---
title: "Divergence and physical significance"
subject: "Electromagnetism"
chapter: 1
key: "semester_3/electromagnetism:divergence-physical-significance-electric"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/divergence-physical-significance-electric
site: BCT Engineering
---

# Divergence and physical significance

Electromagnetism · Chapter 1: Introduction

- Asked in: 2067 Shrawan, 3 marks
- Video: [Divergence and curl:  The language of Maxwell's equations, fluid flow, and more](https://www.youtube.com/watch?v=rB83DpBJQsE) — 3Blue1Brown
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/divergence-physical-significance-electric
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-1

## Question

Define divergence and write down its physical significance as it applies to electric fields.

## Short answer

Divergence of D equals volume charge density ρ_v. Positive divergence indicates a source (positive charge), negative a sink (negative charge). Zero divergence means no net charge, field is solenoidal. It's the point form of Gauss's law.

## Answer

**Definition:** The divergence of a vector field **A** at a point is the net outward flux of **A** per unit volume as the volume shrinks to zero around that point.

```text
∇ · A = lim (Δv → 0)  [ ∮_S A · dS ] / Δv
```

In Cartesian coordinates: ∇ · A = ∂A_x/∂x + ∂A_y/∂y + ∂A_z/∂z (a scalar).

#### Physical significance for electric fields

For the electric flux density **D**, Gauss's law in point form is

```text
∇ · D = ρ_v
```

- Divergence of D at a point equals the **volume charge density** there. Positive divergence means a **source** (positive charge): flux lines start there and spread outward. Negative divergence means a **sink** (negative charge): flux lines end there.
- ∇ · D = 0 means no charge is present at the point: all the flux entering a tiny volume also leaves it. Such a field is called **solenoidal**.
- It converts the integral Gauss law, ∮ D · dS = Q, into a local (point) law, using the divergence theorem ∮ D · dS = ∫ (∇ · D) dv.
- For a point charge, ∇ · D = 0 everywhere except at the charge itself, where it is infinite (a delta function).

Similarly ∇ · B = 0 expresses that magnetic flux lines have no sources (no magnetic monopoles).

## Check yourself

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

1. How is divergence of A defined?
   A. Net circulation per unit area
   B. Net outward flux of A per unit volume as the volume shrinks to zero
   C. Gradient of A
   D. Rate of change of A in time
   Answer: B. ∇·A = lim (Δv→0) (∮A·dS)/Δv. (Key point: Definition)

2. What does ∇·D = ρ_v mean?
   A. Divergence of D at a point equals the volume charge density there
   B. D is zero outside charges
   C. Total charge is zero
   D. D circulates around charge
   Answer: A. The divergence of D at a point equals the local volume charge density, so positive charge is a source and negative charge a sink. (Key point: Point form of Gauss's law)

3. What does ∇·D = 0 at a point imply?
   A. Infinite charge
   B. No charge there; flux entering a tiny volume also leaves, and the field is solenoidal
   C. A negative charge
   D. A strong field
   Answer: B. Zero divergence means no charge at that point, so the field there is solenoidal. (Key point: Zero divergence)

## Related in Electromagnetism

- [Curl and its physical significance](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Acurl-physical-interpretation) — Electromagnetism, chapter 1, asked 2×
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- [Dielectric boundary at y=0, find fields](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adielectric-boundary-y-d1-find-region2) — Electromagnetism, chapter 2

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