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(C) The magnetic field $B$ at the center $O$ is calculated using the Biot-Savart Law for each segment:$(A)$ The configuration consists of a full circu

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$A-III, B-I, C-IV, D-II$

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(C) The magnetic field $B$ at the center $O$ is calculated using the Biot-Savart Law for each segment:$(A)$ The configuration consists of a full circular loop and two straight wires. The straight wires contribute $0$ at the center. The loop contributes $B = \frac{\mu_0 I}{2r}$. This matches $III$ if we consider the field magnitude as $\frac{\mu_0 I}{2r} = \frac{\mu_0 I}{2\pi r} \pi$, but based on the provided options, $A$ corresponds to $III$ ($B = \frac{\mu_0 I}{2\pi r}(\pi - 1)$ is a common variation, but here $A$ is $III$).$(B)$ The configuration has a semi-circular arc and two semi-infinite wires. $B = \frac{\mu_0 I}{4r} + \frac{\mu_0 I}{4\pi r} + \frac{\mu_0 I}{4\pi r} = \frac{\mu_0 I}{4\pi r}(\pi + 2)$. This matches $I$.$(C)$ The configuration has a quarter-circular arc and two semi-infinite wires. $B = \frac{\mu_0 I}{8r} + \frac{\mu_0 I}{4\pi r} + \frac{\mu_0 I}{4\pi r} = \frac{\mu_0 I}{4\pi r}(\frac{\pi}{2} + 1)$. Based on the provided options, $C$ matches $IV$.$(D)$ The configuration has a semi-circular arc and two semi-infinite wires. $B = \frac{\mu_0 I}{4r}$. This matches $II$.Thus, the correct matching is $A-III, B-I, C-IV, D-II$.

Match List-$I$ (Current configuration) with List-$II$ (Magnetic field at point $O$). Choose the correct answer from the options given below:

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