---
title: "Electric dipole potential and field"
subject: "Electromagnetism"
chapter: 2
key: "semester_3/electromagnetism:dipole-potential-and-field-derive"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/dipole-potential-and-field-derive
site: BCT Engineering
---

# Electric dipole potential and field

Electromagnetism · Chapter 2: Electric field

- Asked in: 2074 Asoj, 1+6 marks; 2072 Chaitra, 4 marks; 2068 Chaitra, 2+6 marks; 2067 Shrawan, 1 marks (asked 4 times)
- Video: [Electric Field and Potential due to an electric dipole | B. Tech,  B. Sc,  M. Sc. | EMT](https://www.youtube.com/watch?v=sW4gu7GpLuw) — Tanmay Creations
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/dipole-potential-and-field-derive
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-2

## Question

What is an electric dipole? Derive the expressions for the potential and electric field due to a dipole at a distant point P (large compared to the separation of the charges). Explain the electric dipole moment.

## Short answer

An electric dipole consists of two equal and opposite charges ±Q separated by distance d. Its dipole moment is p = Qd (direction −Q to +Q). For r≫d, V = p·a_r/(4πε₀ r²), and E = (p/(4πε₀ r³))(2cosθ a_r + sinθ a_θ).

## Answer

#### Electric dipole

An **electric dipole** is a pair of equal and opposite point charges +Q and −Q separated by a small distance d. It is a source of an electric field that falls off faster (1/r³) than that of a single charge.

#### Potential at a far point P

Let the charges lie on the z-axis at z = ±d/2, with +Q above. P is at distance r from the centre, at angle θ from the dipole axis, and r ≫ d.

```text
             P
            ╱│╲
        r₂ ╱ │ ╲ r₁         r₁ = distance from +Q
          ╱  │r ╲           r₂ = distance from −Q
        −Q───●───+Q
          ←──d──→
```

```text
V = (Q/4πε₀) (1/r₁ − 1/r₂) = (Q/4πε₀) (r₂ − r₁)/(r₁ r₂)
```

For r ≫ d the lines from the charges to P are almost parallel, so

```text
r₂ − r₁ ≈ d cosθ,      r₁ r₂ ≈ r²
```

```text
V = Q d cosθ / (4πε₀ r²)
```

With the dipole moment **p** = Q**d** (magnitude Qd, directed from −Q to +Q), p · a_r = Qd cosθ, so

```text
V = p · a_r / (4πε₀ r²)  =  p cosθ / (4πε₀ r²)
```

#### Electric field

Use E = −∇V in spherical coordinates:

```text
E_r = −∂V/∂r = 2 p cosθ / (4πε₀ r³)
E_θ = −(1/r) ∂V/∂θ = p sinθ / (4πε₀ r³)
E_φ = −(1/(r sinθ)) ∂V/∂φ = 0
```

```text
E = ( p / (4πε₀ r³) ) ( 2cosθ a_r + sinθ a_θ )
```

Special cases: on the axis (θ = 0), E = 2p / (4πε₀ r³) along a_r. On the equatorial plane (θ = 90°), E = p / (4πε₀ r³) along a_θ, i.e. opposite to **p**.

#### Dipole moment

The **electric dipole moment** **p** = Q**d** is a vector of magnitude charge × separation, directed from the negative to the positive charge. Unit: C·m. It measures the strength of the dipole. The potential and field depend only on the product Qd, not on Q and d separately. In an external field **E**, a dipole experiences a torque **T** = **p** × **E**.

## Check yourself

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

1. What is an electric dipole?
   A. Two equal like charges close together
   B. Two equal and opposite point charges separated by a small distance d
   C. A single charge in a field
   D. Two unequal charges far apart
   Answer: B. A dipole is +Q and -Q separated by a small distance, and its field falls off as 1/r³. (Key point: Definition of dipole)

2. For r >> d, which approximations give V = Qd cosθ/(4πε₀r²)?
   A. r2 - r1 ≈ d cosθ and r1 r2 ≈ r²
   B. r2 - r1 ≈ d and r1 r2 ≈ d²
   C. r1 ≈ r2 ≈ d
   D. r2 - r1 ≈ d sinθ
   Answer: A. Lines from the charges to P are almost parallel, so the path difference is d cosθ and the product of distances is r². (Key point: Far-field approximation)

3. Which is the dipole field?
   A. E = (p/(4πε₀r³))(sinθ a_r + 2cosθ a_θ)
   B. E = (p/(4πε₀r³))(2cosθ a_r + sinθ a_θ)
   C. E = (p/(4πε₀r²))(cosθ a_r)
   D. E = (p/(4πε₀r³))(2sinθ a_r + cosθ a_θ)
   Answer: B. E = -∇V gives E_r = 2p cosθ/(4πε₀r³) and E_θ = p sinθ/(4πε₀r³). (Key point: Field components)

4. What is the dipole moment p?
   A. p = Q/d, in C/m
   B. p = Qd, directed from -Q to +Q, in C·m
   C. p = Qd, directed from +Q to -Q
   D. p = Q + d
   Answer: B. p = Q d points from negative to positive charge, and the field depends only on the product Qd. (Key point: Dipole moment)

## Related in Electromagnetism

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