An electric dipole is placed in an uniform electric field with the dipole axis making an angle `theta` with the direction of the electric field. The orientation of the dipole for stable equilibrium is

Options

A. `(pi)/(6)`
B. `(pi)/(3)`
C. 0
D. `(pi)/(2)`

Answer

To determine the orientation of an electric dipole for stable equilibrium in a uniform electric field, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Dipole in an Electric Field**:
An electric dipole consists of two equal and opposite charges separated by a distance. When placed in an electric field, the dipole experiences a torque that tends to align it with the field.

2. **Torque on the Dipole**:
The torque (\( \tau \)) acting on the dipole in an electric field (\( \vec{E} \)) is given by the formula:
\[ \tau = \vec{p} \times \vec{E} \]
where \( \vec{p} \) is the dipole moment vector. The magnitude of the torque can be expressed as:
\[ |\tau| = pE \sin \theta \]
where \( \theta \) is the angle between the dipole moment and the electric field.

3. **Condition for Equilibrium**:
For the dipole to be in equilibrium, the net torque must be zero. This occurs when:
\[ \sin \theta = 0 \]
This happens at angles \( \theta = 0^\circ \) (dipole aligned with the field) and \( \theta = 180^\circ \) (dipole anti-aligned with the field).

4. **Stable vs. Unstable Equilibrium**:
- **Stable Equilibrium**: The dipole is in stable equilibrium when, after a small displacement, it experiences a restoring torque that brings it back to its equilibrium position. This occurs when \( \theta = 0^\circ \).
- **Unstable Equilibrium**: The dipole is in unstable equilibrium when, after a small displacement, it does not return to its original position. This occurs when \( \theta = 180^\circ \).

5. **Conclusion**:
Therefore, for stable equilibrium, the orientation of the dipole must be parallel to the electric field, which corresponds to \( \theta = 0^\circ \).

### Final Answer:
The orientation of the dipole for stable equilibrium is \( 0^\circ \) (parallel to the electric field).

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