Analysis of the Problem

We have two concentric circular current loops in the xy-plane, centered at the origin.

• Loop 1 (smaller): Radius $R$, current $I_1$ (anti-clockwise).
• Loop 2 (larger): Radius $2R$, current $I_2$ (clockwise).
• Given condition: $I_2 > 2I_1$.
• The magnetic field at a point $(x, y)$ in the xy-plane is denoted by $\overrightarrow B(x, y)$.

We will evaluate each statement systematically.

Evaluation of Option A

Statement A: $\overrightarrow B(x, y)$ is perpendicular to the xy-plane at any point in the plane.

1. Biot-Savart Law: The magnetic field element $d\overrightarrow B$ produced by a current element $I d\overrightarrow l$ at a position $\overrightarrow r'$ is given by: $d\overrightarrow B = \frac{{\mu_0}}{{4\pi}} \frac{{I d\overrightarrow l \times (\overrightarrow r - \overrightarrow r')}}{{|\overrightarrow r - \overrightarrow r'|^3}}$ where $\overrightarrow r$ is the position vector of the point where the field is being calculated.

2. Applying to the setup:

   • Both current loops lie in the xy-plane. This means any current element $d\overrightarrow l$ for either loop is a vector in the xy-plane.
   • The point of observation $(x, y)$ is also in the xy-plane, so its position vector $\overrightarrow r = x\hat i + y\hat j$ is in the xy-plane.
   • The position vector of the current element $\overrightarrow r'$ is also in the xy-plane.
   • Therefore, the vector difference $(\overrightarrow r - \overrightarrow r')$ is also a vector in the xy-plane.

3. Cross Product: The cross product of two vectors in the xy-plane, $d\overrightarrow l$ and $(\overrightarrow r - \overrightarrow r')$, will result in a vector perpendicular to the xy-plane, i.e., a vector along the z-axis (either $+\hat k$ or $-\hat k$). $d\overrightarrow l \times (\overrightarrow r - \overrightarrow r') = (d\overrightarrow B)_z \hat k$

4. Total Field: The total magnetic field $\overrightarrow B$ is the vector sum (integral) of all such elements $d\overrightarrow B$ from both loops. Since every $d\overrightarrow B$ is directed along the z-axis, their sum $\overrightarrow B$ must also be directed along the z-axis. $\overrightarrow B(x, y) = B_z(x, y) \hat k$ This means the magnetic field is perpendicular to the xy-plane at any point in the plane.

5. Conclusion for A: Statement A is correct.

Evaluation of Option B

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

1. Symmetry: The physical system consists of two concentric circular loops centered at the origin. This configuration has cylindrical symmetry about the z-axis.

2. Effect of Rotation: If we rotate the coordinate system (or the observation point) around the z-axis by any angle, the current distribution looks exactly the same. The physics of the situation is unchanged.

3. Field Magnitude: Consequently, the magnitude of the magnetic field, |$\overrightarrow B$|, cannot depend on the azimuthal angle $\phi$. It can only depend on the distance from the axis of symmetry (the z-axis), which is the radial distance $r = \sqrt{x^2 + y^2}$, and the coordinate along the axis, $z$. Since we are in the xy-plane, $z=0$.

4. Conclusion for B: Therefore, |$\overrightarrow B$(x, y)| depends on x and y only through $r$. Statement B is correct.

Evaluation of Option C

Statement C: |$\overrightarrow B$(x, y)| is non-zero at all points for $r < R$.

1. Field Direction: For $r < R$, the point is inside both loops.

   • The field due to loop 1 ($I_1$, anti-clockwise) points in the +z direction (out of the plane). Let's call its z-component $B_{1z}$.
   • The field due to loop 2 ($I_2$, clockwise) points in the -z direction (into the plane). Let's call its z-component $B_{2z}$ (which will be a negative value).
   • The total field's z-component is $B_z(r) = |B_1(r)| - |B_2(r)|$. (Using magnitudes of fields)

2. Field at the Center (r=0):

   • $|B_1(0)| = \frac{{\mu_0 I_1}}{{2R}}$
   • $|B_2(0)| = \frac{{\mu_0 I_2}}{{2(2R)}} = \frac{{\mu_0 I_2}}{{4R}}$
   • The net field at the center is $B_z(0) = \frac{{\mu_0 I_1}}{{2R}} - \frac{{\mu_0 I_2}}{{4R}} = \frac{{\mu_0}}{{4R}}(2I_1 - I_2)$.
   • Given $I_2 > 2I_1$, the term $(2I_1 - I_2)$ is negative. So, $B_z(0) < 0$. The field at the center is non-zero.

3. Field near the inner loop (r → R⁻):

   • As the observation point approaches the wire of the smaller loop ($r \to R$ from the inside), the magnetic field contribution from this loop, $B_1$, becomes very large and tends to infinity ($|B_1(r)| \to \infty$). The direction is +z.
   • The contribution from the larger loop, $B_2$, remains finite at $r=R$.
   • Therefore, as $r \to R^-$, the net field $B_z(r) = |B_1(r)| - |B_2(r)|$ tends to $+\infty$.

4. Intermediate Value Theorem: The function $B_z(r)$ is continuous for $0 \le r < R$. We have found that $B_z(0)$ is negative and $B_z(r)$ becomes positive and large as $r \to R^-$. By the Intermediate Value Theorem, there must exist a radial distance $r_0$ such that $0 < r_0 < R$ where $B_z(r_0) = 0$.

5. Conclusion for C: The magnetic field is zero at some points for $r<R$. Therefore, statement C is incorrect.

Evaluation of Option D

Statement D: $\overrightarrow B$(x, y) points normally outward from the xy-plane for all the points between the two loops.

1. Region of Interest: This statement concerns the region $R < r < 2R$.

2. Field Directions:

   • For a point outside the smaller loop ($r > R$), the field $\overrightarrow B_1$ due to the anti-clockwise current $I_1$ points into the plane, i.e., in the -z direction.
   • For a point inside the larger loop ($r < 2R$), the field $\overrightarrow B_2$ due to the clockwise current $I_2$ also points into the plane, i.e., in the -z direction.

3. Total Field: Since both $\overrightarrow B_1$ and $\overrightarrow B_2$ point in the -z direction in the region $R < r < 2R$, their vector sum $\overrightarrow B = \overrightarrow B_1 + \overrightarrow B_2$ must also point in the -z direction.

4. Conclusion for D: The field points normally inward, not outward. Therefore, statement D is incorrect.

Final Summary

• Statement A is correct.
• Statement B is correct.
• Statement C is incorrect.
• Statement D is incorrect.

The correct statements are A and B.