Question

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 $(0,0,\sqrt{3}R)$ 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 $+\hat{j}$ direction. Which of the following statements regarding the magnetic field $\vec{B}$ is (are) true?

• A. If $I_1=I_2$, then $\vec{B}$ cannot be equal to zero at the origin (0, 0, 0)
• B. If $I_1>0$ and $I_2<0$, then $\vec{B}$ can be equal to zero at the origin (0, 0, 0)
• C. If $I_1<0$ and $I_2>0$, then $\vec{B}$ can be equal to zero at the origin (0, 0, 0)
• D. If $I_1=I_2$, then the z-component of the magnetic field at the centre of the loop is $(-\frac{{\mu }_{0}I}{2R})$

Answer 1

Answer: (A), (B), (D)

(A) At origin, $\vec{B}=0$ due to two wires if $I_1=I_2$ , hence $({\vec{B}}_{net})$ at origin is equal to $\vec{B}$ due to ring, which is non-zero.
(B) If $I_1>0$ and $I_2<0,\vec{B}$ at origin due to wires will be along $+\hat{k}$ direction and $\vec{B}$ due to ring is along $-\hat{k}$ direction and hence $\vec{B}$ can be zero at origin.
(C) If $I_1<0$ and $I_2>0,\vec{B}$ at origin due to wires is along $-\hat{k}$ and also along $-\hat{k}$ due to ring $-\hat{k}$ and also along $-\hat{k}$ due to ring, hence $\vec{B}$ cannot be zero.
(D) ....
At centre of ring, $\vec{B}$ due to wires is along x -axis,
hence z-component is only because of ring which $\vec{B}=\frac{{\mu }_{0}i}{2R}(-\hat{k})$