---
title: "Express a_x field in cylindrical, spherical"
subject: "Electromagnetism"
chapter: 1
key: "semester_3/electromagnetism:express-ax-cylindrical-spherical"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/express-ax-cylindrical-spherical
site: BCT Engineering
---

# Express a_x field in cylindrical, spherical

Electromagnetism · Chapter 1: Introduction

- Asked in: 2072 Chaitra, 2+3 marks
- Video: [16. Numerical Problem on Transformation of Vectors in Coordinate Systems in Electromagnetic Theory](https://www.youtube.com/watch?v=zPYXGpPLMtM) — Electrical Tutorial
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/express-ax-cylindrical-spherical
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-1

## Question

Express the uniform vector field F = 5a_x in (a) cylindrical components (b) spherical components.

## Short answer

In cylindrical: F=5 cosφ a_ρ −5 sinφ a_φ. In spherical: F=5 sinθ cosφ a_r +5 cosθ cosφ a_θ −5 sinφ a_φ.

## Answer

**Given:** F = 5 a_x (uniform field).

#### (a) Cylindrical components

Use the projections of **F** on the cylindrical unit vectors.

```text
F_ρ = F · a_ρ = 5 a_x · a_ρ = 5 cosφ
F_φ = F · a_φ = 5 a_x · a_φ = −5 sinφ
F_z = F · a_z = 0
```

**F = 5 cosφ a_ρ − 5 sinφ a_φ**

#### (b) Spherical components

```text
F_r = F · a_r = 5 sinθ cosφ
F_θ = F · a_θ = 5 cosθ cosφ
F_φ = F · a_φ = −5 sinφ
```

**F = 5 sinθ cosφ a_r + 5 cosθ cosφ a_θ − 5 sinφ a_φ**

(The dot products used: a_x · a_ρ = cosφ, a_x · a_φ = −sinφ; a_x · a_r = sinθ cosφ, a_x · a_θ = cosθ cosφ.)

**Check:** the magnitude must stay 5. Cylindrical: 25cos²φ + 25sin²φ = 25 ✓. Spherical: 25[sin²θ cos²φ + cos²θ cos²φ + sin²φ] = 25[cos²φ + sin²φ] = 25 ✓.

## Check yourself

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

1. What is 5 a_x in cylindrical components?
   A. 5 sinφ a_ρ + 5 cosφ a_φ
   B. 5 cosφ a_ρ - 5 sinφ a_φ
   C. 5 a_ρ
   D. 5 cosφ a_ρ + 5 sinφ a_φ
   Answer: B. F_ρ = F·a_ρ = 5cosφ and F_φ = F·a_φ = -5sinφ, with F_z = 0. (Key point: Cylindrical components)

2. Which is F_θ for F = 5 a_x?
   A. 5 sinθ cosφ
   B. 5 cosθ cosφ
   C. -5 sinφ
   D. 5 cosθ sinφ
   Answer: B. a_x · a_θ = cosθ cosφ, so F_θ = 5 cosθ cosφ. (Key point: Spherical components)

3. Why do the components depend on angles even though the field is uniform?
   A. The field magnitude changes
   B. The unit vectors a_ρ, a_φ, a_r, a_θ change direction with position
   C. The field is not constant
   D. Cartesian components are not constant
   Answer: B. The field is uniform, but curvilinear unit vectors change direction from point to point, so its projections on them vary. The magnitude stays 5. (Key point: Why components vary)

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