8 min readConcept sheet2 or 3-mark derivation of τ = pE sin θ, and 1-mark questions on the net force and equilibrium of a dipole

Torque on a dipole in a uniform electric field

In a uniform field a dipole feels zero net force but a torque τ = p × E, which turns it to line up with the field.

Torqueτ = p × E; size τ = pE sin θ, where θ is the angle between p and E

Net force in a uniform fieldZero, because +qE and −qE cancel

Maximum torqueτ(max) = pE, when p is at 90° to E; unit of torque N m

EquilibriumStable at θ = 0° (p along E); unstable at θ = 180°

Derivation: torque on a dipole

A dipole (−q at A, +q at B, AB = 2a) sits in a uniform field E, with p at an angle θ to E.

1
Force on +q: qE, along E. Force on −q: qE, opposite to E.
2
The forces are equal and opposite, so the net force is zero. The dipole does not move as a whole.
3
But the two forces act along different lines. They form a couple, which turns the dipole.
4
Torque = one force × perpendicular distance between their lines of action = qE × 2a sin θ.
5
Since p = q × 2a: τ = pE sin θ.
6★
So, in vector form τ = p × E. It is perpendicular to the plane of p and E, and it tries to turn p into line with E.

★ key step

How to draw the figure

1
Draw several parallel horizontal arrows for the uniform field E.
2
Draw the dipole AB tilted at angle θ to the field: −q at A, +q at B. Mark p from A to B.
3
At B draw an arrow qE along the field. At A draw an arrow qE opposite to the field.
4
Drop a perpendicular from B to the field line through A, and label it 2a sin θ.

Torque at different angles

Angle θ between p and E	Torque τ = pE sin θ	What it means
0°	0	p along E: stable equilibrium
30°	pE/2	Half the maximum torque
90°	pE (maximum)	p at right angles to E
180°	0	p against E: unstable equilibrium

Dipole in a non-uniform field

In a non-uniform field, the forces on +q and −q are no longer equal. So there is a net force as well as a torque. If p points along E, the dipole is pulled towards the region of stronger field. If p points against E, it is pushed towards the weaker field. This explains why a comb charged by dry hair attracts bits of paper. The comb's field polarises the paper, making tiny dipoles along the field. The comb's field is non-uniform, so the paper is pulled towards the comb.

Worked example: torque on a dipole

Problem

(a) A dipole of moment 4 × 10⁻⁹ C m makes 30° with a uniform field of 5 × 10⁴ N C⁻¹. Find the torque on it. (b) Charges of ±2 μC are 1 cm apart. Find the largest torque on this dipole in a field of 1 × 10⁵ N C⁻¹.

Working

(a) τ = pE sin θ = 4 × 10⁻⁹ × 5 × 10⁴ × sin 30°.
τ = 2 × 10⁻⁴ × 0.5 = 1 × 10⁻⁴ N m.
(b) p = q × 2a = 2 × 10⁻⁶ C × 0.01 m = 2 × 10⁻⁸ C m.
The torque is largest at θ = 90°: τ(max) = pE = 2 × 10⁻⁸ × 1 × 10⁵ = 2 × 10⁻³ N m.

Answer

(a) 1 × 10⁻⁴ N m; (b) 2 × 10⁻³ N m

In real life

In the LCD screen of a calculator or TV, a small electric field turns rod-shaped molecules that behave like dipoles. Turning them changes how light passes through, which forms the digits and pictures. A comb rubbed in dry hair lifting paper bits uses the same pull on dipoles.

Exam angle

3 marks: “Derive an expression for the torque on an electric dipole in a uniform electric field. Write it in vector form.” The forces qE and −qE form a couple with arm 2a sin θ. τ = qE × 2a sin θ = pE sin θ; vector form τ = p × E.

2 marks: “Show that the net force on a dipole in a uniform field is zero. When is the dipole in stable equilibrium?” The forces +qE and −qE are equal and opposite, so they cancel. Stable equilibrium when p is along E (θ = 0°): the torque is zero and a small turn brings it back.

1 marks: “At what angle between p and E is the torque on a dipole maximum?” At 90°, where τ = pE.

Remember it

Uniform field: twist, no push (τ = pE sin θ, F = 0). Non-uniform field: twist and push.