---
title: "Show curl of E is zero"
subject: "Electromagnetism"
chapter: 2
key: "semester_3/electromagnetism:curl-e-zero-static-field"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/curl-e-zero-static-field
site: BCT Engineering
---

# Show curl of E is zero

Electromagnetism · Chapter 2: Electric field

- Asked in: 2073 Shrawan, 2 marks
- Video: [2.2.4 The Curl of E](https://www.youtube.com/watch?v=F2fn9EkEkcU) — Real Physics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/curl-e-zero-static-field
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-2

## Question

Show that ∇×E = 0 for the static electric field.

## Short answer

For a point charge, E is radial and depends only on r, so its curl in spherical coordinates is zero. By superposition, any static charge distribution yields ∇×E=0. Equivalently, ∮E·dl=0 around any closed path, and Stokes' theorem gives ∇×E=0 everywhere.

## Answer

**Statement.** For a static electric field the curl is zero, ∇×E = 0, so the field is conservative.

#### Proof using the point charge field
A static field is the superposition of fields of point charges, so it is enough to show it for one charge Q at the origin:

```text
E = Q/(4πε₀r²) a_r      (E_r depends on r only; E_θ = E_φ = 0)
```
In spherical coordinates:

```text
∇×E = (1/(r sinθ))[∂(sinθ E_φ)/∂θ − ∂E_θ/∂φ] a_r
    + (1/r)[(1/sinθ) ∂E_r/∂φ − ∂(rE_φ)/∂r] a_θ
    + (1/r)[∂(rE_θ)/∂r − ∂E_r/∂θ] a_φ
```
E_r has no θ or φ dependence and E_θ = E_φ = 0, so every term vanishes: ∇×E = 0. By superposition it holds for any static charge distribution.

#### Integral form (Stokes' theorem)
The work done in moving a charge from A to B is −Q∫E·dl = Q(V_B − V_A), which depends only on the end points. Around a closed path A back to A, the work is zero:

```text
∮ E·dl = 0
```
By Stokes' theorem, ∮E·dl = ∫(∇×E)·dS = 0 for every surface, so ∇×E = 0. This is Maxwell's equation for electrostatics, and it allows E = −∇V.

## Check yourself

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

1. Why is it enough to prove ∇×E = 0 for one point charge?
   A. Any static field is a superposition of point-charge fields
   B. Point charges are the only fields
   C. Curl is not linear
   D. Because E is constant
   Answer: A. Superposition extends the result to any static distribution. (Key point: Reduction)

2. For E = Q/(4πε₀r²) a_r, why does the curl vanish?
   A. Because Q is zero
   B. Because E is uniform
   C. Because a_r is constant
   D. E_r depends only on r and E_θ = E_φ = 0, so every term is zero
   Answer: D. All curl terms need θ or φ dependence or E_θ, E_φ. (Key point: Why vanishing)

3. What integral result relates to ∇×E = 0 through Stokes' theorem?
   A. ∮ E · dS = 0
   B. ∮ D · dS = Q
   C. ∮ E · dl = 0 (work around a closed path is zero)
   D. ∫ E dv = 0
   Answer: C. Zero circulation implies zero curl and allows E = −∇V. (Key point: Integral form)

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