---
title: "Cylindrical vector to spherical"
subject: "Electromagnetism"
chapter: 1
key: "semester_3/electromagnetism:cylindrical-vector-to-spherical-rho-sinphi"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/cylindrical-vector-to-spherical-rho-sinphi
site: BCT Engineering
---

# Cylindrical vector to spherical

Electromagnetism · Chapter 1: Introduction

- Asked in: 2068 Chaitra, 5 marks
- Video: [Vector conversion || Rectangular || Cylindrical || Spherical || by Prof. Niraj Kumar VIT Chennai](https://www.youtube.com/watch?v=I3L22NosQg0) — RF Design Basics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/cylindrical-vector-to-spherical-rho-sinphi
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-1

## Question

Transform vector A = ρ sinφ a_z at point (1, 45°, 2) in cylindrical co-ordinate system to a vector in spherical co-ordinate system.

## Short answer

The vector A = ρ sinφ a_z at (1, 45°, 2) has only a z-component of 0.7071. Convert to spherical: A_r = A_z cosθ = 0.6325, A_θ = -A_z sinθ = -0.3162, A_φ = 0. So A = 0.632 a_r - 0.316 a_θ.

## Answer

**Given:** A = ρ sinφ a_z at the point (ρ = 1, φ = 45°, z = 2). Find the spherical components.

#### Step 1: Location in spherical coordinates

```text
r = √(ρ² + z²) = √(1 + 4) = √5 = 2.236
θ = tan⁻¹(ρ/z) = tan⁻¹(1/2) = 26.57°
φ = 45° (same in both systems)
sinθ = ρ/r = 0.4472,  cosθ = z/r = 0.8944
```

#### Step 2: Value of A at the point

Only A_z exists: A_z = ρ sinφ = 1 × sin45° = 0.7071 (A_ρ = A_φ = 0).

#### Step 3: Cylindrical to spherical

```text
A_r = A_ρ sinθ + A_z cosθ = 0 + 0.7071(0.8944) = 0.6325
A_θ = A_ρ cosθ − A_z sinθ = 0 − 0.7071(0.4472) = −0.3162
A_φ = A_φ = 0
```

**Check:** √(0.6325² + 0.3162²) = √(0.4 + 0.1) = 0.7071 ✓

**Answer: A = 0.632 a_r − 0.316 a_θ** at (1, 45°, 2) (A_φ = 0).

(General form: A = ρ sinφ cosθ a_r − ρ sinφ sinθ a_θ, with ρ = r sinθ.)

## Check yourself

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

1. For the point (rho = 1, phi = 45°, z = 2), what are r and theta?
   A. r = 2.236, theta = 26.57°
   B. r = 2.236, theta = 63.43°
   C. r = 3, theta = 26.57°
   D. r = 1, theta = 45°
   Answer: A. r = sqrt(1+4) = 2.236 and theta = tan^-1(rho/z) = tan^-1(1/2) = 26.57°. (Key point: Spherical location)

2. What is the field at the point, in cylindrical components?
   A. A_ρ = 0.7071, A_z = 0
   B. A_ρ = 0, A_φ = 0, A_z = 0.7071
   C. A_φ = 0.7071 only
   D. A_z = 1
   Answer: B. A has only a z-component A_z = rho sinφ = sin45° = 0.7071. (Key point: Field at the point)

3. Which formulas convert cylindrical to spherical components?
   A. A_r = A_ρ cosθ - A_z sinθ
   B. A_r = A_ρ sinθ + A_z cosθ and A_θ = A_ρ cosθ - A_z sinθ
   C. A_r = A_z sinθ and A_θ = A_ρ
   D. A_r = A_ρ + A_z
   Answer: B. The projections give A_r = A_ρ sinθ + A_z cosθ and A_θ = A_ρ cosθ - A_z sinθ, with A_φ unchanged. (Key point: Transformation formulas)

4. What is A in spherical components?
   A. 0.316 a_r + 0.632 a_θ
   B. 0.632 a_r - 0.316 a_θ
   C. -0.632 a_r + 0.316 a_θ
   D. 0.632 a_r + 0.316 a_φ
   Answer: B. A_r = 0.7071(0.8944) = 0.632 and A_θ = -0.7071(0.4472) = -0.316, with A_φ = 0. (Key point: Result)

## Related in Electromagnetism

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- [Transform vector to spherical coordinates](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Atransform-vector-spherical-point) — Electromagnetism, chapter 1, asked 2×
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- [Vector field to cylindrical at point](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Avector-field-transform-cylindrical) — Electromagnetism, chapter 1
- [Convert vector to spherical system](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aconvert-vector-spherical-fxyz) — Electromagnetism, chapter 1
- [Cartesian vector to cylindrical](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Acartesian-vector-to-cylindrical-xxy) — Electromagnetism, chapter 1

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