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

Explanation

Electrostatic potential potential energy of a dipole in a uniform electric field :m Consider a dipole placec in the uniform electric field `vecE`. A dipole experiences a torque when kept in an uniform electric field `vecE`. This torque rotates the dipole to align it with the direction of the electric filed . 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. The work done by the external torque to rotate the dipole from angle `theta` to theta at constant angular velocity is

`W= int_(theta)^(theta) tau_("ext")dtheta `
Since `tau_("ext")` is equal and opposite to `tau_(E) = vecPxxvecE` we have
`|vectau_("ext")|=|vectau_(E)|=|vecPxxvecE|`
Substituting equation (2) in equation (1) we get ,
`W= int_(theta)^(theta)pE sin theta dtheta = p E ( cos theta- cos theta)`
This work done is equal to the potential energy difference between the angular positions `theta` and `theta `
`U ( theta)-(Utheta)=DeltaU=-pEcostheta + pE cos theta `
If the intial angle is `theta = 90^(@)` and is taken as reference point then `U (theta) +pE costheta =0 ` . The potential energy stored in the sytem of dipole kept in the unifrom electric field is given by `-U=- pEcos theta =- vecp. vecE`
In addition to p and E the potential energy also depends on the orientation `theta ` of the electric dipole with respect to the external electric field. The potential energy is maximum when the dipole is aligned anti - parallel `(theta =pi` ) to the external electric field and minimum when the dipole is aligned parallel `(theta =0)` to the external electric field.