Two concentric circular loops, one of radius and the other of radius , lie in the -plane with the origin as their common center, as shown in the figure. The smaller loop carries current in the anti-clockwise direction and the larger loop carries current

Question

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) correct [2021]

Smart Achievers · Accepted Answer

(1, 2)

As per Biot--Savart's law,

$d \vec{B} = \frac{μ_{0} i d \vec{ℓ} \times \vec{r}}{4 π r^{3}}$

i.e., $\vec{B}$ is perpendicular to both $i d \vec{ℓ}$ and $x y$

$d \vec{ℓ}$ is in the $x y$ -plane and $\vec{r}$ is also in the $x y$ -plane.

$∴$ $d \vec{B}$ is perpendicular to the $x y$ -plane.

Due to symmetry, it depends only on the distance from the centre.

Radial distance $r = \sqrt{x^{2} + y^{2}}$

At the centre, $B_{1} = \frac{μ_{0} I_{1}}{2 R}$ and $B_{2} = \frac{μ_{0} I_{2}}{4 R}$

Clearly, $B_{2} > B_{1}$

As we approach towards the first loop, $B_{1}$ increases to infinity; hence $B_{1}$ dominates.

So, it would be zero at some point between the inner loops and the centre.

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

Solution

1 $\vec{B} (x, y)$ is perpendicular to the $x y$ -plane at any point in the plane.

2 $| \vec{B} (x, y) |$ depends on $x$ and $y$ only through the radial distance $r = \sqrt{x^{2} + y^{2}}$

3 $| \vec{B} (x, y) |$ is non-zero at all points for $r < R$

4 $\vec{B} (x, y)$ points normally outward from the $x y$ -plane for all the points between the two loops.

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Solution

1 B→(x,y) is perpendicular to the xy-plane at any point in the plane.

2 |B→(x,y)| depends on x and y only through the radial distance r=x2+y2

3 |B→(x,y)| is non-zero at all points for r<R

4 B→(x,y) points normally outward from the xy-plane for all the points between the two loops.

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1 $\vec{B} (x, y)$ is perpendicular to the $x y$ -plane at any point in the plane.

2 $| \vec{B} (x, y) |$ depends on $x$ and $y$ only through the radial distance $r = \sqrt{x^{2} + y^{2}}$

3 $| \vec{B} (x, y) |$ is non-zero at all points for $r < R$

4 $\vec{B} (x, y)$ points normally outward from the $x y$ -plane for all the points between the two loops.