Consider a permanent dipole of dipole moment p vector in a uniform external field E vector, as shown in Fig. 1.22.

FIGURE 1.22 Dipole in a uniform electric field.

(By permanent dipole, we mean that p vector exists irrespective of E vector; it has not been induced by E vector)

There is a force qE vector on q, and a force −qE vector on −q. The net force on the dipole is zero, since E vector is uniform. However, the charges are separated, so the forces act at different points, resulting in a torque on the dipole. When the net force is zero, the torque (couple) is independent of the origin. Its magnitude equals the magnitude of each force multiplied by the arm of the couple (perpendicular distance between the two antiparallel forces).

Magnitude of torque, τ = q E × 2 a sin θ
= 2 q a E sin θ

The direction of the torque is normal to the plane of the paper, and is directed out of the paper.

The magnitude of p vector ⨯ E vector is also p E sin θ and its direction is normal to the paper, directed out of it. Thus,

Torque, τ vector = p vector ⨯ E vector (1.22)

This torque will tend to align the dipole with the field E vector. When p vector is aligned with E vector, the torque is zero.

Note: The symbol ‘ ⨯ ‘ denotes the cross product of vectors.

What happens if the field is not uniform? In that case, the net force will evidently be non-zero. In addition there will, in general, be a torque on the system as before. The general case is involved, so let us consider the simpler situations when p vector is parallel to E vector or antiparallel to E vector. In either case, the net torque is zero, but there is a net force on the dipole if E vector is not uniform.

Figure 1.23 is self-explanatory.

FIGURE 1.23(a) Electric force on a dipole E vector parallel to p vector.

FIGURE 1.23(b) Electric force on a dipole E vector antiparallel to p vector.

It is easily seen that when p vector is parallel to E vector, the dipole has a net force in the direction of increasing field. When p vector is antiparallel to E vector, the net force on the dipole is in the direction of decreasing field. In general, the force depends on the orientation of p vector with respect to E vector.

This brings us to a common observation in frictional electricity. A comb run through dry hair attracts pieces of paper. The comb, as we know, acquires charge through friction. But the paper is not charged. What then explains the attractive force? Taking the clue from the preceding discussion, the charged comb ‘polarises’ the piece of paper, i.e., induces a net dipole moment in the direction of field. Further, the electric field due to the comb is not uniform. In this situation, it is easily seen that the paper should move in the direction of the comb!