Animated Solution for Physics - Magnetic Effects of Current: 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 $\mathbf{B}$ is (are) true?

The problem presents a beautiful 3D arrangement of current-carrying elements: two infinitely long straight wires and a circular ring. Our goal is to analyze the magnetic field $\mathbf{B}$ at specific points in space under different current conditions. Let's break down the physics step-by-step.

We have two straight wires lying in the $xy$-plane at $x=+R$ and $x=-R$. A positive current in these wires means it flows in the $+\hat{j}$ direction. Above them, a circular ring of radius $R$ is suspended at a height of $z=\sqrt{3}R$, carrying a current $I$ in the clockwise direction when viewed from above.

The first crucial observation is the magnetic field produced by the ring at the origin $(0,0,0)$. Using the Right-Hand Thumb Rule, if we curl our fingers in the clockwise direction, our thumb points downwards. Therefore, the magnetic field due to the ring at the origin, ${\mathbf{B}}_{\text{ring}}$, is directed along the $-\hat{k}$ direction. This field is a constant non-zero vector in our analysis.

Now, let's test the given options by analyzing the magnetic field produced by the two straight wires at the origin, ${\mathbf{B}}_{\text{wires}}$.

Option (a): $I_1=I_2$ If both wires carry equal currents in the same direction, their magnetic fields at the origin will perfectly cancel each other out. Wire 1 (at $x=+R$) produces a field in the $+\hat{k}$ direction, while Wire 2 (at $x=-R$) produces a field in the $-\hat{k}$ direction. However, the net magnetic field is the sum of the fields from the wires and the ring. Since $\mathbf{B}_{\text{ring}} eq 0$, the net field $\mathbf{B}$ cannot be zero. Thus, Option (a) is correct.

Option (b): $I_1>0$ and $I_2<0$ Here, Wire 1 carries current in the $+\hat{j}$ direction, producing a field in the $+\hat{k}$ direction at the origin. Wire 2 carries current in the $-\hat{j}$ direction, which also produces a field in the $+\hat{k}$ direction at the origin. Therefore, ${\mathbf{B}}_{\text{wires}}$ points in the $+\hat{k}$ direction. Since ${\mathbf{B}}_{\text{ring}}$ points in the $-\hat{k}$ direction, it is entirely possible for these two opposing fields to have equal magnitudes and cancel each other out, resulting in a net zero field. Thus, Option (b) is correct.

Option (c): $I_1<0$ and $I_2>0$ In this scenario, the currents are reversed compared to option (b). Both wires now produce magnetic fields in the $-\hat{k}$ direction at the origin. Since ${\mathbf{B}}_{\text{ring}}$ is also in the $-\hat{k}$ direction, all three magnetic field vectors point downwards. They will add up, meaning the net magnetic field can never be zero. Thus, Option (c) is incorrect.

Option (d): $I_1=I_2$ We need to find the $z$-component of the net magnetic field at the center of the loop, $C(0,0,\sqrt{3}R)$. Let's look at the symmetry of the setup. The two wires are placed symmetrically at $x=+R$ and $x=-R$, and they carry equal currents. The magnetic field vectors they produce at point $C$ will tilt symmetrically. Specifically, the $z$-component of the field from Wire 1 will be exactly equal and opposite to the $z$-component of the field from Wire 2. They perfectly cancel each other out! Therefore, the only contribution to the $z$-component of the magnetic field at $C$ comes from the ring itself. The magnetic field at the center of a current-carrying ring is given by: So, the $z$-component is indeed $-\frac{{\mu }_{0}I}{2R}$. Thus, Option (d) is correct.

The correct statements are (a), (b), and (d).