Dipole in a Uniform External Field

When an electric dipole is placed in a uniform electric field, the two charges experience equal but opposite forces. These forces cancel out (zero net force), but because they act at different points, they form a couple that creates a torque — making the dipole rotate to align itself with the field.​​

Think of a bar magnet placed in a magnetic field — it rotates to align with the field lines. An electric dipole behaves exactly the same way in an electric field.​

Configuration:
A dipole of moment \[\vec p\] is placed at angle θ with a uniform electric field \[\vec E\].

Draw two charges +q and −q separated by distance 2l, with the dipole axis at angle θ to \[\vec E\]. Show +qE arrow (right) on +q and −qE arrow (left) on −q. Mark the perpendicular distance from the centre to the line of action of the force as l sin ⁡θ. Draw a curved arc to show the resulting torque \[\vec τ\].

Forces acting:

Step 1: Perpendicular distance from +q to the axis = l sin⁡ θ
Step 2: Perpendicular distance from −q to the axis = l sin ⁡θ
Step 3: Torque due to +q = qE ⋅ l sin ⁡θ
Step 4: Torque due to −q = qE ⋅ l sin ⁡θ
Step 5: Total torque τ = qE ⋅ l sin⁡ θ + qE ⋅ l sin ⁡θ = qE ⋅ 2l ⋅ sin⁡θ

Since p = q.2l: τ = pE sin ⁡θ

Vector Form:

\[\vec τ\] = \[\vec p\] × \[\vec E\]

Direction of \[\vec τ\]: Perpendicular to the plane containing \[\vec p\] and \[\vec E\], determined by the Right-Hand Screw Rule (curl fingers from \[\vec p\] toward \[\vec E\]; thumb points along \[\vec τ\]).

In a non-uniform field:

• The forces on +q and −q are unequal in magnitude.
• Therefore, net force ≠ 0 → the dipole undergoes translational motion as well as rotation.
• Both torque AND net force act on the dipole.

Real-Life Example: When a plastic comb rubbed on hair is brought near small pieces of paper, the comb creates a non-uniform field that polarises the paper and pulls it toward the comb (translation + rotation).