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

# Express W field in cylindrical, spherical

Electromagnetism · Chapter 1: Introduction

- Asked in: 2068 Baisakh, 5 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-w-cylindrical-spherical
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-1

## Question

Express the vector field W = (x − y) a_y in cylindrical and spherical co-ordinates.

## Short answer

W = (x-y) a_y. In cylindrical: W = ρ(cosφ - sinφ)(sinφ a_ρ + cosφ a_φ). In spherical: W = r sinθ (cosφ - sinφ)(sinθ sinφ a_r + cosθ sinφ a_θ + cosφ a_φ).

## Answer

**Given:** W = (x − y) a_y.

Only W_y = x − y is non-zero. Use x = ρ cosφ, y = ρ sinφ (cylindrical) and x = r sinθ cosφ, y = r sinθ sinφ (spherical). The dot products of a_y with the unit vectors are:
a_y · a_ρ = sinφ, a_y · a_φ = cosφ, a_y · a_z = 0; and a_y · a_r = sinθ sinφ, a_y · a_θ = cosθ sinφ.

#### Cylindrical

```text
x − y = ρ (cosφ − sinφ)

W_ρ = (x − y) sinφ = ρ (cosφ − sinφ) sinφ
W_φ = (x − y) cosφ = ρ (cosφ − sinφ) cosφ
W_z = 0
```

**W = ρ(cosφ − sinφ) [ sinφ a_ρ + cosφ a_φ ]**

#### Spherical

```text
x − y = r sinθ (cosφ − sinφ)

W_r = (x − y) sinθ sinφ = r sin²θ (cosφ − sinφ) sinφ
W_θ = (x − y) cosθ sinφ = r sinθ cosθ (cosφ − sinφ) sinφ
W_φ = (x − y) cosφ = r sinθ (cosφ − sinφ) cosφ
```

**W = r sinθ (cosφ − sinφ) [ sinθ sinφ a_r + cosθ sinφ a_θ + cosφ a_φ ]**

**Check (cylindrical):** W_ρ² + W_φ² = ρ²(cosφ − sinφ)² = (x − y)² = |W|² ✓

## Check yourself

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

1. Which dot products are used for W = (x - y) a_y in cylindrical coordinates?
   A. a_y·a_ρ = cosφ, a_y·a_φ = sinφ
   B. a_y·a_ρ = sinφ, a_y·a_φ = cosφ
   C. a_y·a_ρ = 1, a_y·a_φ = 0
   D. a_y·a_ρ = -sinφ, a_y·a_φ = cosφ
   Answer: B. a_y = sinφ a_ρ + cosφ a_φ, so its projections are sinφ and cosφ. (Key point: Dot products with a_y)

2. What is W_ρ in cylindrical coordinates?
   A. ρ(cosφ - sinφ) cosφ
   B. ρ(cosφ - sinφ) sinφ
   C. ρ(cosφ + sinφ) sinφ
   D. (x - y) cosφ
   Answer: B. W_ρ = W_y sinφ = (x - y) sinφ, and x - y = ρ(cosφ - sinφ). (Key point: Cylindrical form)

3. How is x - y written in spherical coordinates?
   A. r(cosφ - sinφ)
   B. r sinθ (cosφ - sinφ)
   C. r cosθ (cosφ - sinφ)
   D. r sinθ (cosφ + sinφ)
   Answer: B. x = r sinθ cosφ and y = r sinθ sinφ, so x - y = r sinθ (cosφ - sinφ). (Key point: Spherical x - y)

4. Which is W_θ?
   A. r sinθ cosθ (cosφ - sinφ) sinφ
   B. r sin²θ (cosφ - sinφ) sinφ
   C. r sinθ (cosφ - sinφ) cosφ
   D. r cosθ (cosφ - sinφ)
   Answer: A. W_θ = (x - y)(a_y·a_θ) = r sinθ (cosφ - sinφ) cosθ sinφ. (Key point: Spherical W_theta)

## Related in Electromagnetism

- [Express a_x field in cylindrical, spherical](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aexpress-ax-cylindrical-spherical) — Electromagnetism, chapter 1
- [Cylindrical vector to spherical](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Acylindrical-vector-to-spherical-rho-sinphi) — Electromagnetism, chapter 1
- [Vector field to cylindrical at point](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Avector-field-transform-cylindrical) — Electromagnetism, chapter 1
- [Transform vector to spherical coordinates](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Atransform-vector-spherical-point) — Electromagnetism, chapter 1, asked 2×
- [Transform vector to cylindrical coordinates](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Atransform-vector-cylindrical-yxz) — Electromagnetism, chapter 1, asked 3×
- [Vector between two points in cylindrical](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Avector-a-to-b-cylindrical) — Electromagnetism, chapter 1

## Related in other subjects

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- [Cartesian, cylindrical, spherical conversion](https://www.bctengineer.com/question.md?key=semester_4%2Fcomputer_aided_design%3Acoordinate-systems-conversion) — Computer Aided Design, chapter 2