An electric dipole is placed in uniform electric field as shown in the figure. The dipole is slightly rotated about its centre and released. Select the correct statement(s).

Options

A. The motion of dipole is oscillatory but not SHM
B. The motion of dipole is SHM
C. The motion of dipole is neither oscillatory nor SHM
D. The motion of dipole is oscillatory

Answer

To solve the problem regarding an electric dipole placed in a uniform electric field and slightly rotated, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Electric Dipole**:
An electric dipole consists of two equal and opposite charges, +Q and -Q, separated by a distance L. The dipole moment \( \mathbf{P} \) is given by:
\[ \mathbf{P} = Q \cdot \mathbf{L} \]
where \( \mathbf{L} \) is the vector pointing from the negative charge to the positive charge.

2. **Torque on the Dipole**:
When the dipole is placed in a uniform electric field \( \mathbf{E} \), it experiences a torque \( \tau \) given by:
\[ \tau = \mathbf{P} \times \mathbf{E} \]
The magnitude of the torque can be expressed as:
\[ \tau = P E \sin \theta \]
where \( \theta \) is the angle between the dipole moment and the electric field.

3. **Small Angle Approximation**:
If the dipole is slightly rotated, the angle \( \theta \) is small. For small angles, we can use the approximation:
\[ \sin \theta \approx \theta \quad (\text{in radians}) \]
Thus, the torque becomes:
\[ \tau \approx P E \theta \]

4. **Relating Torque to Angular Acceleration**:
The torque is also related to the moment of inertia \( I \) and angular acceleration \( \alpha \) by:
\[ \tau = I \alpha \]
For small oscillations, we can express angular acceleration as:
\[ \alpha = \frac{d^2\theta}{dt^2} \]

5. **Equating the Two Expressions for Torque**:
Setting the two expressions for torque equal gives:
\[ P E \theta = I \frac{d^2\theta}{dt^2} \]

6. **Forming the Equation of Motion**:
Rearranging the equation leads to:
\[ \frac{d^2\theta}{dt^2} + \frac{P E}{I} \theta = 0 \]
This is the standard form of simple harmonic motion (SHM).

7. **Finding the Time Period**:
The time period \( T \) of SHM is given by:
\[ T = 2\pi \sqrt{\frac{I}{P E}} \]

8. **Conclusion**:
Since the dipole undergoes oscillations in the electric field, we conclude that:
- The dipole exhibits simple harmonic motion (SHM).
- The motion is oscillatory in nature.

### Final Answer:
The correct statements are:
- The dipole undergoes simple harmonic motion (SHM).
- The motion is oscillatory.