Electric Dipole

• Dipole moment, p₁ = 1.2 × 10^–30 C-m
• Electric field, E₁ = 5 × 10⁴ NC^–1
• Maximum torque, τ₁ = p₁ E₁ = (1.2 × 10^–30 C-m) × (5 × 10⁴ NC^–1) = 6 × 10^–26 Nm

• Dipole moment, p₂ = 2.4 × 10^–30 C-m
• Electric field, E₂ = 15 × 10⁴ NC^–1
• Maximum torque, τ₂ = p₂ E₂ = (2.4 × 10^–30 C-m) × (15 × 10⁴ NC^–1) = 36 × 10^–26 Nm

We are tasked with analyzing the motion of an electric dipole in a uniform electric field and determining the time period of its oscillations. The solution proceeds as follows:

1. Electric Dipole in a Uniform Field:
The dipole moment is denoted by , and the electric field is . The angle between the dipole moment and the electric field is .

2. Torque on the Dipole:
The torque acting on the dipole is given by:

This equation holds when is small, as the torque is proportional to the angular displacement .

3. Moment of Inertia:
The moment of inertia of the dipole depends on the mass and the length of the dipole. For a dipole consisting of two point masses separated by a distance , the moment of inertia is:

4. Time Period of Oscillation:
For small angular displacements, the motion of the dipole is simple harmonic. The time period of oscillation is given by:

Here, is the restoring torque constant, which is equal to . Substituting and , we get:

Simplify the expression:

Final Answer:
The time period of oscillation of the dipole is:

The potential energy of a dipole in an electric field is given by: Initially, the dipole is aligned with the field ( ), so the initial energy is: When the dipole is flipped opposite to the field ( ), the final energy is: The work required to rotate the dipole is: Substituting values: