A bar magnet of magnetic moment M is placed in a uniform magnetic field B in such a manner that is angle between M and B. Select the correct option(s)

Options

A. Minimum potential energy is -MB
B. Maximum potential energy is MB
C. Bar magnet is in stable equilibrium when `theta = 0^@`
D. Bar magnet is in unstable equilibrium when ` theta = 180^@ `

Explanation

To solve the problem, we need to analyze the behavior of a bar magnet in a uniform magnetic field, particularly focusing on the potential energy and the conditions for stable and unstable equilibrium.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have a bar magnet with a magnetic moment \( M \).
- It is placed in a uniform magnetic field \( B \).
- The angle between the magnetic moment \( M \) and the magnetic field \( B \) is denoted as \( \theta \).

2. **Potential Energy of the Magnet**:
- The potential energy \( U \) of a magnetic moment \( M \) in a magnetic field \( B \) is given by the formula:
\[ U = -\vec{M} \cdot \vec{B} = -MB \cos(\theta) \]
- This means:
- When \( \theta = 0^\circ \) (the magnet is aligned with the field), \( U = -MB \) (minimum potential energy).
- When \( \theta = 90^\circ \) (the magnet is perpendicular to the field), \( U = 0 \) (potential energy is zero).
- When \( \theta = 180^\circ \) (the magnet is anti-aligned with the field), \( U = +MB \) (maximum potential energy).

3. **Conditions for Equilibrium**:
- **Stable Equilibrium**: Occurs when the potential energy is at a minimum. This happens at \( \theta = 0^\circ \).
- **Unstable Equilibrium**: Occurs when the potential energy is at a maximum. This happens at \( \theta = 180^\circ \).

4. **Conclusion**:
- The minimum potential energy is \( -MB \) when \( \theta = 0^\circ \).
- The maximum potential energy is \( +MB \) when \( \theta = 180^\circ \).
- Therefore, the bar magnet is in stable equilibrium when \( \theta = 0^\circ \) and in unstable equilibrium when \( \theta = 180^\circ \).

### Final Answers:
- Minimum potential energy: \( -MB \) (correct)
- Maximum potential energy: \( +MB \) (correct)
- Stable equilibrium at \( \theta = 0^\circ \) (correct)
- Unstable equilibrium at \( \theta = 180^\circ \) (correct)