Derive equation of potential energy of an electric dipole in a uniform electric field .

Explanation

Consider a dipole with charges -q and +q placed in a uniform electric field `vecE` as shown in figure.

Dipole placed in a uniform electric field `vecE` at an angle `theta`
Equal and opposite forces `+qvecE` and `-qvecE` are acting on charges -q and +q .
The forces acting on dipole constitute moment of force and hence dipole experiences a torque
`tau= vecpxx vecE` where `vecp=(2veca)q` which will tend to rotate it ( unless `vecp` is parallel or antiparallel to `vecE`).
Suppose an external torque acting on dipole and it rotates with small angle `Deltatheta = theta_(1) - theta_(0)` the small work done, ,
`DeltaW= tau Deltatheta = p E sin theta d theta`
Total work done during rotation from `theta_(0)` to `theta_(1)`
`W= int_(theta0)^(theta) p E sin thetad theta`
`= p E[ -cos theta]_(theta0)^(theta1)`
`:. W = p E [ cos theta_(0)- cos theta_(1)]`
This work is stored as the potential energy of the system .
`:.` Potential energy of dipole,
`U = pE[ cos theta_(0)- cos theta_(1)]`
Initially a dipole placed at an angle `theta_(0)=(pi)/(2)` and if it makes angle 6, = 0 from that position by rotating then the potential energy of dipole is
`U=pE[cos""(pi)/(2)-costheta]`
`=pE[0-cos theta]`
`:. U=-pEcostheta`
`:. U = -(vecp.vecE)`