Derive an expression for potential energy of an electric dipole placed in a uniform electric field.
Or
Derive an expression for the total work done in rotating an electric dipole through an angle `theta` in a uniform electric field.

Explanation

Potential Energy of dipole : It is defined as the energy possessed by the dipole by virtue of its position in the electric field. Potential energy of the dipole is a scalar quantity and its SI unit is joule.
Consider a dipole placed in a uniform electric field `vecE`, making an angle `theta` with `vecE`, then
`pi = pE sin theta`
Work done in rotating the dipole through small angle `dtheta` is given by
`dW =pi.dtheta" " implies " "dW= pE sin theta d theta`·
By integrating between limits `theta_(1)`, and `theta_(2)`, we get,
`intdW=underset(theta1)overset(theta_(2))int pE sin theta d theta=pE[-costheta]_(theta_(1))^(theta_(2))=pE(cos theta_(2)-costheta_(1))`
Suppose, initially the dipole is at an angle 90° with `vecE(i.e.theta_(1) = 90°)` and we have to set it at an angle `theta` with `vecE` `(ie., theta_(2)= theta)`, then
`W=-pE(costheta-cos 90^(@))=-pE(costheta-0)=pE cos theta=-vecp.vecE`
This work done is stored in the dipole as potential energy. Potential energy, `U =- thetaP. thataE = -pE cos theta`.