Calculation of Magnetic Field at Origin due to Current Distribution

To determine the magnetic field at the origin (point O) due to a given current distribution, we need to apply the Biot-Savart law. This law states that the magnetic field at a point in space due to a current element is directly proportional to the magnitude of the current, the length of the current element, and the sine of the angle between the current element and the line connecting the element to the point of interest.

Step 1: Define the Current Distribution
Firstly, we need to understand the given current distribution. It could be a wire carrying a current, a loop, or any other shape. Let's assume that the current distribution is a wire carrying a current I.

Step 2: Divide the Current Distribution into Infinitesimal Current Elements
Next, we divide the current distribution into infinitesimally small current elements. Each element has a length dl and carries a current dI. The direction of dl is along the wire's direction.

Step 3: Determine the Magnetic Field due to Each Current Element
For each infinitesimal current element, we can calculate the magnetic field it produces at the origin using the Biot-Savart law. The formula for the magnetic field due to a current element is:

dB = (μ₀ / 4π) * (dI * sinθ) / r²

Where:
- dB is the magnetic field produced by the current element
- μ₀ is the permeability of free space (a constant equal to 4π × 10^-7 Tm/A)
- dI is the current element magnitude
- θ is the angle between the current element and the line connecting it to the origin
- r is the distance between the current element and the origin

Step 4: Integrate the Contributions from All Current Elements
To determine the total magnetic field at the origin, we integrate the contributions from all the infinitesimal current elements along the entire current distribution. This integration sums up the magnetic fields produced by each element.

Step 5: Evaluate the Magnetic Field at the Origin
After performing the integration, we can evaluate the magnetic field at the origin by summing up the contributions from all the current elements. This will give us the final magnetic field at point O.

Conclusion
By following these steps, we can calculate the magnetic field at the origin (point O) due to the given current distribution. The Biot-Savart law allows us to determine the magnetic field produced by each infinitesimal current element, and integrating these contributions provides the total magnetic field at the origin.