Derive an expression for the potential energy of an electric dipole of dipole moment `vecp` in an electric field `vecE`.

Explanation

Consider an electric dipole having charges `q_1 = +q and q_2=-q` located at `vecr_1 and vecr_2`, respectively.
Now potential difference between 1 and 2 equals the work done in bringing a unit + ve charge against the electric field from position 2 to 1.
`:. V (vecr_1) - V (vecr_2) = -E`
(displacement parallel to the field `vecE`).
If 2a be the length of electric dipole and dipole be aligned at an angle `theta` from direction of electric field,
then
`V(vecr_1) - V (vecr_2) = -E .2 a cos theta`
`:.` Potential energy of the dipole in a uniform electric field `vecE` is given by
`u = q[ V(vecr_1) - V (vecr_2)] = -E. 2a cos theta = - pE cos theta = - vecp. vecE`.
where p = q.2a = dipole moment of the given electric dipole.