A dipole of dipole moment \( \vec{p} \) is kept in a uniform electric field \( \vec{E} \) such that \( \vec{p} \) and \( \vec{E} \) are in the same direction. Find the work done by an external agent in rotating the dipole slowly through an angle \( \pi \).
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( W=2pE \)
In a uniform field, a dipole feels a torque \( \vec{p}\times\vec{E} \) but no net force. An external agent that rotates it slowly applies a torque that just balances this torque. The work done by the external torque in turning the dipole from angle \( \theta_0 \) to \( \theta_1 \) is \( W=pE\,(\cos\theta_0-\cos\theta_1) \).
Here \( \vec{p} \) starts along \( \vec{E} \), so \( \theta_0=0 \), and after a rotation through \( \pi \) the angle is \( \theta_1=\pi \).
\( \begin{aligned} W &= pE\,(\cos0-\cos\pi) \\ &= pE\,[1-(-1)] \\ &= 2pE \end{aligned} \)
The work is positive. The dipole is turned against the torque of the field, from its stable position (along \( \vec{E} \)) to its unstable position (opposite to \( \vec{E} \)).
NCERT (2026-27):
Potential Energy of a Dipole in an External Field – “… Suppose an external torque τext is applied in such a manner that it just neutralises this torque and rotates it in the plane of paper from angle θ0 to angle θ1 at an infinitesimal angular speed and without angular acceleration. The amount of work done by the external torque will be given by
\( W=pE\,(\cos\theta_0-\cos\theta_1) \)\( \qquad (2.31) \)”