A nonpolar loop of conducting wire carrying a current i is placed as shown. Each of straight section of loop is length L mahnetic field at point P(L/2,L/2,0) is

To determine the magnetic field at point P due to a nonpolar loop of conducting wire carrying a current, we can use the Biot-Savart Law. This law helps us calculate the magnetic field generated by a current-carrying conductor at a specific point in space.

Understanding the Setup

Imagine a square loop of wire lying in the xy-plane, with each side of length L. The current i flows through the wire, and we want to find the magnetic field at the point P, located at coordinates (L/2, L/2, 0). This point is at the center of the loop.

Applying the Biot-Savart Law

The Biot-Savart Law states that the magnetic field dB at a point in space due to a small segment of current-carrying wire is given by:

• dB = (μ₀/4π) * (i * dl × r̂) / r²

Here, μ₀ is the permeability of free space, dl is the length vector of the wire segment, r̂ is the unit vector pointing from the wire segment to the point where we are calculating the field, and r is the distance from the wire segment to that point.

Calculating Contributions from Each Segment

For our square loop, we can break it down into four straight segments. Let's analyze the contributions from each segment:

• Segment 1 (Bottom Side): From (0, 0) to (L, 0)
• Segment 2 (Right Side): From (L, 0) to (L, L)
• Segment 3 (Top Side): From (L, L) to (0, L)
• Segment 4 (Left Side): From (0, L) to (0, 0)

For each segment, we can calculate the magnetic field contribution at point P. However, due to symmetry, the contributions from the top and bottom segments will cancel out in the x-direction, while the left and right segments will add up in the y-direction.

Finding the Total Magnetic Field

Let’s focus on the contributions from the vertical segments (Segment 2 and Segment 4). The distance from these segments to point P is L/2, and the angle between the current direction and the line connecting the segment to point P is 90 degrees. Thus, the sine of the angle is 1.

Calculating the magnetic field from one vertical segment:

• Using the Biot-Savart Law, we find that the contribution from each vertical segment is:
• dB = (μ₀/4π) * (i * L * sin(90°)) / (L/2)² = (μ₀ * i * L) / (2π * (L/2)²) = (2μ₀ * i) / (π * L)

Since there are two vertical segments contributing to the magnetic field at point P, we multiply this result by 2:

• Total B = 2 * (2μ₀ * i) / (π * L) = (4μ₀ * i) / (π * L)

Final Result

The magnetic field at point P due to the current-carrying loop is:

B = (4μ₀ * i) / (π * L)}

This result shows how the geometry of the loop and the current flowing through it directly influence the magnetic field at a specific point in space. The symmetry of the loop simplifies the calculations, allowing us to focus on the contributions from each segment effectively.