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 B

The problem involves two concentric circular loops, one of radius \( R \) and the other of radius \( 2R \), lying in the \( xy \)-plane with the origin as their common center. The smaller loop carries a current \( I_1 \) in the anti-clockwise direction, and the larger loop carries a current \( I_2 \) in the clockwise direction. It is given that \( I_2 > 2I_1 \). We are tasked with determining the behavior of the magnetic field \( \vec{B}(x, y) \) at a point \( (x, y) \) in the \( xy \)-plane and its dependence on the radial distance from the center of the loops.

Step 1: Magnetic Field Due to a Current-Carrying Loop

The magnetic field at a point due to a current-carrying loop is determined using the Biot-Savart law. For a loop of radius \( R \) carrying a current \( I \), the magnetic field produced at any point depends on the geometry of the loop, the current in the loop, and the location of the point relative to the loop. For a point on the axis of the loop, the magnetic field is directed along the axis, and its magnitude is influenced by both the radius of the loop and the distance from the loop to the point of interest. However, for points in the plane of the loop, the situation becomes more complex, but the magnetic field can still be calculated using the principle of superposition.

Step 2: Symmetry of the Problem

In this specific problem, we are dealing with two concentric loops. The key feature of this setup is the symmetry of the configuration. The two loops lie in the same plane and share a common center. Since the loops are concentric and circular, the magnetic field at any point in the plane can be considered as the sum of the contributions from each loop. Because both loops are centered at the origin, and the system exhibits radial symmetry, the magnetic field at any point in the \( xy \)-plane will depend on the radial distance \( r \) from the center, but not on the individual coordinates \( x \) and \( y \). This is a consequence of the fact that the system is rotationally symmetric around the origin.

Step 3: Dependence of the Magnetic Field on the Radial Distance

Since both loops are concentric, the magnetic field produced at any point in the plane will have radial symmetry. This means that the magnetic field vector at a given point will point either directly towards or away from the center of the loops, and its magnitude will only depend on the radial distance \( r = \sqrt{x^2 + y^2} \) from the center. The contributions from both loops will add up depending on the distance from each loop, but the resulting magnetic field will only vary with the distance \( r \), not with the individual values of \( x \) or \( y \).

Step 4: Resulting Magnetic Field at Any Point

The magnetic field \( \vec{B}(x, y) \) at any point in the \( xy \)-plane due to the two concentric current-carrying loops depends on the radial distance \( r = \sqrt{x^2 + y^2} \) and is the result of the combined contributions from the two loops. The field produced by each loop is symmetric, and thus the net field depends on the radial distance from the center of the loops alone. This means that the magnitude of the magnetic field at any point will vary with \( r \) but not directly with the specific values of \( x \) and \( y \) individually.

Final Answer:

The magnitude of the magnetic field \( |\vec{B}(x, y)| \) depends on \( x \) and \( y \) only through the radial distance \( r = \sqrt{x^2 + y^2} \). This means that the magnetic field is a function of the distance from the center of the loops and does not depend on the individual coordinates \( x \) and \( y \).