There is a flat circular current coil placed in a uniform magnetic field in such a manner that its magnetic moment is opposite to the direction of the magnetic field. The coil is

Options

A. in neutral equilibrium
B. in stable equilibrium
C. in unstable equilibrium
D. not in equilibrium

Explanation

To solve the problem, we need to analyze the situation of a flat circular current coil placed in a uniform magnetic field with its magnetic moment opposite to the direction of the magnetic field. We will determine the type of equilibrium the coil is in.

### Step-by-Step Solution:

1. **Identify the Magnetic Moment Direction**:
- The magnetic moment \( \vec{M} \) of the coil is given by the formula:
\[ \vec{M} = n \cdot I \cdot A \cdot \hat{n} \]
where \( n \) is the number of turns, \( I \) is the current, \( A \) is the area of the coil, and \( \hat{n} \) is the unit vector normal to the plane of the coil.
- Since the magnetic moment is opposite to the magnetic field \( \vec{B} \), we can denote the magnetic moment as \( \vec{M} = -M \hat{k} \) and the magnetic field as \( \vec{B} = B \hat{k} \).

2. **Calculate the Potential Energy**:
- The potential energy \( U \) of the magnetic moment in a magnetic field is given by:
\[ U = -\vec{M} \cdot \vec{B} \]
- Substituting the values:
\[ U = -(-M \hat{k}) \cdot (B \hat{k}) = MB \]
- Since \( M \) is positive, \( U \) is positive, indicating that the potential energy is at a maximum.

3. **Determine the Torque**:
- The torque \( \vec{\tau} \) acting on the coil is given by:
\[ \vec{\tau} = \vec{M} \times \vec{B} \]
- Since \( \vec{M} \) and \( \vec{B} \) are in opposite directions, the cross product will yield:
\[ \vec{\tau} = (-M \hat{k}) \times (B \hat{k}) = 0 \]
- The torque is zero, indicating that there is no tendency for the coil to rotate.

4. **Analyze the Type of Equilibrium**:
- The coil is in equilibrium since the net torque is zero. However, since the potential energy is at a maximum, this indicates that the equilibrium is unstable.
- In unstable equilibrium, any small displacement will result in a force that moves the system away from the equilibrium position.

5. **Conclusion**:
- Therefore, the coil is in an **unstable equilibrium**.

### Final Answer:
The coil is in unstable equilibrium.