1/0.03)² = 1111.1
sum    = 1511.1,   √ = 38.87 m⁻¹

f_c(TM₁₁) = 1.5×10⁸ × 38.87 = 5.83×10⁹ Hz
```

**Answer: f_c(TM₁₁) = 5.83 GHz** (λ_c = 5.14 cm). For comparison the dominant TE₁₀ has f_c = 3 GHz.

## Check yourself

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

1. Which describes a TM mode?
   A. E_z = 0, H_z non-zero
   B. H_z = 0, E_z non-zero
   C. Both E_z and H_z zero
   D. H_z and E_z both non-zero
   Answer: B. Transverse magnetic means the magnetic field has no component along propagation. (Key point: TM definition)

2. What are the allowed indices of TM modes and the lowest TM mode?
   A. m, n ≥ 0; TM10
   B. m, n ≥ 1; TM11
   C. m ≥ 1, n ≥ 0; TM10
   D. m, n ≥ 2; TM22
   Answer: B. If either index is zero all field components vanish, so the lowest TM mode is TM11. (Key point: TM indices)

3. What is f_c(TM11) for a = 5 cm, b = 3 cm?
   A. 3 GHz
   B. 5.83 GHz
   C. 8.75 GHz
   D. 3.89 GHz
   Answer: B. √(400 + 1111.1) = 38.87; × 1.5×10^8 = 5.83 GHz. (Key point: TM11 cut-off value)

4. What is the correct way to combine the dimensions?
   A. Add m/a and n/b, then multiply by c/2
   B. Add (m/a)^2 and (n/b)^2, take the root, multiply by c/2
   C. Multiply a by b
   D. Use only the larger side
   Answer: B. f_c = (c/2)√[(m/a)^2 + (n/b)^2]. (Key point: Method)

## Related in Electromagnetism

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- [Cut-off and single-mode range, 4×2 cm](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Awaveguide-4x2cm-cutoff-single-mode-range) — Electromagnetism, chapter 6
- [Define cutoff frequency, dominant mode](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Acutoff-frequency-dominant-mode-define) — Electromagnetism, chapter 6
- [Cut-off of dominant and TE11, 2.5×1.2 cm](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Awaveguide-2-5x1-2-cutoff-te11) — Electromagnetism, chapter 6
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- [Single-mode range, air-filled 2×1 cm](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Awaveguide-2x1cm-single-mode-range) — Electromagnetism, chapter 6

## Related in other subjects

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- [Rectangular waveguide TM mode field equations](https://www.bctengineer.com/question.md?key=semester_7%2Frf_and_microwave_engineering%3Arectangular-waveguide-tm-equations) — RF and Microwave Engineering, chapter 4, asked 11×
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