---
title: "Transform vector B to cylindrical"
subject: "Electromagnetism"
chapter: 1
key: "semester_3/electromagnetism:transform-vector-b-cylindrical"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/transform-vector-b-cylindrical
site: BCT Engineering
---

# Transform vector B to cylindrical

Electromagnetism · Chapter 1: Introduction

- Asked in: 2067 Shrawan, 4 marks
- Video: [EMT/ VECTOR ALGEBRA LEC-13/Conversion of  Any vector into CYLINDRICAL COORDINATE SYSTEM](https://www.youtube.com/watch?v=QVucuAy41mw) — CHITRA EDUCATION
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/transform-vector-b-cylindrical
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-1

## Question

Transform the vector B = y a_x − x a_y + z a_z into cylindrical co-ordinates.

## Short answer

B = y a_x - x a_y + z a_z. In cylindrical, B_ρ = 0, B_φ = -ρ, B_z = z. So B = -ρ a_φ + z a_z.

## Answer

**Given:** B = y a_x − x a_y + z a_z.

#### Transformation formulas

```text
B_ρ = B_x cosφ + B_y sinφ
B_φ = −B_x sinφ + B_y cosφ
B_z = B_z
```

Here B_x = y, B_y = −x, B_z = z. Substitute x = ρ cosφ and y = ρ sinφ:

```text
B_ρ = y cosφ − x sinφ = ρ sinφ cosφ − ρ cosφ sinφ = 0
B_φ = −y sinφ − x cosφ = −ρ sin²φ − ρ cos²φ = −ρ
B_z = z
```

**Answer: B = −ρ a_φ + z a_z**

**Check:** |B|² in Cartesian = y² + x² + z² = ρ² + z², and in cylindrical = ρ² + z² ✓. The field circulates around the z-axis in the −φ (clockwise) sense, with magnitude proportional to ρ.

## Check yourself

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

1. What are B_x, B_y, B_z for B = y a_x - x a_y + z a_z, and what is substituted?
   A. B_x = y, B_y = -x, B_z = z; x = ρ cosφ, y = ρ sinφ
   B. B_x = x, B_y = y; x = ρ
   C. B_x = -y, B_y = x
   D. B_x = y, B_y = x
   Answer: A. The Cartesian components are read directly, then x and y are replaced by ρ cosφ and ρ sinφ. (Key point: Substitution)

2. What is B_ρ?
   A. ρ
   B. 0
   C. -ρ
   D. z
   Answer: B. B_ρ = y cosφ - x sinφ = ρ sinφ cosφ - ρ cosφ sinφ = 0. (Key point: B_rho)

3. What is B in cylindrical coordinates?
   A. ρ a_φ + z a_z
   B. -ρ a_φ + z a_z
   C. -ρ a_ρ + z a_z
   D. ρ a_ρ - z a_z
   Answer: B. B_φ = -y sinφ - x cosφ = -ρ, so B = -ρ a_φ + z a_z, a clockwise circulation about the z-axis. (Key point: Final result)

## Close variants set in other exams

- [Transform vector to cylindrical coordinates](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Atransform-vector-cylindrical-yxz) — Electromagnetism, chapter 1, asked 3×

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