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(A) The magnetic field at the origin due to the wire at $x=R$ is $\vec{B}_1 = \frac{\mu_0 I_1}{2 \pi R} (\hat{k})$ and due to the wire at $x=-R$ is $\

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$A, B, D$

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(A) The magnetic field at the origin due to the wire at $x=R$ is $\vec{B}_1 = \frac{\mu_0 I_1}{2 \pi R} (\hat{k})$ and due to the wire at $x=-R$ is $\vec{B}_2 = \frac{\mu_0 I_2}{2 \pi R} (-\hat{k})$.$(A)$ If $I_1=I_2$, the magnetic field due to the wires at the origin is $\vec{B}_{wires} = \vec{B}_1 + \vec{B}_2 = 0$. However, the circular loop creates a non-zero magnetic field at the origin. Thus, the net magnetic field $\vec{B}_{net} 
eq 0$. Statement $(A)$ is true.$(B)$ If $I_1 > 0$ and $I_2  0$ and $I_2  0$, so $\vec{B}_{wires}$ is along $+\hat{k}$. The magnetic field due to the loop at the origin is along $-\hat{k}$. Since both fields are along the $z$-axis, they can cancel each other out. Statement $(B)$ is true.$(C)$ If $I_1  0$, $\vec{B}_{wires}$ is along $-\hat{k}$. The magnetic field due to the loop at the origin is also along $-\hat{k}$. They cannot cancel each other out. Statement $(C)$ is false.$(D)$ The magnetic field due to the wires at the center of the loop $(0, 0, \sqrt{3}R)$ is purely along the $x$-axis. The $z$-component of the magnetic field at the center is solely due to the loop, which is $\left(-\frac{\mu_0 I}{2 R}ight)$. Statement $(D)$ is true.

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