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

# Scalar potential, Laplace, magnetic flux

Electromagnetism · Chapter 3: Magnetic field

- Asked in: 2072 Chaitra, 1+2 marks
- Video: [EMF50 Scalar Magnetic Potential](https://www.youtube.com/watch?v=6eJMnBfQXGc) — Lectures in Electrical Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/scalar-magnetic-potential-laplace
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-3

## Question

Define scalar magnetic potential and show that it satisfies the Laplace's equation.

## Short answer

In a current-free region, H = -∇V_m defines scalar magnetic potential V_m. From ∇·B=0 and B=μH, ∇·H=0, so substitution gives ∇²V_m=0, satisfying Laplace's equation.

## Answer

#### Scalar magnetic potential
In a region where there is no current (J = 0), Ampere's law gives ∇×H = 0. Since the curl of the gradient of any scalar is zero, H can be written as the negative gradient of a scalar function V_m:

**H = −∇V_m**

V_m is the scalar magnetic potential. It is the work done per unit pole strength in moving a magnetic pole between two points: V_m(ab) = −∫ H·dl. Its unit is the ampere (A). It is valid only in current-free regions, and is multi-valued if the path encircles a current.

#### Proof that ∇²V_m = 0
From the magnetic Gauss law, ∇·B = 0. In a linear homogeneous medium B = μH:

```text
∇·B = μ ∇·H = 0  ⇒  ∇·H = 0
```

Substitute H = −∇V_m:

```text
∇·(−∇V_m) = 0
⇒ ∇²V_m = 0
```

So the scalar magnetic potential satisfies Laplace's equation, exactly like the electrostatic potential in a charge-free region. The magnetic flux through a surface can then be found from B = −μ∇V_m, using the boundary conditions to solve for V_m.

## Check yourself

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

1. Why can H be written as -grad V_m in a current-free region?
   A. Because div H = 0
   B. Because curl H = 0 and the curl of a gradient is zero
   C. Because B = mu H
   D. Because the region has no magnetic material
   Answer: B. With J = 0, curl H = 0, and since curl(grad V) = 0 always, H can be the negative gradient of a scalar. (Key point: Why H = -grad V_m is allowed)

2. Which law supplies the condition that leads to Laplace's equation for V_m?
   A. Div B = 0 (magnetic Gauss law)
   B. Faraday's law
   C. Coulomb's law
   D. Div D = rho_v
   Answer: A. div B = 0 with B = mu H gives div H = 0; substituting H = -grad V_m gives div grad V_m = 0. (Key point: Which law gives Laplace)

3. Which statement about scalar magnetic potential is correct?
   A. It is valid everywhere including inside conductors
   B. It is single-valued around a current
   C. Its unit is the tesla
   D. It is valid only where J = 0 and can be multi-valued around a current
   Answer: D. V_m (unit ampere) exists only in current-free regions and is multi-valued if a path encircles a current. (Key point: Limitations of V_m)

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