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 centre, 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 clock wise 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?

• A. $\vec{ B }( x , y )$ is perpendicular to the $xy$-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 $r < R$
• D. $\vec{ B }( x , y )$ points normally outward from the $xy$-plane for all the points between the two loops

Answer 1

The Correct Option is A, B

Step 1: Given Information
Two concentric circular loops are given, one with radius \( R \) and the other with radius \( 2R \). The loops are in the \( xy \)-plane, sharing the origin as their common center.
- The smaller loop carries current \( I_1 \) in the anti-clockwise direction.
- The larger loop carries current \( I_2 \) in the clockwise direction, and we are given that \( I_2 > 2I_1 \).
- We are asked to determine the magnetic field \( \vec{B}(x, y) \) at a point \( (x, y) \) in the \( xy \)-plane.

Step 2: Magnetic Field Due to Circular Current Loop
The magnetic field at a point due to a current-carrying circular loop is given by the Biot-Savart law. The field produced by each loop at any point in the plane of the loop depends on the distance from the point to the center of the loop.
- For a point in the plane of the loop, the magnetic field produced by each loop is perpendicular to the plane of the loop, pointing either upward or downward, depending on the direction of the current.
- The contribution of each loop to the magnetic field at any point depends on the radial distance \( r = \sqrt{x^2 + y^2} \) from the center.

Step 3: Analyzing Statement A
We are asked to verify whether \( \vec{B}(x, y) \) is perpendicular to the \( xy \)-plane at any point in the plane.
- The magnetic field produced by each loop is directed along the axis perpendicular to the plane of the loop, which is the \( z \)-axis in this case.
- Therefore, the magnetic field at any point in the \( xy \)-plane will be along the \( z \)-axis, meaning it will be perpendicular to the \( xy \)-plane.
Hence, statement (A) is correct.

Step 4: Analyzing Statement B
We are asked to verify whether \( |\vec{B}(x, y)| \) depends on \( x \) and \( y \) only through the radial distance \( r = \sqrt{x^2 + y^2} \).
- From the Biot-Savart law and the symmetry of the problem, the magnetic field at any point in the \( xy \)-plane due to both loops will depend only on the radial distance \( r \) from the center of the loops.
- Since the magnetic field produced by each loop is a function of the radial distance \( r \), and because the loops are symmetric about the origin, the field will only depend on \( r \), which is a function of \( x \) and \( y \) as \( r = \sqrt{x^2 + y^2} \).
Hence, statement (B) is correct.

Final Answer:
Both statements (A) and (B) are correct:
- (A) \( \vec{B}(x, y) \) is perpendicular to the \( xy \)-plane at any point in the plane.
- (B) \( |\vec{B}(x, y)| \) depends on \( x \) and \( y \) only through the radial distance \( r = \sqrt{x^2 + y^2} \).