---
title: "State and explain Gauss's law"
subject: "Electromagnetism"
chapter: 2
key: "semester_3/electromagnetism:gauss-law-statement-explain"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/gauss-law-statement-explain
site: BCT Engineering
---

# State and explain Gauss's law

Electromagnetism · Chapter 2: Electric field

- Asked in: 2067 Shrawan, 2 marks
- Video: [Gauss's Law for Electric Field Explained: Detailed Case Study](https://www.youtube.com/watch?v=qerz-Z-OQiI) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/gauss-law-statement-explain
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-2

## Question

State and explain Gauss's law.

## Short answer

Gauss's law: ∮ D·dS = Q_enc; total electric flux through a closed surface equals enclosed free charge. It simplifies field calculation for symmetric distributions; differential form is ∇·D = ρ_v. Example: for a point charge, D = Q/(4πr²) a_r.

## Answer

**Statement.** The total electric flux passing through any closed surface equals the total free charge enclosed by that surface.

```text
Ψ = ∮ D·dS = Q_enclosed
```
Here D = εE is the electric flux density (C/m²).

#### Explanation
- The closed surface is called a Gaussian surface; it can be any shape, real or imaginary.
- Flux out of the surface counts as positive; charge outside the surface contributes no net flux, because every line that enters also leaves.
- If the enclosed charge is a volume distribution, Q = ∫ρ_v dv, giving ∮D·dS = ∫ρ_v dv.
- Applying the divergence theorem gives the point (differential) form, **∇·D = ρ_v**, the first Maxwell equation for statics.

#### Use
For symmetric charge distributions (point, line, sheet, sphere, cylinder), choose a Gaussian surface on which D is constant and normal (or tangential), so D comes out of the integral. Example for a point charge Q: take a sphere of radius r, so D(4πr²) = Q, giving D = Q/(4πr²) a_r and E = Q/(4πε₀r²) a_r (Coulomb's law).

## Check yourself

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

1. Gauss's law states that the total flux through a closed surface equals:
   A. The total charge outside the surface
   B. The free charge enclosed
   C. The charge on the surface only
   D. Zero always
   Answer: B. ∮D·dS = Q_enclosed. (Key point: Statement)

2. What is the net flux contribution of charge outside the surface?
   A. Positive
   B. Negative
   C. Zero, since every line entering also leaves
   D. Equal to the inside charge
   Answer: C. Flux lines from outside charges enter and leave, so net flux is zero. (Key point: Outside charge)

3. Applying the divergence theorem to Gauss's law gives:
   A. ∇×D = ρ_v
   B. ∇·D = ρ_v
   C. ∇·E = 0
   D. ∇²D = ρ_v
   Answer: B. ∮D·dS = ∫ρ_v dv together with the divergence theorem gives ∇·D = ρ_v. (Key point: Differential form)

## Related in Electromagnetism

- [E due to line charge using Gauss's law](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aline-charge-field-gauss-law) — Electromagnetism, chapter 2, asked 3×