Stable equilibrium of a magnetic dipole in a uniform magnetic field will be at orientation when:

Options

A. Dipole moment vector and magnetic field vector are in same direction
B. Dipole moment vector and magnetic field vector are in opposite direction
C. Dipole moment vector is perpendicular to magnetic field vector
D. Both a and c

Explanation

To determine the stable equilibrium of a magnetic dipole in a uniform magnetic field, we need to analyze the potential energy associated with the dipole's orientation relative to the magnetic field.

### Step-by-Step Solution:

1. **Understanding the Magnetic Dipole**:
A magnetic dipole consists of two equal and opposite magnetic charges separated by a small distance. The dipole moment \(\mathbf{M}\) is a vector quantity that points from the negative to the positive charge.

2. **Magnetic Field**:
A uniform magnetic field \(\mathbf{B}\) is represented as a vector that has a constant magnitude and direction throughout the space.

3. **Potential Energy of a Magnetic Dipole**:
The potential energy \(U\) of a magnetic dipole in a magnetic field is given by the formula:
\[ U = -\mathbf{M} \cdot \mathbf{B} = -M B \cos \theta \]
where \(\theta\) is the angle between the dipole moment vector \(\mathbf{M}\) and the magnetic field vector \(\mathbf{B}\).

4. **Analyzing Potential Energy**:
- The potential energy is minimized when \(\cos \theta\) is maximized.
- The maximum value of \(\cos \theta\) occurs when \(\theta = 0^\circ\) (i.e., when the dipole moment is aligned with the magnetic field).
- At \(\theta = 0^\circ\), the potential energy \(U\) becomes:
\[ U = -M B \cos(0) = -M B \]
- This is the minimum potential energy, indicating a stable equilibrium.

5. **Conditions for Stable Equilibrium**:
- For stable equilibrium, the dipole must be aligned with the magnetic field, which corresponds to \(\theta = 0^\circ\).
- In this orientation, the dipole experiences a torque that tends to keep it aligned with the magnetic field.

6. **Evaluating Options**:
- **Option A**: Dipole moment vector and magnetic field vector are in the same direction (Correct).
- **Option B**: Dipole moment vector and magnetic field vector are in the opposite direction (Incorrect).
- **Option C**: Dipole moment vector is perpendicular to the magnetic field vector (Incorrect).
- **Option D**: Both A and C (Incorrect).

7. **Conclusion**:
The stable equilibrium of a magnetic dipole in a uniform magnetic field occurs when the dipole moment vector is aligned with the magnetic field vector, which corresponds to Option A.

### Final Answer:
**The correct answer is Option A: Dipole moment vector and magnetic field vector are in the same direction.**