Current flows through uniform square frames as shown. In which case is the magnetic field at the centre of the frame not zero ?

Options

A.
B.
C.
D.

Answer

To determine in which case the magnetic field at the center of a uniform square frame is not zero, we can analyze the situation step by step.

### Step-by-Step Solution:

1. **Understanding the Configuration**:
- We have a uniform square frame through which current is flowing. The magnetic field at the center of the frame depends on the direction of the current in relation to the center point.

2. **Analyzing Case A**:
- In this case, if the current flows along one side of the square frame and we extend that line, the center point lies directly on the same axis as the current.
- According to the Biot-Savart law, the magnetic field produced by a straight current-carrying conductor at a point on its axis is zero.
- **Conclusion**: The magnetic field at the center is zero.

3. **Analyzing Case B**:
- Similar to Case A, if the current flows along another side of the square frame and we extend that line, the center point again lies on the same axis as the current.
- Again, the magnetic field at the center will be zero for the same reason as in Case A.
- **Conclusion**: The magnetic field at the center is zero.

4. **Analyzing Case C**:
- In this case, if the current flows in a way that it creates a loop around the center, the magnetic field at the center can be determined using the right-hand thumb rule.
- The right-hand thumb rule states that if you curl the fingers of your right hand in the direction of the current, your thumb points in the direction of the magnetic field.
- Here, the magnetic field at the center will point inward, and it will have a non-zero value.
- **Conclusion**: The magnetic field at the center is not zero.

5. **Analyzing Case D**:
- If there are two wires carrying current in opposite directions, the magnetic fields produced by each wire at the center will cancel each other out.
- The net magnetic field at the center will be zero because the magnitudes are equal and the directions are opposite.
- **Conclusion**: The magnetic field at the center is zero.

### Final Answer:
The magnetic field at the center of the square frame is not zero in **Case C**.