Question

Derive the torque produce in electric dipole

Answer 1

Consider an electric dipole of dipole momment p in a uniform electic field E such that the dipole makes an angle  α with the direction of electric field. Under this condition, a positive charge experiences a force in the direction of field and negative charge opposite to the field. The two forces constitute a couple. 

Moment of couple or torque is 
force * perpendicular distance
ζ = q*e*p*sinα
  = 2qeaEsinα
  =p×E.

magnitude is  pEsinα

Answer 2

See the diagram.

AOB is an electric dipole, of length 2a, and centered and fixed at O.  There an electric charge +q at B and -q at A. 

Let us say that the dipole is in a uniform electric field of strength E in a direction as shown.

The force F on B (+q) will be in the direction of E.     F = q E
Force on A (-q) will be in the direction opposite to E and  = -F = -q E

If the dipole is parallel to E, then the two forces cancel and there is no torque. They are in equilibrium.

If the dipole makes an angle Ф with the direction of E, then the forces constitute a couple of forces.  The forces on q and -q rotate the dipole in the same direction.

Torque = sum of moments =  Σ Force * arm length (perpendicular from O to the force)
 
   Torque = F a Sin Φ  +   (-F) (-a sin Φ ) = 2 q E a sin Φ
              = 2 a q * E * sin Φ
               = p χ E  = cross product of dipole moment and electric field

      ,   where    p is dipole moment vector with magnitude = 2 a q
                direction of p is from negative charge towards +ve charge.
  
   The direction of Torque is given by Right Hand Thumb rule.

In the diagram shown, the direction of torque is into the diagram  and is perpendicular to the plane of the diagram.