Question

An electric dipole is placed at a distance of 2 cm from an infinite plane sheet having positive charge density \( \sigma \). Choose the correct option from the following.

• A. Torque on dipole is zero and net force is directed away from the sheet.
• B. Torque on dipole is zero and net force acts towards the sheet.
• C. Potential energy of dipole is minimum and torque is zero.
• D. Potential energy and torque both are maximum.

Hint:

In the case of a uniform electric field, the torque on a dipole is zero, and the dipole's potential energy is minimum when aligned with the electric field.

Answer 1

The Correct Option is C

The electric field due to an infinite plane sheet of charge is given by: \[ E = \frac{\sigma}{2 \epsilon_0} \] The electric field is uniform, and there is no variation in the field across the dipole, meaning the torque on the dipole is zero.
Also, the dipole is in the minimum potential energy configuration when aligned with the electric field, and the net force on the dipole due to the uniform electric field is zero.
Therefore, the potential energy of the dipole is minimum, and the torque is zero.

Answer 2

Step 1: Understanding the situation
An electric dipole is placed at a distance of 2 cm from an infinite plane sheet that carries a uniform positive surface charge density \( \sigma \). The electric field produced by an infinite sheet of charge is uniform and acts perpendicular to the plane. The magnitude of this field is given by:
\[ E = \frac{\sigma}{2\varepsilon_0} \] This electric field always points away from the sheet since the charge on the sheet is positive.

Step 2: Behavior of a dipole in a uniform electric field
When an electric dipole with dipole moment \( \vec{p} \) is placed in a uniform electric field \( \vec{E} \):
1. It experiences a torque \( \tau = pE \sin\theta \), which tends to align the dipole along the direction of the field.
2. The potential energy of the dipole is given by \( U = -pE \cos\theta \), where \( \theta \) is the angle between \( \vec{p} \) and \( \vec{E} \).

Step 3: Conditions for equilibrium and energy state
For equilibrium, the torque on the dipole must be zero. This occurs when \( \sin\theta = 0 \), i.e., \( \theta = 0^\circ \) or \( 180^\circ \).
- When \( \theta = 0^\circ \): The dipole is aligned with the electric field (the positive charge of the dipole faces away from the positively charged sheet). In this case, the potential energy is minimum because \( U = -pE \).
- When \( \theta = 180^\circ \): The dipole is anti-parallel to the field, and the potential energy is maximum because \( U = +pE \).

Step 4: Application to the given situation
Since the infinite sheet produces a uniform field, the dipole experiences no net translational force but a torque that tends to align it with the field. Once it becomes aligned, the torque vanishes (\( \tau = 0 \)), and the configuration corresponds to the minimum potential energy state.

Step 5: Conclusion
At equilibrium, the dipole is aligned with the electric field due to the infinite charged sheet. Therefore:
- The potential energy of the dipole is minimum.
- The torque on the dipole is zero.

Final Answer:
Potential energy of dipole is minimum and torque is zero.