Two infinitely long straight wires lie in the xy - plane along the lines $x=\pm R$. The wire located at $x = +R$ carries a constant current $I_1$ and the wire located at $x =-R$ carries a constant current $I_2$. A circular loop of radius R is suspended with its centre at $\left(0, 0\sqrt{3}R\right)$ and in a plane parallel to the xy - plane. This loop carries a constant current $I$ 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 $+\widehat{j}$ direction. Which of the following statements regarding the magnetic field $\overrightarrow{B}$ is (are) true?

• Correct answer
  A

  If $I_1=I_2, \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n} \overrightarrow{B}$ cannot be equal to zero at the origin $\left(0, 0, 0\right)$
• Correct answer
  B

  If $I_1>0 \mathrm{a}\mathrm{n}\mathrm{d} I_2<0, \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n} \overrightarrow{B}$ can be equal to zero at the origin $\left(0, 0, 0\right)$
• C

  If $I_1<0 \mathrm{a}\mathrm{n}\mathrm{d} I_2>0, \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n} \overrightarrow{B}$ can be equal to zero at the origin $\left(0, 0, 0\right)$
• Correct answer
  D

  If $I_1=I_2$ , Then the z-component of the magnetic field at the centre of the loop is \((-\mu 0 \mathrm{I} / 2 \mathrm{R})\)

Solution

$\left(A\right)$ At origin, $\overrightarrow{B}=0$ due to two wires if $I_1=I_2, hence \left({\overrightarrow{B}}_{net}\right)$ at origin is equal to $\overrightarrow{B}$ due to ring, which is non-zero. $\left(B\right)$ If $I_1>0 \mathrm{a}\mathrm{n}\mathrm{d} I_2<0,\overrightarrow{B}$ at origin due to wires will be along $+\widehat{k}$ direction and $\overrightarrow{B}$ due to ring is along $-\widehat{k}$ direction and hence $\overrightarrow{B}$ can be zero at origin $\left(C\right)$ If $I_1<0 \mathrm{a}\mathrm{n}\mathrm{d} I_2>0,\overrightarrow{B}$ at origin due to wires is along $-\widehat{k}$ and also along $-\widehat{k}$ due to ring, hence $\overrightarrow{B}$ cannot be zero $\left(D\right)$

Hence, the correct answer is options: 'A', 'B', 'D'.