Question

Which of the following angles between an electric dipole moment and an applied electric field will result in the most stable state? a) \(0 \mathrm{rad}\) b) \(\pi / 2 \mathrm{rad}\) c) \(\pi \mathrm{rad}\) d) The electric dipole moment is not stable under any condition in an applied electric field.

Short answer

Answer: The electric dipole moment is in its most stable state when it is parallel (\(0 \mathrm{rad}\)) or antiparallel (\(\pi \mathrm{rad}\)) to the applied electric field, as there is no torque experienced.

Step-by-step solution

Evaluate the torque at \(\theta = 0 \mathrm{rad}\)

At this angle, the electric field is parallel to the electric dipole moment. The torque experienced by the electric dipole is given by \(\tau = pE \sin{0} = 0\). No torque is experienced, and this is a stable state.

Evaluate the torque at \(\theta = \frac{\pi}{2} \mathrm{rad}\)

At this angle, the electric field is perpendicular to the electric dipole moment. The torque experienced by the electric dipole is given by \(\tau = pE \sin{\frac{\pi}{2}} = pE\). The torque is at its maximum, and this is an unstable state.

Evaluate the torque at \(\theta = \pi \mathrm{rad}\)

At this angle, the electric field is antiparallel to the electric dipole moment. The torque experienced by the electric dipole is given by \(\tau = pE \sin{\pi} = 0\). No torque is experienced, and this is a stable state.

Evaluate the torque for "not stable under any condition"

This option is incorrect, as we have found that there are stable states when the electric dipole moment is either parallel (\(0 \mathrm{rad}\)) or antiparallel (\(\pi \mathrm{rad}\)) to the applied electric field.

Determine the most stable state

Comparing the values obtained for each angle, we see that the most stable state occurs when the electric dipole moment is parallel or antiparallel to the applied electric field, as there is no torque experienced. This corresponds to both the angles \(\theta = 0 \mathrm{rad}\) and \(\theta = \pi \mathrm{rad}\). Thus, the correct answer is a) \(0 \mathrm{rad}\) and c) \(\pi \mathrm{rad}\).