Derive an expression for electric potential energy of an electric dipole placed in uniform electric field.

Explanation

Let an electric dipole of dipole moment `vecp` be placed in an electric field `vecE` making an angle `theta` with the direction of electric field intensity `vecE`.
The torque acting on the dipole is given by
`pi=pEsin theta`
work done to rotate the dipole through an angle do is given by,
`dW=pi d theta`
`dW=pE sin theta d theta`
work done in rotating the dipole from `theta_(1) to theta_(2)` is given by
`W=overset(theta_(2))underset(theta_(1))intdW=overset(theta_(2))underset(theta_(1))intpEsin theta d theta`
`=pEoverset(theta_(2))underset(theta_(1))int sin theta d theta`
`pE[-costheta_(2)]_(theta_(1))^(theta_(2))`
`=-pE[costheta_(2)-costheta_(1)]`
If `theta_(1)=90^(@) and theta_(2)=0^(@)` then
`W=-pE[costheta-cos90^(@)]`
`W=-pE[costheta-0]`
`W=-pEcostheta`
This work done is stored as electric potential energy (U) of dipole.
`:." "U=-pEcostheta`
`U=-vecp.vecE`
Special Case
When `theta = 90^(@)=-pE`
(ii) When `theta=90^(@)`, (dipole is parallel to electric field)
`U=-pE cos 90^(@)`
`U=0`