Dipole in Uniform Electric Field

In a uniform field E, the force on the positive charge is F₊ = +qE (in the direction of E) and on the negative charge is F₋ = −qE (opposite to E). These forces are equal in magnitude and opposite in direction, so the net force on the dipole is zero. A uniform field cannot translate a dipole — it can only rotate it. (In a non-uniform field there would also be a net force, but that is a separate topic.)

Consider the dipole at angle θ to the field. The perpendicular distance between the lines of action of the two forces is d·sinθ, where d is the separation between the charges. Each force contributes a torque arm of (d/2)sinθ about the centre, so the total torque is τ = qE·(d/2)sinθ + qE·(d/2)sinθ = qEd sinθ. Since the dipole moment is p = qd, this simplifies to τ = pE sinθ. In vector form, τ = p × E, where the cross product automatically gives both the magnitude and the direction (by the right-hand rule) of the torque axis.

The torque τ = p × E always acts to decrease θ — it rotates the dipole toward alignment with E. When θ = 0° (p parallel to E), the torque is zero and this is stable equilibrium: a small nudge restores the dipole to alignment. When θ = 180° (p anti-parallel to E), the torque is also zero, but this is unstable equilibrium: any tiny perturbation causes the dipole to flip all the way to θ = 0°. The maximum torque occurs at θ = 90°, giving τ_max = pE.

Rotating the dipole against the torque requires work, which is stored as potential energy. Integrating the torque from the reference angle (θ = 90°, where U is defined as zero) gives U = −pE cosθ = −p·E (dot product). The minimum energy U = −pE occurs at θ = 0° (stable alignment), and the maximum U = +pE occurs at θ = 180° (unstable anti-alignment). This energy landscape explains why polar molecules spontaneously orient themselves along an applied electric field, releasing energy to their surroundings in the process.

• \vec{\tau} — Torque (N·m): The net rotational effect of the field on the dipole, directed perpendicular to both p and E by the right-hand rule.
• \vec{p} — Electric dipole moment (C·m): Vector of magnitude p = qd pointing from the negative charge to the positive charge; measures the strength of the dipole.
• \vec{E} — Electric field (N/C (or V/m)): The uniform external electric field in which the dipole is placed.
• \theta — Angle between p and E (degrees or radians): The angle that the dipole moment vector makes with the direction of the electric field.
• U — Potential energy (J): Energy stored in the orientation of the dipole relative to the field; minimum at θ = 0°, maximum at θ = 180°.

Problem. A water molecule (H₂O) has an electric dipole moment of p = 6.2 × 10⁻³⁰ C·m. It is placed in a uniform electric field of magnitude E = 1.5 × 10⁵ N/C at an angle of 30° to the field. (a) Calculate the torque on the dipole. (b) Find the potential energy of the dipole in this orientation, taking U = 0 at θ = 90°.

Answer. Torque ≈ 4.65 × 10⁻²⁵ N·m (rotating dipole toward alignment); Potential energy ≈ −8.05 × 10⁻²⁵ J

• Electric Dipole Moment — The dipole moment p = qd is the fundamental quantity that determines the magnitude of both the torque and the potential energy of a dipole in a field.
• Uniform Electric Field — A uniform field is the specific condition under which a dipole experiences pure torque without any net translational force — the central assumption of this concept.
• Torque and Rotational Equilibrium — The cross product definition of torque (τ = r × F) and the conditions for rotational equilibrium are the mechanical concepts directly applied when analysing dipole rotation.
• Electric Potential Energy — The potential energy U = −p·E is derived from the work done against the torque, connecting dipole orientation to energy storage in the electric field.