A uniform magnetic field B of 0.3 T is along the positive Z-direction. A rectangular loop (abcd) 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 the rectangular loop corresponds to stable equilibrium in a uniform magnetic field, we can follow these steps:

### Step 1: Understand the Concept of Magnetic Moment and Torque
The magnetic moment (\( \mathbf{m} \)) of a current-carrying loop is given by:
\[ \mathbf{m} = I \cdot \mathbf{A} \]
where \( I \) is the current and \( \mathbf{A} \) is the area vector of the loop. The torque (\( \mathbf{\tau} \)) experienced by the loop in a magnetic field (\( \mathbf{B} \)) is given by:
\[ \mathbf{\tau} = \mathbf{m} \times \mathbf{B} \]
For stable equilibrium, the torque must be zero, which occurs when the angle \( \theta \) between \( \mathbf{m} \) and \( \mathbf{B} \) is zero degrees (i.e., they are parallel).

### Step 2: Analyze the Given Magnetic Field
The magnetic field \( \mathbf{B} \) is given as \( 0.3 \, \text{T} \) in the positive Z-direction. This means:
\[ \mathbf{B} = 0.3 \, \hat{k} \, \text{T} \]

### Step 3: Determine the Magnetic Moment Direction for Each Orientation
1. **Orientation A**: Current flows anticlockwise when viewed from above.
- The area vector \( \mathbf{A} \) will point in the positive X-direction.
- The angle \( \theta \) between \( \mathbf{m} \) (in X-direction) and \( \mathbf{B} \) (in Z-direction) is \( 90^\circ \).
- **Not stable equilibrium** (Torque is non-zero).

2. **Orientation B**: Current flows anticlockwise when viewed from below.
- The area vector \( \mathbf{A} \) will point in the positive Z-direction.
- The angle \( \theta \) between \( \mathbf{m} \) (in Z-direction) and \( \mathbf{B} \) (in Z-direction) is \( 0^\circ \).
- **Stable equilibrium** (Torque is zero).

3. **Orientation C**: Current flows clockwise when viewed from above.
- The area vector \( \mathbf{A} \) will point in the negative X-direction.
- The angle \( \theta \) between \( \mathbf{m} \) (in -X-direction) and \( \mathbf{B} \) (in Z-direction) is \( 90^\circ \).
- **Not stable equilibrium** (Torque is non-zero).

4. **Orientation D**: Current flows clockwise when viewed from below.
- The area vector \( \mathbf{A} \) will point in the negative Z-direction.
- The angle \( \theta \) between \( \mathbf{m} \) (in -Z-direction) and \( \mathbf{B} \) (in Z-direction) is \( 180^\circ \).
- **Not stable equilibrium** (Torque is non-zero).

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

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