---
title: "Axial field of dipole and quadrupole"
subject: "Engineering Physics"
chapter: 7
key: "semester_1/engineering_physics:dipole-quadrupole-axial-field-conclusion"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/dipole-quadrupole-axial-field-conclusion
site: BCT Engineering
---

# Axial field of dipole and quadrupole

Engineering Physics · Chapter 7: Electrostatics

- Asked in: 2071 Shrawan, 5 marks
- Video: [Electric field  in any direction due to dipole and quadrupole 2](https://www.youtube.com/watch?v=gJO9sh9WX_4) — Physics Diva
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/dipole-quadrupole-axial-field-conclusion
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-7

## Question

Calculate the electric field at any point at an axial distance due to a dipole and a quadrupole. What conclusion can you draw from your results?

## Short answer

On the axis, dipole field E = (1/4πε₀)·2p/r³; linear quadrupole field E = (1/4πε₀)·3Q/r⁴ with Q=2qa². Conclusion: field decays faster for higher multipoles — point charge 1/r², dipole 1/r³, quadrupole 1/r⁴.

## Answer

#### Dipole on its axis
Charges +q and −q separated by 2a, centre at origin; point P at distance r on the axis.

E = (q/4πε0)[1/(r − a)² − 1/(r + a)²] = (q/4πε0) · 4ar/(r² − a²)²

For r ≫ a, with p = 2aq:

**E_dipole = (1/4πε0) · 2p / r³**

#### Linear quadrupole on its axis
Charges +q at x = −a and x = +a, and −2q at the origin (net charge zero, dipole moment zero).

```text
  +q          -2q          +q               P
  (+)-------- (-) -------- (+) -------------.
 -a            0            +a               r
```

1. E = (q/4πε0)[1/(r − a)² + 1/(r + a)² − 2/r²]
2. Expand for a ≪ r: 1/(r ∓ a)² = (1/r²)(1 ± 2a/r + 3a²/r² + …)
3. Sum: 2/r² + 6a²/r⁴ + … − 2/r² = 6a²/r⁴

With quadrupole moment Q = 2qa²:

**E_quad = (1/4πε0) · 3Q / r⁴**

#### Conclusion
| Source | Field falls as |
|---|---|
| Point charge | 1/r² |
| Dipole | 1/r³ |
| Quadrupole | 1/r⁴ |

As charges are arranged so that their individual fields cancel more completely (net charge zero, then dipole moment zero), the field decays faster with distance. Each step multiplies the field by a factor of about a/r, so higher multipoles are important only close to the source.

## Check yourself

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

1. How does the axial field of a dipole vary for r >> a?
   A. As p/(4πε0 r^2)
   B. As 2p/(4πε0 r^2)
   C. As 2p/(4πε0 r^3)
   D. As 3p/(4πε0 r^4)
   Answer: C. E = (q/4πε0)·4ar/(r^2 - a^2)^2 ≈ 2p/(4πε0 r^3). (Key point: Dipole field)

2. Why does the quadrupole's axial field start at the 1/r^4 term?
   A. The 6a^2/r^4 terms cancel
   B. The 2/r^2 terms cancel against -2/r^2, leaving 6a^2/r^4
   C. There is no -2q charge
   D. The dipole term is added
   Answer: B. Expanding 1/(r ∓ a)^2 and subtracting the -2q contribution leaves 6a^2/r^4. (Key point: Quadrupole cancellation)

3. What is the quadrupole axial field in terms of Q = 2qa^2?
   A. E = 3Q/(4πε0 r^4)
   B. E = Q/(4πε0 r^3)
   C. E = 3Q/(4πε0 r^3)
   D. E = 2Q/(4πε0 r^2)
   Answer: A. (q/4πε0)(6a^2/r^4) = 3(2qa^2)/(4πε0 r^4). (Key point: Quadrupole result)

4. What conclusion follows from the point charge, dipole and quadrupole fields?
   A. Higher multipoles dominate at large distance
   B. All fall as 1/r^2
   C. Cancellation makes the field decay more slowly
   D. More complete cancellation gives faster decay (1/r^2, 1/r^3, 1/r^4)
   Answer: D. Each step multiplies the field by about a/r, so higher multipoles matter only close to the source. (Key point: Conclusion)

## Related in Engineering Physics

- [Axial dipole field twice equatorial](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adipole-axial-twice-equatorial) — Engineering Physics, chapter 7, asked 3×
- [Dipole field on axial line](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adipole-axial-field-inverse-cube) — Engineering Physics, chapter 7
- [Potential of electric quadrupole on axis](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aquadrupole-axis-potential) — Engineering Physics, chapter 7, asked 3×
- [Field of short linear quadrupole on axis](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aquadrupole-axis-field) — Engineering Physics, chapter 7
- [Potential of a dipole at any point](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adipole-potential-general-point) — Engineering Physics, chapter 7, asked 2×
- [Dipole field on equatorial line](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adipole-equatorial-field) — Engineering Physics, chapter 7

## Related in other subjects

- [Electric dipole potential and field](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adipole-potential-and-field-derive) — Electromagnetism, chapter 2, asked 4×
- [Define electric dipole and polarization](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adefine-dipole-and-polarization) — Electromagnetism, chapter 2
- [Potential gradient gives electric field](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Apotential-gradient-gives-electric-field) — Electromagnetism, chapter 2
- [Basic propagation models and far-field distance](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Apropagation-models-far-field) — Wireless Communication, chapter 3