Torque and Workdone by a dipole

Torque on a dipole :- When a dipole placed in electric field then torque will generate. It is generate when dipole present at any angle θ.
Let a dipole AB is placed at angle θ in an uniform electric field E.

we know that torque = force*perpendicular distance between line of action
so τ = F* AC     ....(1)
here AC = perpendicular distance
since E = F / q
so F =qE           ....(2)
and from triangle ABC
Sinθ = AC/AB
Sinθ = AC/2a
AC = 2a Sinθ    ....(3)

so from equation (1)
  τ = qE * 2a Sinθ
since 2aq = p
so τ = pE Sinθ   ....(4)
It is the torque due to dipole.

Work done by rotating a dipole in electric field :-
Let a dipole AB is placed at angle θ1 in uniform electric field E. It is rotate from θ1 to θ2 by torque and angular displacement dθ.
we know that work done in rotational motion = torque * angular displacement
dW = τ * dθ
since τ = pE Sinθ
so dW =  pE Sinθ dθ
integrating both sides
∫dW = ∫pE Sinθ dθ
since ∫dW = W
W = pE ∫ Sinθ dθ (here limit of integral is θ1 to θ2)
since ∫ Sin dθ  = -Cosθ
W =  -pE [Cosθ] (limit θ1 to θ2]
W = - pE[Cos θ2 - Cosθ1]
W = pE [ Cos θ1 - Cosθ2]
It is the work done by rotating a dipole in electric field.

if θ1 = 0 & θ2 = θ
W = pE [Cos0 - Cosθ]
since Cos0 = 1
so W = pE [1 - Cosθ]
this work is stored as Potential energy of dipole.
U = W = pE[1 - Cosθ]