Electric Dipole and Dipole in a Uniform Field

Two equal and opposite charges a tiny distance apart form a dipole — the model behind polar molecules like water and the behaviour of matter in fields.

Definition — Electric dipole: a pair of equal and opposite charges +q and −q separated by a small distance 2a.
Definition — Dipole moment: p = q × 2a, a vector pointing from the negative to the positive charge. SI unit: C·m.

Field of a dipole

• On the axial line (along p), at distance r ≫ a: E_axial = k·2p / r³, directed along p.
• On the equatorial line (perpendicular bisector): E_eq = k·p / r³, directed opposite to p.
• The axial field is twice the equatorial field at the same distance, and both fall off as 1/r³ — faster than a point charge's 1/r².

Dipole in a uniform external field

The two charges feel equal and opposite forces qE, so the net force is zero, but they form a couple that produces a torque:
τ = p × E, magnitude τ = pE·sinθ,
where θ is the angle between p and E. The torque tends to align p with E. The potential energy is U = −p·E = −pE·cosθ, minimum (stable) at θ = 0 and maximum (unstable) at θ = 180°.

Exam Tricks & Tips

• 🎯 Axial field is DOUBLE the equatorial field at equal distance (2p vs p) — a favourite one-marker.
• 🎯 Dipole field ∝ 1/r³, so it dies faster than a single charge's 1/r².
• 🎯 Torque is maximum at θ = 90° (τ = pE) and zero at θ = 0° or 180°.
• 🎯 p points from − to + — get this direction right before any cross product.
• 🎯 Stable equilibrium at θ = 0 (U = −pE); unstable at θ = 180° (U = +pE).
• ❌ Common mistake: saying a dipole in a uniform field feels a net force. It feels only a torque; a net force needs a NON-uniform field.

Expected exam pattern

A 2-mark derivation or statement of axial/equatorial fields, a torque numerical τ = pE·sinθ, or a conceptual question on why the net force is zero and where equilibrium is stable.

Quick recap

Dipole moment p = q·2a (− to +). Axial field 2kp/r³, equatorial kp/r³; both ∝ 1/r³. In a uniform field: net force zero, torque τ = p × E (max at 90°), energy U = −p·E (stable at θ = 0°).

Cover the answer, recall, then check. 10 cards on the electric dipole.

Q1. Define electric dipole moment.
A1. p = q × 2a, a vector from the negative to the positive charge; unit C·m.

Q2. Field on the axial line of a dipole (r ≫ a)?
A2. E_axial = k·2p/r³, directed along p.

Q3. Field on the equatorial line?
A3. E_eq = k·p/r³, directed opposite to p.

Q4. How do axial and equatorial fields compare?
A4. The axial field is twice the equatorial field at the same distance.

Q5. How does a dipole's field fall off with distance?
A5. As 1/r³ — faster than a point charge's 1/r².

Q6. Net force on a dipole in a uniform field?
A6. Zero — the two equal and opposite forces cancel.

Q7. Torque on a dipole in a uniform field?
A7. τ = p × E, magnitude pE·sinθ.

Q8. At what angle is the torque maximum?
A8. At θ = 90°, where τ = pE.

Q9. Potential energy of a dipole in a field?
A9. U = −p·E = −pE·cosθ.

Q10. Where is the dipole in stable equilibrium?
A10. At θ = 0° (p parallel to E), where U = −pE is minimum.

A water molecule is bent, with oxygen slightly negative and the two hydrogens slightly positive. That tiny separation of charge — an electric dipole — is why water dissolves salt, why microwaves heat food, and why antennas radiate. The dipole is one of the most exam-heavy ideas in electrostatics.

Electric dipole — the core idea

An electric dipole is a pair of equal and opposite point charges (+q and −q) separated by a small distance 2a.

Its strength is the dipole moment:

p = q × (2a)

a vector pointing from the negative charge to the positive charge, with SI unit coulomb-metre (C·m). The magnitude tells you how "strong" the dipole is; the direction tells you how it is oriented.

Beginner: field on the axial line

On the axis of the dipole (the end-on position), at distance r from the centre (r ≫ a):

E_axial = 2kp / r³

directed parallel to p (in the direction of p).

Intermediate: field on the equatorial line

On the equatorial line (the broadside position), at distance r:

E_equatorial = kp / r³

directed anti-parallel to p. Note two things: the equatorial field is half the axial field at the same distance, and the dipole field falls as 1/r³ — faster than a single charge's 1/r², because the two opposite charges partly cancel at large distance.

Advanced: dipole in a uniform external field

Place a dipole in a uniform field E. The two charges feel equal and opposite forces (+qE and −qE), so the net force is zero — the dipole does not translate. But the forces form a couple that produces a torque:

τ = pE sin θ, or in vector form τ = p × E

where θ is the angle between p and E. The torque tends to align p with E. The potential energy of the dipole is:

U = −pE cos θ = −p·E

Minimum energy (U = −pE, stable) when p is parallel to E (θ = 0); maximum energy (U = +pE, unstable) when anti-parallel (θ = 180°).

Worked example

A dipole of moment 4 × 10⁻⁹ C·m is placed in a uniform field of 5 × 10⁴ N/C at 30° to the field. Find the torque.

τ = pE sin θ = (4 × 10⁻⁹)(5 × 10⁴)(sin 30°)
τ = (4 × 10⁻⁹)(5 × 10⁴)(0.5) = 1 × 10⁻⁴ N·m.

Real-world / exam application

The torque-alignment mechanism explains how polar molecules line up in a field (dielectric polarisation), how a microwave oven's oscillating field flips water dipoles to generate heat, and how dipole antennas radiate. Exam questions almost always ask for axial vs equatorial fields, torque, or work done in rotating a dipole (W = pE(cos θ₁ − cos θ₂)).

Exam tricks & shortcuts

• Axial field is twice the equatorial field at the same distance — remember the "2".
• Dipole field ∝ 1/r³ (not 1/r²) — a quick way to spot dipole problems.
• Work to rotate from θ₁ to θ₂: W = pE(cos θ₁ − cos θ₂).
• Mnemonic "N-to-P": dipole moment points Negative to Positive.

Key formulas

• Dipole moment: p = q·2a (points − to +), unit C·m.
• Field on axial line: E = 2kp/r³ (r ≫ a).
• Field on equatorial line: E = kp/r³.
• Torque in uniform field: τ⃗ = p⃗ × E⃗; τ = pE sin θ.
• Potential energy: U = −p⃗·E⃗ = −pE cos θ.

A dipole experiences a torque (but no net force) in a uniform field, tending to align with it.

Worked Example

Problem: An electric dipole consists of charges +2 μC and −2 μC separated by 5 cm. It is placed in a uniform electric field of 1 × 10⁵ N/C at an angle of 30° to the field. Find (a) the dipole moment and (b) the torque acting on it.

Solution:
Step 1 — Compute the dipole moment p = q × (2a), where 2a is the charge separation:
p = (2 × 10⁻⁶ C) × (0.05 m) = 1 × 10⁻⁷ C·m.

Step 2 — Recall the torque on a dipole in a uniform field:
τ = p E sin θ.

Step 3 — Substitute p = 1 × 10⁻⁷, E = 1 × 10⁵, θ = 30° (sin 30° = 0.5):
τ = (1 × 10⁻⁷)(1 × 10⁵)(0.5).

Step 4 — Evaluate:
= (1 × 10⁻²)(0.5) = 5 × 10⁻³ N·m.

Step 5 — The torque tends to align the dipole with the field (θ → 0).

Answer: Dipole moment p = 1 × 10⁻⁷ C·m; torque τ = 5 × 10⁻³ N·m.