5) An electric dipole of moment \(\vec{P}\) is lying along a uniform electric field \(\vec{E}\). The work done in rotating the dipole through \(\frac{\pi}{3}\) is:
Solution:
The potential energy \(U\) of a dipole in a uniform electric field is given by: \[ U = -pE \cos \theta, \] where \(\theta\) is the angle between the dipole moment \(\vec{P}\) and the field \(\vec{E}\). Initially, when the dipole is aligned with the field, \(\theta=0\) and the energy is: \[ U_i = -pE. \] After rotating by \(\frac{\pi}{3}\), the angle becomes \(\frac{\pi}{3}\) and the energy is: \[ U_f = -pE \cos\frac{\pi}{3} = -pE \left(\frac{1}{2}\right) = -\frac{pE}{2}. \] The work done by an external agent in rotating the dipole is the increase in potential energy: \[ \Delta U = U_f - U_i = \left(-\frac{pE}{2}\right) - (-pE) = \frac{pE}{2}. \] Hence, the work done in rotating the dipole is \(\frac{pE}{2}\), so option D is the correct answer.
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