Biot Savart Law

Step 1: Concept
The total field at O is the sum of the fields from the straight wires and the circular arc.
Step 2: Analysis
- Field from the horizontal semi-infinite wire: . - Field from the 3/4 circular arc: . - The other linear wire passes through O, so its field is zero.
Step 3: Calculation
.
Step 4: Conclusion
Hence, the magnetic induction is .
Final Answer:(D)

Step 1: Understanding the Concept:
This problem requires the application of the Biot-Savart Law to find the magnetic field produced by a small current-carrying element. The Biot-Savart Law gives the magnetic field at a point in space due to a current element.

Step 2: Key Formula or Approach:
The magnitude of the magnetic field due to a current element at a distance is given by the Biot-Savart Law: where is the permeability of free space ( ), is the current, is the length of the element, is the distance from the element to the point, and is the angle between the direction of the current element and the position vector to the point.

Step 3: Detailed Explanation:
Given data:
Current, .
Length of the element, .
Distance to the point on the y-axis, .
The current element is along the x-axis, and the point is on the y-axis. Therefore, the angle between the current element and the position vector is . Thus, .
The constant .
Calculation:
Substitute the values into the Biot-Savart Law formula:

Step 4: Final Answer:
The magnitude of the magnetic field due to the element at the given point is T.