Question

Obtain the expression for the torque acting on an electric dipole placed in a uniform electric field.

Answer 1

In a uniform electric field E, a dipole experiences a torque τ given by τ = p × E, but experiences no net force. 

Proof : 

Consider a permanent dipole of dipole moment p in a uniform external field E, as shown in Fig. There is a force qE on q and a force –qE on –q. The net force on the dipole is zero, since E 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 = p E sin θ 

Its direction is normal to the plane of the paper, coming out of it. The magnitude of p × E is also p E sin θ and its direction is normal to the paper, coming out of it. In vector notation,

This torque will tend to align the dipole with the field E. When p is aligned with E, the torque is zero. When p is parallel to E or antiparallel to E, then the net torque is zero, but there is a net force on the dipole if E is not uniform. i.e. if θ = 0º, τ = 0; if θ = 90º, τ = pE (maximum); if θ = 180º, τ = 0; S.I. Unit of Torque is Newton.meter (Nm). 

Note : If the dipole is placed in a non−uniform electric field at an angle θ, in addition to a torque, it also experiences a force. 

Physical significance of dipoles : 

In most molecules, the centres of positive charges and of negative charges lie at the same place. Therefore, their dipole moment is zero. CO₂ and CH₄ are of this type of molecules. However, they develop a dipole moment when an electric field is applied. But in some molecules, the centres of negative charges and of positive charges do not coincide. Therefore they have a permanent electric dipole moment, even in the absence of an electric field. Such molecules are called polar molecules. Water molecules, H₂O, is an example of this type. Various materials give rise to interesting properties and important applications in the presence or absence of electric field.