(a) (i) Derive an expression for potential energy of an electric dipole p in an external uniform electric field E. When is the potential energy of…

The amount of work done in rotating the dipole from \(\displaystyle \theta=\theta_{0}\) to \(\displaystyle \theta=\theta_{1}\) by the external torque \[\begin{aligned} & \mathrm{W}=\int_{\theta_{0}}^{\theta_{0}} \tau_{e \mathrm{x} t} d \theta \\ & \qquad \begin{aligned} & =\int_{\theta_{0}}^{\theta_{0}} \mathrm{p} \mathrm{E} \sin \theta d \theta \\ \mathrm{~W} & =\mathrm{p} \mathrm{E}\left(\cos \theta_{0}-\cos \theta_{1}\right) \end{aligned} \\ & \text { For } \theta_{0}=\frac{\pi}{2} \text { and } \theta_{1}=\theta \\ & =\mathrm{p} \mathrm{E}\left(\cos \frac{\pi}{2}-\cos \theta\right) \\ & \begin{aligned} \mathrm{U}(\theta) & =-\mathrm{p} \mathrm{E} \cos \theta \\ & =-\vec{\mathrm{p}} \cdot \vec{\mathrm{E}} \end{aligned} \end{aligned} \]