Two infinitely long straight wires lie in the $\mathrm{xy}$-plane along the lines $\mathrm{x}=+R$. The wire located at $\mathrm{x}=+\mathrm{R}$ carries a constant current $\mathrm{I}_1$ and the wire located at $\mathrm{x}=-R$ carries a constant current $\mathrm{I}_2$. A circular loop of radius $R$ is suspended with its centre at $(0,0, \sqrt{3} R)$ and in a plane parallel to the xy-plane. This loop carries a constant current $/$ in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the $+\hat{\mathrm{j}}$ direction. Which of the following statements regarding the magnetic field $\dot{B}$ is (are) true?
$(A)$ If $I_1=I_2$, then B' cannot be equal to zero at the origin $(0,0,0)$
$(B)$ If $\mathrm{I}_1>0$ and $\mathrm{I}_2<0$, then $\mathrm{B}$ can be equal to zero at the origin $(0,0,0)$
$(C)$ If $\mathrm{I}_1<0$ and $\mathrm{I}_2>0$, then $\mathrm{B}$ can be equal to zero at the origin $(0,0,0)$
$(D)$ If $\mathrm{I}_1=\mathrm{I}_2$, then the $\mathrm{z}$-component of the magnetic field at the centre of the loop is $\left(-\frac{\mu_0 \mathrm{I}}{2 \mathrm{R}}\right)$
IIT 2018, Advanced
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$(A)$ At the origin, $\mathrm{B}=0$ due to two wires if $\mathrm{I}_1=\mathrm{I}_2$, hence $\overrightarrow{\mathrm{B}}_{\text {net }}$ at the origin is equal to $\mathrm{B}$, due to loop, which is non zero.
$(B)$ If $\mathrm{I}_1>0$ and $\mathrm{I}_2<0$, $\dot{\mathrm{B}}$ at the origin due to wires will be along $+\hat{\mathrm{k}}$ direction and $\dot{\mathrm{B}}$ due to loop is along $-\hat{\mathrm{k}}$ direction, hence $\dot{\mathrm{B}}$ can be zero at the origin.
$(C)$ If $\mathrm{I}_1<0$ and $\mathrm{I}_2>0, \overrightarrow{\mathrm{B}}$ at the origin due to wires is along $-\hat{\mathrm{k}}$ and also due to ring is along $-\hat{k}$ so, $\vec{B}$ can not be zero.
$(D)$ At the centre of the loop B due to wires is along $\mathrm{x}$-axis. Hence the z-component of the magnetic field at the center of the loop is $\left(\frac{\mu_0 I}{2 R}\right)(-\hat{k})$.
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