Question

Two concentric circular loops, one of radius $R$ and the other of radius $2R,$ lie in the $xy-$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 $I_2>2I_1$. $\overrightarrow{B}(x,y)$ denotes the magnetic field at a point $(x,y)$ in the $xy-$ plane. Which of the following statement(s) is (are) current ?

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

Answer 1

Answer: (A), (B)

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


Magnetic field at the centre due to circular loops will be,

$B_1=\frac{{\mu }_0{I}_1}{2R};B_2=\frac{{\mu }_0{I}_2}{4R}$

$\because I_2>2I_1$

$\therefore B_2>B_2$

$\left(1\right)$ Magnetic field due to a circular loop at any point in its plane will be perpendicular to the plane, but it may be outwards or inwards depending on the position as shown in the diagram.

$\left(2\right)$ Due to symmetry it will depend only on distance from centre which is $r=\sqrt{x^2+y^2}$.

$\left(3\right)$ Field will be in opposite direction inside and outside the loop, due to opposite sense of flow of currents.

$\left(4\right)$ Field may or may not be non-zero for $r<R$, as it is in opposite direction due to both the loops.

Hence, options $\left(\text{A}\right)$ and $\left(\text{B}\right)$ are the correct answer.