---
title: "Dipole field on equatorial line"
subject: "Engineering Physics"
chapter: 7
key: "semester_1/engineering_physics:dipole-equatorial-field"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/dipole-equatorial-field
site: BCT Engineering
---

# Dipole field on equatorial line

Engineering Physics · Chapter 7: Electrostatics

- Asked in: 2074 Chaitra, 5 marks
- Video: [Electric field due to dipole on the equator | Electric charges & fields | Physics | Khan Academy](https://www.youtube.com/watch?v=EmbzBpan-jI) — Khan Academy India - English
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/dipole-equatorial-field
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-7

## Question

What is electrical dipole and dipole moment? Derive an expression of the electric field intensity at a point due to dipole at equatorial line.

## Short answer

An electric dipole is a pair of equal opposite charges ±q separated by 2a, with dipole moment p = 2aq. On the equatorial line, the field is E = (1/4πε₀) p/(r²+a²)^(3/2), antiparallel to p; for r ≫ a, E = (1/4πε₀) p/r³.

## Answer

An **electric dipole** is a pair of equal and opposite charges +q and −q separated by a small distance 2a. The **dipole moment** is p = q(2a), a vector from −q to +q, unit C·m.

#### Field on the equatorial (perpendicular bisector) line
Let P be at distance r from the centre O along the perpendicular bisector.

```text
              P
              .
            / | \
        r1 /  |  \ r2
          /   | r  \
   -q ---/----O-----\--- +q
        <-a->  <-a->
```

1. Each charge is at distance r' = √(r² + a²) from P.
2. Magnitudes are equal: E₊ = E₋ = (1/4πε0) q/(r² + a²).
3. E₊ points away from +q; E₋ points towards −q. Each makes angle θ with the line parallel to the dipole axis, where cosθ = a/√(r² + a²).
4. Components perpendicular to the axis (along OP) cancel; components parallel to the axis add. Both point from +q to −q, opposite to p:

E = 2 E₊ cosθ = 2 · (1/4πε0) · q/(r² + a²) · a/√(r² + a²)

5. With p = 2aq:

**E = (1/4πε0) · p / (r² + a²)^(3/2)**, directed antiparallel to p.

6. For a short dipole (r ≫ a):

**E = (1/4πε0) · p / r³**

This is half the magnitude of the axial-line field (2p/4πε0r³) at the same distance, and in the opposite direction.

## Check yourself

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

1. Why are the magnitudes of E+ and E- equal on the equatorial line?
   A. Each charge is at the same distance √(r^2 + a^2) from P
   B. The charges are of different size
   C. P is on the axis
   D. The distances are r + a and r - a
   Answer: A. E = (1/4πε0) q/(r^2 + a^2) for both. (Key point: Equal magnitudes)

2. Which components survive, and what factor does each carry?
   A. Components along OP add, each with cos θ = r/√(r^2 + a^2)
   B. All components cancel
   C. Components parallel to the axis cancel; those along OP add
   D. Components parallel to the dipole axis add, each with cos θ = a/√(r^2 + a^2)
   Answer: D. Perpendicular parts cancel; the surviving parts point from +q toward -q. (Key point: Resolving)

3. What is the field at P on the equatorial line?
   A. (1/4πε0)·2p/r^3, along p
   B. (1/4πε0)·p/(r^2 + a^2), along p
   C. (1/4πε0)·p/(r^2 + a^2)^(3/2), antiparallel to p, → p/(4πε0 r^3) for r >> a
   D. Zero everywhere on the line
   Answer: C. E = 2E+ cos θ = 2kqa/(r^2 + a^2)^(3/2) with p = 2aq. (Key point: Result)

4. How does the equatorial field compare with the axial field at the same distance?
   A. Twice the magnitude and in the same direction
   B. Half the magnitude and in the opposite direction
   C. Equal magnitude and direction
   D. Half the magnitude and in the same direction
   Answer: B. Equatorial p/(4πε0 r^3) against axial 2p/(4πε0 r^3). (Key point: Comparison with axial)

## 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
- [Dipole potential and resultant field](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Adipole-potential-and-resultant-field) — Engineering Physics, chapter 7
- [Field of long charged rod, equatorial](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Acharged-rod-field-potential-continuous) — 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
- [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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