---
title: "Magnetic flux from vector potential"
subject: "Electromagnetism"
chapter: 3
key: "semester_3/electromagnetism:magnetic-flux-from-vector-potential-a"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/magnetic-flux-from-vector-potential-a
site: BCT Engineering
---

# Magnetic flux from vector potential

Electromagnetism · Chapter 3: Magnetic field

- Asked in: 2072 Chaitra, 5 marks
- Video: [EMF - Unit III - Problem on Vector Potential - Magnetic Flux](https://www.youtube.com/watch?v=yTnVbLwNltU) — Padmasri Naban
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/magnetic-flux-from-vector-potential-a
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-3

## Question

Given the vector magnetic potential A = −(ρ²/4) a_z Wb/m, calculate the total magnetic flux crossing the surface φ = π/2, 1 ≤ ρ ≤ 2 m and 0 ≤ z ≤ 5 m.

## Short answer

B = ∇×A = (ρ/2) a_φ Wb/m². The surface at φ=π/2 has dS = dρ dz a_φ. Flux Ψ = ∫₀⁵∫₁² (ρ/2) dρ dz = 3.75 Wb.

## Answer

Magnetic flux density follows from the vector potential as B = ∇×A, and the flux through a surface is Ψ = ∫ B·dS.

**Step 1: find B.** A = A_z a_z with A_z = −ρ²/4, a function of ρ only. In cylindrical coordinates:

```text
B = ∇×A = (1/ρ ∂A_z/∂φ) a_ρ − (∂A_z/∂ρ) a_φ
        = −∂/∂ρ (−ρ²/4) a_φ
        = (ρ/2) a_φ   Wb/m²
```

**Step 2: surface.** The surface φ = π/2 is a plane (the yz-plane, y > 0) with 1 ≤ ρ ≤ 2 and 0 ≤ z ≤ 5. Its normal is a_φ and dS = dρ dz a_φ.

**Step 3: integrate.**

```text
Ψ = ∫₀⁵ ∫₁² (ρ/2) a_φ · (dρ dz a_φ)
  = ∫₀⁵ dz × ∫₁² (ρ/2) dρ
  = 5 × [ρ²/4]₁²
  = 5 × (4 − 1)/4
  = 5 × 0.75
  = 3.75 Wb
```

**Answer: Ψ = 3.75 Wb** (in the +a_φ direction).

## Check yourself

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

1. How is B obtained from the vector potential A?
   A. B = grad A
   B. B = div A
   C. B = curl A
   D. B = -dA/dt
   Answer: C. The vector magnetic potential is defined by B = curl A. (Key point: B from A)

2. For A = -(rho^2/4) a_z, what is B?
   A. -(rho/2) a_phi
   B. (rho/2) a_phi
   C. (rho^2/4) a_rho
   D. (rho/2) a_z
   Answer: B. B = -(dA_z/d rho) a_phi = -d/d rho(-rho^2/4) a_phi = (rho/2) a_phi. (Key point: Evaluating the curl)

3. On the surface phi = pi/2, what is dS?
   A. rho d rho d phi a_z
   B. rho dphi dz a_rho
   C. d rho dz a_phi
   D. rho d rho dz a_phi
   Answer: C. The surface is a plane of constant phi spanned by rho and z, so dS = d rho dz a_phi (no extra rho factor). (Key point: Surface element)

4. What is the total flux?
   A. 7.5 Wb
   B. 3.75 Wb
   C. 1.875 Wb
   D. 15 Wb
   Answer: B. Flux = 5 x integral from 1 to 2 of (rho/2) d rho = 5 x 0.75 = 3.75 Wb. (Key point: Flux result)

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