Question

Derive an expression for the torque experienced by a dipole due to a uniform electric field.

Answer 1

1. Consider an electric dipole of dipole moment `vec"p"` placed in a uniform electric field. The charge +q will experience a force q `vec"E"` in the direction of the field and charge - q `vec"E"` will experience a force -qE in a direction opposite to the field. Since the external field `vec"E"` is uniform, the total force acting on the dipole is zero.
2. These two forces acting at different points will constitute a couple and the dipole experience a torque. This torque tends to rotate the dipole.
3. The total torque on the dipole about the point 0, is given by
   `vectau = vec"0A" xx (- "q"vec"E") + vec"0B" xx "q"vec"E"`
   Torque on dipole.
4. Total torque is perpendicular to the plane of the paper and is directed into it. The magnitude of the total torque
   `vectau = |vec"0A"| (- "q"vec"E") |sin theta + |vec"0B"| "q" vec"E"| sin theta`
   
   Torque on dipole
   τ = qE.2a sin θ
5. Where θ is the angle made by `vec"p"` with `vec"E"`. Since p = 2aq, the torque is written in terms of the vector product as
   `vec tau = vec"p" xx vec"E"`
6. The magnitude of the torque is τ = pE sin θ and is maximum when θ = 90^o.
   This torque tends to rotate the dipole and align it with the electric field `vec"E"`. once `vec"p"` is aligned with, the total torque on the dipole becomes zero.
7. If the electric field is nonuniform, then the force experienced by +q is different from that experienced by -q. In addition to the torque, there will be a net force acting on the dipole.