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

Explanation

The torque `tau` acting on an electric dipole of magnetic moment p placed in uniform electric field E is given by
`tau=pEsintheta` . . (i)
Where `theta` is the angle made by the dipole with the electric field E. the torque tends to align the magnetic dipole along the direction of the field. if the dipole is rotated through a small angle `d theta` against the torque, work has to be done, which is stored in the form of potential energy of the dipole.

Work done in rotating the dipole through an angle `d theta` against the torque `tau` is given by
`dW=taud theta=pEsinthetad theta`
If the dipole is rotated from `theta_(1)` to `theta_(2)`, then
Total work done,
`W=intdW=pEint_(theta_(2))^(theta_(2))sinthetad theta`
`=-pE|(costheta)|_(theta_(1))^(theta_(2))`
`W=-pE(costheta_(2)-costheta_(1))`
this work done is stored as potential energy U.
`therefore U=-pE(costheta_(2)-costheta_(1))`
if the dipole is rotated from `theta_(1)=(pi)/(2)` to `theta_(2)=theta`, then
`U=-pE(costheta-0)=-pE cos theta=-vec(p).vec(E)`.