Two concentric circular loops, one of radius $R$ and the other of radius $2 R$, lie in the $x y$-plane with the origin as their common center, as shown in the figure. The smaller loop carries current $I_1$ in the anti-clockwise direction and the larger loop carries current $I_2$ in the clockwise direction, with $I_2>2 I_1 . \vec{B}(x, y)$ denotes the magnetic field at a point $(x, y)$ in the $x y$-plane. Which of the following statement($s$) is(are) current?
$(A)$ $\vec{B}(x, y)$ is perpendicular to the $x y$-plane at any point in the plane
$(B)$ $|\vec{B}(x, y)|$ depends on $x$ and $y$ only through the radial distance $r=\sqrt{x^2+y^2}$
$(C)$ $|\vec{B}(x, y)|$ is non-zero at all points for $r$
$(D)$ $\vec{B}(x, y)$ points normally outward from the $x y$-plane for all the points between the two loops
IIT 2021, Advanced
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$(A)$ $d \overrightarrow{ B }=\frac{\mu_0 1 \overrightarrow{ d \ell} \times \overrightarrow{ r }}{4 \pi I ^5}$
$\overrightarrow{ d \ell}$ is in xy plane & $\overrightarrow{ r }$ is also in xy plane
so $d \overrightarrow{ B }$ is perpendicular to xy plane
$(B)$ Due to symmetry it depends only on $r=\sqrt{x^2+y^2}$
$(C)$ At centre $B_1=\frac{\mu_0 I_1}{2 R} ; B_2=\frac{\mu_0 I_2}{4 R} \Rightarrow B_2>B_1$
bu as we approach towards first loop $B_1$ increases to infinity hence $B_1$ dominates.
So it would be zero at some point between inner loops and centre.
Ans. $(A,B)$
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