Derive an expression for potential energy …

Considera dipole with charges +q and -q placed in a uniform eletric field as shown in figure.

In unifor electric field, dipole experiences a torque τ, which is given by

τ = p×E

where p is electric dipole moment. Torque τ will tend to rotate the dipole if p is not parallel to E.

Suppose an external torque τ_ext is applied to neutalize this torque and rotates the dipole from an angle θo to θ₁

at an infinitesimal angular speed and without angular acceleration. The amount of workdon by the external torque will be given by

W = 

This work is stored as the potential energy of the system.

We can then assosciate potential energy U(θ) with an inclination θ of the dipole. Inorder to choose the

reference angular position where potential Energy is considered as zero, we select θ_o as π/2 as the reference angular position.

We can then write U(θ) = pE( cos(π/2) - cosθ) = -pE cosθ = -p·E

when θ = 0 , i.e., p is aligned in the direction of E, potential energy is minimum.

when θ = π , i.e., p is aligned in the oposite direction of E, potential energy is maximum