Solution

Our x axis is along the wire with the origin at the midpoint. The current flows in the positive x direction. All segments of the wire produce magnetic fields at $$P_1$$ that are out of the page. According to the Biot-Savart law, the magnitude of the field any (infinitesimal) segment produces at $$P_1$$ is given by $$dB=\dfrac{\mu_{0}i}{4\pi}\dfrac{sin{\theta}}{r^2}dx$$ where $$\theta$$ (the angle between the segment and a line drawn from the segment to $$P_{1}$$ and $$r$$ (the length of that line) are functions of $$x$$. Replacing $$r$$ with $$\sqrt{x^2+R^2}$$ with $$sin{\theta}$$ with $$R/r=R/\sqrt{x^2+R^2}$$, we integrate from $$x=-L/2$$ to $$x=L/2$$. The total field is $$B=\dfrac{\mu_{0}iR}{4\pi} \int_{-L/2}^{L/2}\dfrac{dx}{(x^2+R^2)^{3/2}}=\dfrac{\mu_{0}iR}{4\pi}\dfrac{1}{R^2}\dfrac{x}{(x^2+R^2)^{1/2}} \bigg|_{-L/2}^{L/2}=\dfrac{\mu_{0}i}{2\pi R}\dfrac{L}{\sqrt{L^2+4R^2}}$$=\dfrac{(4\pi\times 10^{-7} T.m/A)(0.0582A)}{2\pi (0.131m)}\dfrac{0.180m}{\sqrt{(0.180m)^2+4(0.131m)^2}}=5.03\times 10^{-8}T$$