Find the magnetic field at the point P in figure. The curved portion is a semicircle and the straight wires are long

To find the magnetic field at point P, we analyze the contributions from each segment of the current-carrying wire: the two long straight wires and the semicircular arc.

Step 1: Magnetic Field Due to the Straight Wires

The magnetic field produced by a long, straight current-carrying wire at a perpendicular distance d is given by:

B = (μ₀ i) / (2πd)

For the upper straight wire, the distance from point P is d. The magnetic field at point P, B₁, due to this wire is:

B₁ = μ₀ i / (2πd)

Similarly, the lower straight wire contributes an equal magnetic field, B₂, since it is at the same distance d from point P. Therefore:

B₂ = μ₀ i / (2πd)

Step 2: Magnetic Field Due to the Semicircular Arc

To find the magnetic field at point P due to the semicircular arc of radius d/2, we use the formula for the magnetic field at the center of a current-carrying circular arc:

B = (μ₀ i θ) / (4πR)

Here, θ is the angle subtended by the arc at the center (π radians for a semicircle), and R is the radius (d/2). Substituting these values:

B_a = μ₀ i / (2d)

Step 3: Net Magnetic Field at Point P

To find the magnetic field at point P, we sum up the contributions from all segments. Since all magnetic fields are perpendicular to the plane and directed outwards, they add up:

B = B₁ + B₂ + B_a

Substituting the values:

B = (μ₀ i / 2πd) + (μ₀ i / 2πd) + (μ₀ i / 2d)

Simplifying further:

B = (μ₀ i / d) (1 + 1/π)

Conclusion

Thus, the net magnetic field at point P is given by:

B = (μ₀ i / d) (1 + 1/π)

This completes our calculation to find the magnetic field at point P.