A uniform magnetic field B of`0.3` T is along the positive Z-direction. A rectangular loop (abed) of sides `10 cm xx 5 cm` carries a current I of 12 A. Out of the following different orientations which one corresponds to stable equilibrium?

Options

A.
B.
C.
D.

Explanation

To determine which orientation of a rectangular loop carrying a current in a uniform magnetic field corresponds to stable equilibrium, we can follow these steps:

### Step 1: Understand the Magnetic Moment
The magnetic moment \( \vec{M} \) of a current-carrying loop is given by:
\[ \vec{M} = I \cdot A \]
where \( I \) is the current and \( A \) is the area vector of the loop. The direction of \( \vec{M} \) is determined by the right-hand rule: if you curl the fingers of your right hand in the direction of the current, your thumb points in the direction of \( \vec{M} \).

### Step 2: Determine the Torque on the Loop
The torque \( \vec{\tau} \) experienced by the loop in a magnetic field \( \vec{B} \) is given by:
\[ \vec{\tau} = \vec{M} \times \vec{B} \]
For stable equilibrium, the torque should be zero, which occurs when \( \vec{M} \) and \( \vec{B} \) are parallel (0 degrees) or anti-parallel (180 degrees).

### Step 3: Analyze Each Orientation
1. **First Orientation**: Current is anti-clockwise.
- \( \vec{M} \) points out of the plane (positive Z-direction).
- \( \vec{B} \) is also in the positive Z-direction.
- Since both are parallel, torque \( \vec{\tau} = 0 \). This is a stable equilibrium.

2. **Second Orientation**: Current flows in a different direction.
- Determine the direction of \( \vec{M} \) using the right-hand rule.
- If \( \vec{M} \) is in the negative Y-direction and \( \vec{B} \) is in the positive Z-direction, then \( \vec{M} \) and \( \vec{B} \) are not parallel, resulting in a non-zero torque. This is not stable.

3. **Third Orientation**: Current flows in a way that \( \vec{M} \) is in the positive Z-direction and \( \vec{B} \) is also in the positive Z-direction.
- Thus, \( \vec{M} \) and \( \vec{B} \) are parallel, resulting in zero torque. This is stable.

4. **Fourth Orientation**: Current flows such that \( \vec{M} \) is in the negative Z-direction and \( \vec{B} \) is in the positive Z-direction.
- Here, \( \vec{M} \) and \( \vec{B} \) are anti-parallel, resulting in zero torque, but this is an unstable equilibrium because the system has higher potential energy.

### Conclusion
The orientation that corresponds to stable equilibrium is the **first orientation**, where the magnetic moment is parallel to the magnetic field.