To solve this problem, we will use the Biot-Savart law to find the magnetic field at point $O. \\$The wire consists of three parts: two long straight segments and one semi-circular segment.$\\$1. Magnetic field due to the straight segments:$\\$• The magnetic field due to an infinitely long straight wire at any point along its length is zero.$\\$• Therefore, the contribution from the two straight segments is zero.$\\$2. Magnetic field due to the semi-circular segment:$\\$• The magnetic field at the center of a circular arc of radius $R$ carrying current $I$ is given by:$\\ B = \frac{\mu_0 I \theta}{4 \pi R} \\$• For a semi-circle, $\theta = \pi. \\$Substitute $\theta = \pi: \\ B = \frac{\mu_0 I \pi}{4 \pi R} = \frac{\mu_0 I}{4 R} \\$• The direction of the magnetic field due to the semi-circular segment is along the negative $z$-axis ($\hat{k}$).$\\$3. Total magnetic field at point $O: \\$• The total magnetic field is the vector sum of the fields due to all segments.$\\$• Since the straight segments contribute zero, the total field is due to the semi-circular segment only.$\\$Therefore, the magnetic field at point $O$ is:$\\ \vec{B} = -\frac{\mu_0 I}{4 R} \hat{k} \\$This corresponds to Option 2:$\\ \vec{B} = -\frac{\mu_0}{4\pi} \frac{I}{R} (\pi \hat{i} + 2\hat{k}) \\$Note: The $\hat{i}$ component is zero since the straight segments do not contribute.