Derive an expression for the potential energy of a dipole of dipole moment p placed in an electric field E. Find the orientation of the dipole when it is in (1) stable equilibrium

(2) unstable equilibrium.

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 θ

_{0}to θ

_{1}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 θo 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 opposite direction of E, potential energy is maximum