---
title: "Scalar vs vector magnetic potential"
subject: "Electromagnetism"
chapter: 3
key: "semester_3/electromagnetism:scalar-vs-vector-magnetic-potential"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/scalar-vs-vector-magnetic-potential
site: BCT Engineering
---

# Scalar vs vector magnetic potential

Electromagnetism · Chapter 3: Magnetic field

- Asked in: 2076 Chaitra, 2 marks; 2075 Asoj, 3 marks; 2070 Chaitra, 3 marks; 2069 Chaitra, 3 marks (asked 4 times)
- Video: [Scalar and Vector Magnetic Potential - Explanation](https://www.youtube.com/watch?v=9QBh2VwqXrs) — SP Engg. ACADEMY
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/scalar-vs-vector-magnetic-potential
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-3

## Question

Differentiate between scalar and vector magnetic potential.

## Short answer

Scalar magnetic potential V_m satisfies H = −∇V_m and applies only in current-free regions. Vector magnetic potential A satisfies B = ∇×A everywhere, governed by ∇²A = −μJ, and has units of Wb/m.

## Answer

Both potentials are defined to find H or B more easily. The **scalar magnetic potential** V_m is defined only where J = 0, with H = −∇V_m. The **vector magnetic potential** A is defined everywhere, with B = ∇×A.

| Point | Scalar magnetic potential V_m | Vector magnetic potential A |
|---|---|---|
| Definition | H = −∇V_m | B = ∇×A |
| Nature | Scalar (amp) | Vector (Wb/m) |
| Region of validity | Only where J = 0 (∇×H = 0) | Everywhere, with or without current |
| Governing equation | ∇²V_m = 0 (since ∇·B = 0) | ∇²A = −μJ |
| Source | Current-free region | Current density J |
| Single-valued? | Not necessarily (multi-valued around a current) | Yes |
| Expression | V_m = −∫H·dl | A = (μ/4π)∫ J dv / R |
| Unit | A (ampere) | Wb/m |

Note that B = ∇×A is valid because ∇·B = 0, and ∇·(∇×A) = 0 always.

## Check yourself

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

1. Which pair of definitions is correct?
   A. B = −∇V_m and H = ∇×A
   B. H = −∇V_m and B = ∇×A
   C. H = ∇×V_m and B = −∇A
   D. B = ∇V_m and H = A
   Answer: B. Scalar potential defines H = −∇V_m; vector potential defines B = ∇×A. (Key point: Definitions)

2. Where is V_m defined?
   A. Everywhere
   B. Only where J = 0
   C. Only inside conductors
   D. Only at infinity
   Answer: B. V_m needs ∇×H = 0, so only current-free regions; A works everywhere. (Key point: Region of validity)

3. Why can B always be written as ∇×A?
   A. ∇·B = 0 and ∇·(∇×A) = 0 always
   B. ∇×B = 0
   C. B is a scalar
   D. A is single-valued only if J = 0
   Answer: A. Divergence of a curl is zero, consistent with ∇·B = 0. (Key point: Why curl A works)

## Related in Electromagnetism

- [Scalar potential, Laplace, magnetic flux](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Ascalar-magnetic-potential-laplace) — Electromagnetism, chapter 3
- [Magnetic flux from vector potential](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Amagnetic-flux-from-vector-potential-a) — Electromagnetism, chapter 3
- [H of infinite filament via vector potential](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Ah-infinite-filament-vector-potential) — Electromagnetism, chapter 3

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