Question

1. Define torque acting on a dipole of dipole moment \[\vec{p}\] placed in a uniform electric field \[\vec{E}\] Express it in the vector from and point out the direction along which it acts. Express it in the vector from and point out the direction along which it acts.
2. What happens if the field is non-uniform?
3. What would happen if the external field
   \[\vec{E}\]  is increasing (i) parallel to \[\vec{p}\] and (ii) anti-parallel to \[\vec{p}\]?

Answer 1

a. The torque `vecτ` acting on a dipole of dipole moment `vecp`​ in a uniform electric field `vecE` is given by:

`vecτ = vecpxxvecE`

Direction of Torque:

The direction of `vecτ` is perpendicular to the plane formed by `vecp`​​ and `vecE`, as given by the right-hand rule for the cross product. This torque tends to rotate the dipole to align it with the field direction (i.e., make `vecp` point in the direction of `vecE`).

b. If the electric field is non-uniform, the dipole experiences:

1. Torque (just like in the uniform field), and
2. A net translational force.

This net force arises because the forces on the two charges (positive and negative) are no longer equal and opposite in a non-uniform field. As a result, the dipole experiences motion in addition to rotation.

c. Effect of Increasing Electric Field `vecE`:

1. When `vecE` is Increasing Parallel to `vecp`:
   • The potential energy of the dipole, given by `U = -vecp xx vecE`, becomes more negative.
   • The dipole is already aligned with the field, and an increase in E further stabilizes this alignment.
   • No torque acts because `vecp`​ is parallel to `vecE`; the dipole remains in equilibrium.
2. When `vecE` is Increasing Anti-parallel to `vecp`​:
   • The potential energy becomes more positive, making the dipole increasingly unstable.
   • A torque acts to rotate the dipole so that it aligns in the direction of the field.
   • The dipole will tend to flip to align with the direction of increasing field.