Work done in turning a electric dipole on uniform electric field

*“work done in turning a electric dipole on uniform electric field”:

In adjacent figure shows the uniform electric field E and electric dipole AB is played in this field. The dipole moment of dipole is P. Initially the angle formed by the dipole axis with the direction of electric field is θ1 and some force of moment (torque) is applied on the dipole then it  change the position at angle θ2·

let the work done against dipole torque τ = P·E sinθ  in turning a dipole by small angle dθ is dw·

work done = dipole torque × angular displacement
              dw = τ·dθ
              dw = P·Esinθ                            taking integration of both side in the limit  θ1 to θ2

∫w =∫θ1​θ2
​​pEsinθdθ


​W = - P·E [cosθ2 - cosθ1]            
 W = P·E [ cosθ1 - cosθ2]
if  θ1 = 0 and θ2 = θ 
then   w = P· E ( cos0 - cosθ)
∴  w = P ·E ( 1 - cosθ)

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