Field at center due to very long wire carrying current $dI$

$d B=\frac{\mu_{0}}{2 \pi}=\frac{I}{R}=\frac{\mu_{0}}{4 \pi^{2}} \frac{I}{R} d t h \eta$

$\vec{B}=\int d \vec{B}=\left[\int_{0}^{\pi / 2} d B \cos \theta \hat{i}+\int_{0}^{\pi / 20 i} d B \sin \theta \hat{i}\right]$

$=\frac{\mu_{0}}{4 \pi^{2} R} \frac{I}{R}\left[\int_{0}^{\pi / 2} \cos 0 d t h \eta \hat{i}+\int_{0}^{\pi / 20 i} \sin 0 d \theta \hat{j}\right]$

$\vec{B}=\frac{\mu_{0}}{4 \pi^{2} R} \frac{I}{R}[\hat{i}+\hat{j}] \Rightarrow B=\frac{\mu_{0}}{4 \pi^{2} R} \frac{I}{R} \sqrt{2}$