Question

The torque on an electric dipole consisting of charges $q$ and $-q$ of dipole moment $P$ in a uniform electric field $E$ is ________.

• A. $qE$
• B. $-qE$
• C. Zero
• D. $P \cdot E$
• E. $P \times E$

Hint:

$\tau = P E \sin \theta$, where $\theta$ is the angle between $P$ and $E$.

Answer 1

Answer: (E)

Step 1: Understanding the Concept
This question asks for the expression for the torque experienced by an electric dipole when placed in a uniform external electric field. A dipole consists of two equal and opposite charges. In a uniform field, the forces on the two charges are equal and opposite, creating a couple that results in a torque, causing the dipole to rotate.
Step 2: Key Formula or Approach
The torque (\(\vec{\tau}\)) on an electric dipole is given by the cross product of the electric dipole moment vector (\(\vec{P}\)) and the electric field vector (\(\vec{E}\)).
\[ \vec{\tau} = \vec{P} \times \vec{E} \] The magnitude of the torque is given by \(\tau = PE \sin\theta\), where \(\theta\) is the angle between \(\vec{P}\) and \(\vec{E}\).
Step 3: Detailed Explanation
1. Definition of Electric Dipole Moment (\(\vec{P}\)).
The electric dipole moment vector \(\vec{P}\) has a magnitude equal to the charge \(q\) times the separation distance \(d\) between the charges (\(P = qd\)). Its direction is from the negative charge (-q) to the positive charge (+q).
2. Forces on the Dipole.
- The positive charge \(+q\) experiences a force \(\vec{F}_+ = q\vec{E}\) in the direction of the field.
- The negative charge \(-q\) experiences a force \(\vec{F}_- = -q\vec{E}\) in the direction opposite to the field.
Since the field is uniform, the net force on the dipole is \(\vec{F}_{net} = q\vec{E} - q\vec{E} = 0\). There is no translational motion.
3. Calculating the Torque.
The two forces form a couple. The torque is calculated about a pivot point (e.g., the center of the dipole). The magnitude of the torque is the magnitude of one force times the perpendicular distance between the forces. If \(\theta\) is the angle between the dipole moment \(\vec{P}\) and the field \(\vec{E}\), the perpendicular distance is \(d \sin\theta\).
\[ \tau = F \times (\text{perpendicular distance}) = (qE)(d \sin\theta) = (qd)E \sin\theta \] Since the magnitude of the dipole moment is \(P = qd\), we have:
\[ \tau = PE \sin\theta \] This is the magnitude of the cross product \(\vec{P} \times \vec{E}\). Therefore, the torque vector is given by:
\[ \vec{\tau} = \vec{P} \times \vec{E} \] Let's analyze the options:
(D) P.E: This is the dot product, \(PE\cos\theta\). The potential energy of the dipole is \(U = -\vec{P} \cdot \vec{E}\), so this is incorrect.
(E) P \(\times\) E: This is the cross product, which correctly represents the torque vector.
Step 4: Final Answer
The torque on the electric dipole is \(\vec{P} \times \vec{E}\).