---
title: "Potential of electric quadrupole on axis"
subject: "Engineering Physics"
chapter: 7
key: "semester_1/engineering_physics:quadrupole-axis-potential"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/quadrupole-axis-potential
site: BCT Engineering
---

# Potential of electric quadrupole on axis

Engineering Physics · Chapter 7: Electrostatics

- Asked in: 2074 Asoj, 5 marks; 2070 Asar, 5 marks; 2067 Asoj, 5 marks (asked 3 times)
- Video: [potential due to an electric quadrupole | imran abid](https://www.youtube.com/watch?v=zu3Jo83Z8_U) — Circus of Physics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/quadrupole-axis-potential
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-7

## Question

What is an electric quadrupole? Calculate the electric potential of a linear quadrupole (of separation 2a) at an axial distance r from its center.

## Short answer

An electric quadrupole is a charge arrangement with zero net charge and dipole moment, e.g., +q, −2q, +q on a line. Its axial potential at distance r from centre is V = 2qa²/[4πε₀ r(r²−a²)], which ≈ Q/(4πε₀ r³) for large r, where quadrupole moment Q = 2qa².

## Answer

#### Electric quadrupole
An electric quadrupole is a system of charges with zero net charge and zero net dipole moment. A **linear quadrupole** consists of two equal and opposite dipoles placed end to end: charges +q, −2q, +q on a line with equal spacing.

```text
   +q          -2q          +q
    o-----a-----o-----a-----o----------> P
   x=-a         0          x=+a     r
```
#### Potential on the axis
Take the −2q at the origin and +q at x = ± a (distance between the two end charges = 2a). Point P is on the axis at distance r from the centre (r > a).

1. The distances of P from the charges are (r − a), r and (r + a).
2. By superposition:
 V = (1/4πε₀) [ q/(r − a) − 2q/r + q/(r + a) ]
3. Combine the first and third terms:
 q/(r − a) + q/(r + a) = 2qr / (r² − a²)
4. V = (q/4πε₀) [ 2r/(r² − a²) − 2/r ]
 = (q/4πε₀) · [2r² − 2(r² − a²)] / [r (r² − a²)]
 = (q/4πε₀) · 2a² / [r (r² − a²)]

 **V = 2qa² / [4πε₀ r (r² − a²)]**

5. For r ≫ a, r² − a² ≈ r²:

 **V ≈ 2qa² / (4πε₀ r³) = Q / (4πε₀ r³)**, where Q = 2qa² is the quadrupole moment.

The potential falls off as 1/r³, faster than a dipole (1/r²) and a monopole (1/r).

## Check yourself

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

1. What defines a linear electric quadrupole?
   A. Two equal charges of opposite sign (+q, -q)
   B. Net charge of 2q with zero dipole moment
   C. Zero net charge and zero net dipole moment, e.g. +q, -2q, +q in a line
   D. Zero net charge but a non-zero dipole moment
   Answer: C. A quadrupole is two opposite dipoles end to end, so both net charge and dipole moment vanish. (Key point: Definition)

2. How is the potential at P on the axis obtained?
   A. Add vector fields and integrate
   B. Add the scalar potentials of +q, -2q and +q at distances r - a, r and r + a
   C. Use only the two end charges
   D. Use the -2q charge only
   Answer: B. V = (1/4πε0)[q/(r - a) - 2q/r + q/(r + a)]. (Key point: Superposition)

3. What is the exact axial potential of the quadrupole?
   A. V = 2qa^2/[4πε0 r(r^2 - a^2)]
   B. V = 2qa/[4πε0 (r^2 - a^2)]
   C. V = qa^2/(4πε0 r^2)
   D. V = 2q/(4πε0 r)
   Answer: A. Combining the terms gives (q/4πε0)·2a^2/[r(r^2 - a^2)]. (Key point: Result)

4. For r >> a, how does the quadrupole potential vary with r?
   A. As 1/r^2, like a dipole
   B. As 1/r, like a point charge
   C. As 1/r^4
   D. As 1/r^3, faster than the dipole (1/r^2) and monopole (1/r)
   Answer: D. V ≈ Q/(4πε0 r^3) with Q = 2qa^2. (Key point: Falloff with distance)

## Related in Engineering Physics

- [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
- [Axial field of dipole and quadrupole](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adipole-quadrupole-axial-field-conclusion) — Engineering Physics, chapter 7
- [Ring on axis: potential then field](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aring-axial-potential-then-field) — Engineering Physics, chapter 7, asked 2×
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- [Electron on ring axis performs SHM](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Aring-electron-shm-axial) — 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×

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