Two concentric circular loops, one of radius R and the other of radius 2R, lie in the x y-plane with the origin as their common center, as shown in the figure. The smaller loop carries current I₁ in the anti-clockwise direction and the larger loop carries current I₂ in the clockwise direction, with I₂>2 I₁,{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) correct? [JEE Adv 2021 P2]

$\overline{B}(x,y)$ is perpendicular to the $x y$ -plane at any point in the plane
$|\vec{B}(x,y)|$ depends on $x$ and $y$ only through the radial distance $r-\sqrt{x^{2}+y^{2}}$
$|\overline{B}(x,y)|$ is non-zero at all point for $r < R$
$\vec{B}(x,y)$ points normally outward from the $x y$ -plane for all the points between the two loops

Solution:

As per Biot-savart's law,

d\vec{B}=\;\frac{\mu_0 i \overrightarrow{d\ell} \times \vec{r}}{4 \pi r^{3}}

i.e.,

\vec{B}

is perpendicular to both

i \overrightarrow{d\ell}

and

\vec{r}

\overrightarrow{d\ell}

is in

x y

plane

\vec{r}

and is also in

x y

plane

\therefore\;\; d\vec{B}

is perpendicular to

x y

plane

Due to symmetry it depends only on distance form cantre, radial distance

r=\sqrt{x^{2}+y^{2}}

At centre

B_{1}=\;\frac{\mu_0 I_1}{2R}

and

B_{2}=\;\frac{\mu_0 I_2}{4R}

clearly

B_{2}>B_{1}

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.

Field will be in opposite direction inside and outside the loop.

Go to Question:

Prev Question

Next Question