The Electric Dipole

A pair of opposite charges: its field, and how it twists in a field.

12 minUpdated 22 October 2026electric dipoledipole momenttorque on a dipole

Lesson at a glanceShow the key points

p⃗ = qd⃗, from −q to +q (29.15).
V = p cos θ/4πε₀r² (29.16) for r ≫ d.
E_r = 2p cos θ/4πε₀r³, E_θ = p sin θ/4πε₀r³; E = (p/4πε₀r³)√(3cos²θ + 1), tan α = ½ tan θ (29.17–29.18).
End-on: E = 2p/4πε₀r³; broadside-on: E = p/4πε₀r³ (opposite to p), V = 0.
Uniform field: torque Γ⃗ = p⃗ × E⃗ (29.19); energy U = −p⃗·E⃗ (29.20).

A water molecule is neutral overall but slightly positive at one end and negative at the other. Such a pair of separated charges, a dipole, has a field of its own and turns in an applied field.

Dipole moment

Definition · Electric dipole

Charges +q and −q separated by a small distance d.

Dipole moment(29.15)

\vec p = q\vec d

d⃗ points from the negative to the positive charge; its line is the dipole axis

Potential of a dipole

At P, distance r ≫ d from the dipole’s centre, at angle θ to its axis, the distances to the charges are about r ∓ (d/2) cos θ. Adding the two potentials and neglecting d² against r²:

(29.16)

V = \frac{1}{4\pi\varepsilon_0}\frac{p\cos\theta}{r^2}

falls as 1/r² and depends on direction, unlike a point charge’s 1/r

In fact any charge distribution whose potential has this form is called a dipole, with dipole moment p along the direction θ is measured from.

Field of a dipole

The radial and transverse field components come from differentiating V:

E_r = -\frac{\partial V}{\partial r} = \frac{1}{4\pi\varepsilon_0}\frac{2p\cos\theta}{r^3}, \qquad E_\theta = -\frac1r\frac{\partial V}{\partial\theta} = \frac{1}{4\pi\varepsilon_0}\frac{p\sin\theta}{r^3}

(29.17–29.18)

E = \frac{1}{4\pi\varepsilon_0}\frac{p}{r^3}\sqrt{3\cos^2\theta + 1}, \qquad \tan\alpha = \tfrac12\tan\theta

α = angle between E⃗ and the radial direction

Interactive

The field of an electric dipole

Far from a dipole (r ≫ d), the potential is p cos θ/4πε₀r² and the field has a radial part 2p cos θ/4πε₀r³ and a transverse part p sin θ/4πε₀r³. On the axis (θ = 0) the field is 2p/4πε₀r³ along p; on the perpendicular bisector (θ = 90°) the potential is zero and the field is p/4πε₀r³, opposite to p. (Values in units of p/4πε₀.)

V (× p/4πε₀)0.0786

E_r, E_θ0.0524, 0.0262

E = \frac{p}{4\pi\varepsilon_0 r^3}\sqrt{3\cos^2\theta + 1}

0.0586

\tan\alpha = \tfrac12\tan\theta

α = 26.6°

Angle from the dipole axis

\theta

45°

Distance

r

3.00

Two special positions

End-on (on the axis, θ = 0): V = p/4πε₀r², E = 2p/4πε₀r³ along p⃗.
Broadside-on (on the bisector, θ = 90°): V = 0, E = p/4πε₀r³, opposite to p⃗.

Worked example

Field of a molecule-sized dipole

A dipole of moment 6.0 × 10⁻³⁰ C m (like a water molecule). Find the field 1.0 nm away on its axis, and on its bisector.

A dipole in a uniform field

Torque

Forces qE⃗ and −qE⃗ cancel, but they don’t act along the same line. Their torques about the centre add to

Torque on a dipole(29.19)

\vec\Gamma = \vec p\times\vec E, \qquad \Gamma = pE\sin\theta

Potential energy

Turning the dipole by dθ against the torque changes its energy by pE sin θ dθ. Taking U = 0 when the dipole is perpendicular to the field:

(29.20)

U = -\vec p\cdot\vec E = -pE\cos\theta

least (−pE) when aligned with the field, greatest (+pE) when opposed

Interactive

A dipole in a uniform field

In a uniform field the forces qE and −qE on the two ends are equal and opposite: no net force, but a torque τ = p × E, of size pE sin θ, turning the dipole toward the field. Its potential energy U = −p·E = −pE cos θ is least when aligned. Release it and it swings about the field direction.

angle to the field60°

\tau = pE\sin\theta

0.87 pE

U = -pE\cos\theta

-0.50 pE

Angle between p and E

\theta

60°

What to remember

1p⃗ = qd⃗, from − to +.
2V = p cos θ/4πε₀r²; E_r = 2p cos θ/4πε₀r³, E_θ = p sin θ/4πε₀r³.
3Axis: E = 2p/4πε₀r³; bisector: E = p/4πε₀r³, V = 0.
4Uniform field: no net force, torque p⃗ × E⃗, energy −p⃗·E⃗.

Check yourself

A dipole points opposite to a uniform field. It is in…

Doubling the distance from a dipole changes its field by…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

ReadElectric dipolesOpenStax University Physics Vol. 2 §5.7 ReadElectric dipoleHyperPhysics ReadElectric dipole momentWikipedia

PreviousField from Potential & Equipotentials Next Conductors, Insulators & Semiconductors

The Electric Dipole

A pair of opposite charges: its field, and how it twists in a field.

12 minUpdated 22 October 2026electric dipoledipole momenttorque on a dipole

Lesson at a glanceShow the key points

p⃗ = qd⃗, from −q to +q (29.15).
V = p cos θ/4πε₀r² (29.16) for r ≫ d.
E_r = 2p cos θ/4πε₀r³, E_θ = p sin θ/4πε₀r³; E = (p/4πε₀r³)√(3cos²θ + 1), tan α = ½ tan θ (29.17–29.18).
End-on: E = 2p/4πε₀r³; broadside-on: E = p/4πε₀r³ (opposite to p), V = 0.
Uniform field: torque Γ⃗ = p⃗ × E⃗ (29.19); energy U = −p⃗·E⃗ (29.20).

A water molecule is neutral overall but slightly positive at one end and negative at the other. Such a pair of separated charges, a dipole, has a field of its own and turns in an applied field.

Dipole moment

Definition · Electric dipole

Charges +q and −q separated by a small distance d.

Dipole moment(29.15)

\vec p = q\vec d

d⃗ points from the negative to the positive charge; its line is the dipole axis

Potential of a dipole

At P, distance r ≫ d from the dipole’s centre, at angle θ to its axis, the distances to the charges are about r ∓ (d/2) cos θ. Adding the two potentials and neglecting d² against r²:

(29.16)

V = \frac{1}{4\pi\varepsilon_0}\frac{p\cos\theta}{r^2}

falls as 1/r² and depends on direction, unlike a point charge’s 1/r

In fact any charge distribution whose potential has this form is called a dipole, with dipole moment p along the direction θ is measured from.

Field of a dipole

The radial and transverse field components come from differentiating V:

E_r = -\frac{\partial V}{\partial r} = \frac{1}{4\pi\varepsilon_0}\frac{2p\cos\theta}{r^3}, \qquad E_\theta = -\frac1r\frac{\partial V}{\partial\theta} = \frac{1}{4\pi\varepsilon_0}\frac{p\sin\theta}{r^3}

(29.17–29.18)

E = \frac{1}{4\pi\varepsilon_0}\frac{p}{r^3}\sqrt{3\cos^2\theta + 1}, \qquad \tan\alpha = \tfrac12\tan\theta

α = angle between E⃗ and the radial direction

Interactive

The field of an electric dipole

V (× p/4πε₀)0.0786

E_r, E_θ0.0524, 0.0262

E = \frac{p}{4\pi\varepsilon_0 r^3}\sqrt{3\cos^2\theta + 1}

0.0586

\tan\alpha = \tfrac12\tan\theta

α = 26.6°

Angle from the dipole axis

\theta

45°

Distance

r

3.00

Two special positions

End-on (on the axis, θ = 0): V = p/4πε₀r², E = 2p/4πε₀r³ along p⃗.
Broadside-on (on the bisector, θ = 90°): V = 0, E = p/4πε₀r³, opposite to p⃗.

Worked example

Field of a molecule-sized dipole

A dipole of moment 6.0 × 10⁻³⁰ C m (like a water molecule). Find the field 1.0 nm away on its axis, and on its bisector.

A dipole in a uniform field

Torque

Forces qE⃗ and −qE⃗ cancel, but they don’t act along the same line. Their torques about the centre add to

Torque on a dipole(29.19)

\vec\Gamma = \vec p\times\vec E, \qquad \Gamma = pE\sin\theta

Potential energy

Turning the dipole by dθ against the torque changes its energy by pE sin θ dθ. Taking U = 0 when the dipole is perpendicular to the field:

(29.20)

U = -\vec p\cdot\vec E = -pE\cos\theta

least (−pE) when aligned with the field, greatest (+pE) when opposed

Interactive

A dipole in a uniform field

angle to the field60°

\tau = pE\sin\theta

0.87 pE

U = -pE\cos\theta

-0.50 pE

Angle between p and E

\theta

60°

What to remember

1p⃗ = qd⃗, from − to +.
2V = p cos θ/4πε₀r²; E_r = 2p cos θ/4πε₀r³, E_θ = p sin θ/4πε₀r³.
3Axis: E = 2p/4πε₀r³; bisector: E = p/4πε₀r³, V = 0.
4Uniform field: no net force, torque p⃗ × E⃗, energy −p⃗·E⃗.

Check yourself

A dipole points opposite to a uniform field. It is in…

Doubling the distance from a dipole changes its field by…

Keep exploring

Hand-picked videos, simulations and readings that explain this idea from a different angle.

ReadElectric dipolesOpenStax University Physics Vol. 2 §5.7 ReadElectric dipoleHyperPhysics ReadElectric dipole momentWikipedia

PreviousField from Potential & Equipotentials Next Conductors, Insulators & Semiconductors