Potential Energy of a Dipole in a uniform Electric Field

Consider a dipole of moment p=2dq placed in a uniform electric field E.p makes an angle θ with the field direction. The charge +q experience a force qE in the direction. The charge –q experiences an equal force in the opposite direction. Thus the net linear force on the dipole is zero. The two forces are not collinear. So they form a clockwise torque which tends to set the dipole in the direction of the electric field. The magnitude of the torque is

    t = qE (2d sin θ)
      = pE sind θ

The torque can be written in vector form as
    t = pxE
suppose we choose the potential energy of a dipole in an external field to be zero when θ = 90. Let W be the work done by an external agent to turn the dipole from its reference orientation to angle θ. Then

W = ∫^θ₉₀  t dθ = ∫^θ₉₀  pE sin θ dθ = -pE cosθ

This work done is stored as potential energy in the dipole.

. : U (θ) = -pE cos θ = -p.E