Derive an expression for electric potential energy of a systemm of charges in an electric field.

Explanation

Consider an electric dipole placed in a uniform electric field `vecE` with its dipole moment `vecP` making an angle `theta` with the field. Two equal and opposite force `+ q vecE and - q vecE` act on its two ends. The two forces form couple. The torque experience by the dipole is
`tau =PE sin theta `
If the dipole is rotated through a small angle `dtheta` against the torque acting on it, them the small work does is
`dW = tau d theta = pE sin theta d theta`
The total work done in roatting the dipole from its orientation making an angle `theta _(1),` with the direction of the field to `theta _(2)` is
`W = int _(theta ) ^(theta) pE sin theta d theta `
`=p E[- cos theta ] _(theta ) ^(theta) =p E [cos theta_(1- cos theta _(2) ]`
THis work done is tored as the potential energy U of the dipole.
`therefore U = pE [ cos theta _(1) -cos theta _(2)]`
If initially the dipole is oriented perpendicular to the direction of the field `(theta _(1) =90^(@))` and brought to some orientation making an angle `theta` with the field `(theta _(2) =theta),` then the potential energy of the dipole will be
`U =pE [ cos 90^(@) - cos theta ] = pE [0- cos theta]`
`U = pE cos theta =- vecp.vecE `