JEE Advanced · 2021 · 03 Oct 2021 · Paper 2 · Magnetic Effects of Current

Using Biot-Savart law: $d\vec{B}=\frac{{\mu }_{0}id\vec{\ell }\times \vec{r}}{4\pi r^3}$

(A) Since both $d\vec{\ell }$ and $\vec{r}$ lie in the xy-plane, their cross product $d\vec{\ell }\times \vec{r}$ is perpendicular to the xy-plane. Therefore $\vec{B}(x,y)$ is perpendicular to the xy-plane at any point. ✓

(B) Due to circular symmetry about the z-axis, the magnetic field depends only on the radial distance $r=\sqrt{x^2+y^2}$ from the center. ✓

(C) At the center (r = 0):


Since $I_2>2I_1$, we have $B_2>B_1$, so the net field at center points downward.

As we move toward the inner loop, $B_1$ increases (approaching infinity at the wire), while $B_2$ decreases. Therefore, there exists a point between the center and inner loop where $B_1=B_2$, making the net field zero. ✗

(D) The direction of $\vec{B}$ between the loops depends on which field dominates. Since $I_2>2I_1$, the field from the outer loop (pointing into the plane by right-hand rule) dominates, so $\vec{B}$ points into the plane, not outward. ✗

Correct answers: (A) and (B)