Question

The magnitude of the torque experienced by an electric dipole of dipole moment \(P\) placed at an angle of \(30^\circ\) in a uniform electric field \(E\) is

• A. \(PE\)
• B. \(\frac{\sqrt{3}}{2}PE\)
• C. \(\frac{PE}{2}\)
• D. \(\frac{PE}{\sqrt{2}}\)
• E. \(\frac{PE}{\sqrt{3}}\)

Hint:

Torque on a dipole is maximum at \(90^\circ\) and zero at \(0^\circ\). Always use \(\tau = PE\sin\theta\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
When an electric dipole is placed in a uniform external electric field, it experiences a torque. This torque tends to align the dipole with the direction of the electric field. The magnitude of the torque depends on the magnitude of the dipole moment, the strength of the electric field, and the angle between the dipole moment vector and the electric field vector.
Step 2: Key Formula or Approach:
The torque (\( \tau \)) on an electric dipole is given by the vector product of the 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:
\[ \tau = |\vec{P}| |\vec{E}| \sin\theta = PE \sin\theta \] where \( \theta \) is the angle between \( \vec{P} \) and \( \vec{E} \).
Step 3: Detailed Explanation:
We are given:
- Dipole moment = P
- Electric field = E
- Angle, \( \theta = 30^\circ \)
We use the formula for the magnitude of the torque:
\[ \tau = PE \sin\theta \] Substitute the value of the angle \( \theta = 30^\circ \):
\[ \tau = PE \sin(30^\circ) \] We know that \( \sin(30^\circ) = \frac{1}{2} \).
\[ \tau = PE \left(\frac{1}{2}\right) = \frac{PE}{2} \] Step 4: Final Answer:
The magnitude of the torque is \( \frac{PE}{2} \). This corresponds to option (C).