Question

Derive an expression for potential energy of an electric dipole $\vec{p}$ in an external uniform electric field $\vec{E}$. When is the potential energy of the dipole (1) maximum, and (2) minimum ? 
 

Hint:

- The potential energy of an electric dipole in a uniform electric field is given by \( U = -\vec{p} \cdot \vec{E} = - pE \cos \theta \).
- The potential energy is maximum (\( U = +pE \)) when the dipole is antiparallel to the field (\( \theta = 180^\circ \)).
- The potential energy is minimum (\( U = -pE \)) when the dipole is parallel to the field (\( \theta = 0^\circ \)).
- Understanding this concept helps in analyzing dipole behavior in external fields and potential energy changes due to rotations.

Answer 1

(i) The amount of work done in rotating the dipole from \(\theta = \theta_0\) to \(\theta = \theta_1\) by the external torque is: \[ W = \int_{\theta_0}^{\theta_1} \tau_{\text{ext}} \, d\theta \] Since the torque \(\tau_{\text{ext}} = pE \sin \theta\), we can write: \[ W = \int_{\theta_0}^{\theta_1} pE \sin \theta \, d\theta \] The work done is then: \[ W = pE (\cos \theta_0 - \cos \theta_1) \] For \(\theta_0 = \frac{\pi}{2}\) and \(\theta_1 = \theta\), we get: \[ W = pE \left( \cos \frac{\pi}{2} - \cos \theta \right) = -pE \left( \cos \theta \right) \] The potential energy is given by: \[ U(\theta) = - pE \cos \theta = - \vec{p} \cdot \vec{E} \] (1) The potential energy is maximum when: \[ \vec{p} \text{ is antiparallel to } \vec{E} \] Alternatively: \[ \theta = 180^\circ \quad \text{or} \quad \pi \, \text{radians} \] (2) The potential energy is minimum when: \[ \vec{p} \text{ is parallel to } \vec{E} \] Alternatively: \[ \theta = 0^\circ \]