Dipole in a uniform electric field & Dipole in a non uniform electric field

Dipole in a uniform electric field :

Consider a dipole AB of dipole moment p placed at an angle θ in an uniform electric field E (Fig.). The charge +q experiences a force qE in the direction of the field. The charge –q experiences an equal force in the opposite direction. Thus the net force on the dipole is zero.

Fig.  Dipole in a uniform electric field

The two equal and unlike parallel forces are not passing through the same point, resulting in a torque on the dipole, which tends to set the dipole in the direction of the electric field.

The magnitude of torque is,

τ = One of the forces x perpendicular distance between the forces

τ = F x 2d sin θ

τ = qE x 2d sin θ = pE sin θ    (∵ q × 2d = P)

In vector notation,  τ = p × E

Dipole in a non-uniform electric field : 

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 is parallel to E or antiparallel to E. 

In either case, the net torque is zero, but there is a net force on the dipole if E is not uniform.

It is easily seen that when p is parallel to E, the dipole has a net force in the direction of increasing field. 

When p is antiparallel to E, the net force on the dipole is in the direction of decreasing field. 

In general, the force depends on the orientation of p with respect to E.