Derive an expression for the torque acting on an electric dipole suspended freely in a uniform electric field. How will you determine the direction of torque?

Explanation

Potential energy of an electric dipole in a uniform electric field : The energy possessed by an electric dipole due to its position (orientation) in an electric field is called potential energy of the dipole. Suppose that an electric dipole having electric dipole moment p is placed along a direction making an angle `theta` with the direction of an external uniform electric field E. Then, torque acting on the dipole is given by
`tau = p E sin theta`
If the dipole is rotated through an infinitesimally small angle `d theta` against the torque acting on it, the small work done
`dW = tau d theta = pE sin theta d theta`
Therefore, work done in rotating the dipole from its orientation making angle `theta_1` with the direction of electric field to `theta_2` given by
`W = int pE sin theta = p E|-cos theta| (theta_2)/(theta_1)`
`W = pE (cos theta_1 - cos theta_2)`
This work done is stored in the dipole in the form of its potential energy.
Therefore, when an electric dipole is rotated inside the electric field, potential energy stored in the electric dipole,
`U = pE (cos theta_1 - cos theta_2)`
Let us assume that dipole is initially oriented perpendicular to the direction of the electric field and is then brought to the orientation making angle with the field direction.
`implies theta_1 = 90^@ ` and `theta_2 = 0`
`therefore U = -pE cos theta`
`U = -pE cos theta`
In vector notation
`U = - vecp. vecE`