Derive an expression for electrostatic potential energy of the dipole in a uniform electric field .

Explanation

Consider a dipole placed in the uniform electric field `vec(E)`. A dipole experiences a torque when kept in an uniform electric field `vec(E)`. This torque rotates the dipole to align it with the direction of the electric field. To rotate the dipole (at constant angular velocity ) from its initial angle `theta` to another angle `theta` against the torque exerted by the electric field, an equal and opposite external torque must be applied on the dipole.

(ii) The work done by the external torque to rotate the dipole from angle `theta` to `theta` at constant angular velocity is
`W=underset(theta)overset(theta)inttau_(ext)d""theta" ".......(1)`
(iii) Since `vec(tau)_(ext)` is equal and opposite to `vec(tau)_(E)=vecp=vec(E)` We have
`|vec(tau)_(ext)|=|vectau_(E)|=|vecpxxvecE|" ".........(2)`
Substituting equation (2) in equation (1), we get
`W=underset(theta)overset(theta)intpEsinthetad""theta`
`W=pE(costheta.-costheta)`
(iv) This work done is equal to the potential energy difference between the angular positions `theta` to `theta`.
`U(theta)-U(theta.)=DeltaU=-pEcostheta+pEcostheta.`. If the initial anlge is `theta.=90^(@)` and is taken as reference point then `U(theta.)=pEcos90^(@)=0`. The potential energy stored in the system of dipole kept in the uniform electric field is given by
`U=-pEcostheta=-vecp.vecE" "......(3)`
(v) The potential energy is maximum when the dipole is aligned anti-parallal `(theta=pi)` to the external electric field minimum when the dipole is aligned parallel `(theta=0)` to the external electric field.