Magnetic Field Due to Two Concentric Circular Loops

To determine the correct options, let us analyze the magnetic field $\vec{B}(x, y)$ produced by the two concentric circular loops placed in the $xy$-plane with their common center at the origin $(0, 0)$.

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1. Direction of Magnetic Field (Option A)

By the Biot-Savart Law, the magnetic field at a position vector $\vec{r} = x\hat{i} + y\hat{j}$ in the $xy$-plane due to a current element $I d\vec{\ell}$ at $\vec{r}' = x'\hat{i} + y'\hat{j}$ is given by: $d\vec{B}(\vec{r}) = \frac{\mu_0 I}{4\pi} \frac{d\vec{\ell} \times (\vec{r} - \vec{r}')}{|\vec{r} - \vec{r}'|^3}$

Since both the loops and the position vector $\vec{r}$ lie entirely in the $xy$-plane: $d\vec{\ell} = dx'\hat{i} + dy'\hat{j}$ $\vec{r} - \vec{r}' = (x - x')\hat{i} + (y - y')\hat{j}$

The cross product $d\vec{\ell} \times (\vec{r} - \vec{r}')$ has components only along the $z$-axis ($\hat{k}$). Integrating over both loops gives: $\vec{B}(x, y) = B_z(x, y) \hat{k}$

Thus, $\vec{B}(x, y)$ is oriented perpendicular to the $xy$-plane at every point in the plane.

Option A is CORRECT.

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2. Radial Symmetry (Option B)

Both loops are circular and centered at the origin $(0, 0)$. The current distribution possesses continuous rotational symmetry about the $z$-axis.

Consequently, the magnitude and direction of the magnetic field depend solely on the distance $r = \sqrt{x^2 + y^2}$ from the origin: $|\vec{B}(x, y)| = B(r)$

Option B is CORRECT.

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3. Magnetic Field inside the Smaller Loop ($r < R$) (Option C)

Let the unit vector perpendicular to the $xy$-plane pointing outwards be $+\hat{k}$.

• The smaller loop carries an anti-clockwise current $I_1$, producing a magnetic field in the $+\hat{k}$ direction inside $r < R$.
• The larger loop carries a clockwise current $I_2$, producing a magnetic field in the $-\hat{k}$ direction inside $r < 2R$.

At the center $(r = 0)$, the net magnetic field is: $B_z(0) = \frac{\mu_0 I_1}{2R} - \frac{\mu_0 I_2}{2(2R)} = \frac{\mu_0}{2R} \left( I_1 - \frac{I_2}{2} \right)$

Given that $I_2 > 2I_1$, we have $I_1 - \frac{I_2}{2} < 0$, which implies: $B_z(0) < 0 \quad \implies \quad \vec{B}(0) \text{ points along } -\hat{k}$

As $r \to R^-$, the point approaches the wire of the smaller loop carrying current $I_1$. The contribution from $I_1$ diverges to $+\infty$ along $+\hat{k}$, while the contribution from $I_2$ remains finite: $\lim_{r \to R^-} B_z(r) = +\infty$

Since $B_z(r)$ is a continuous function of $r$ on the interval $[0, R)$, by the Intermediate Value Theorem, there must exist at least one radial distance $r_0 \in (0, R)$ where: $B_z(r_0) = 0 \quad \implies \quad |\vec{B}(r_0)| = 0$

Therefore, $|\vec{B}(x, y)|$ does NOT remain non-zero for all points satisfying $r < R$.

Option C is INCORRECT.

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4. Magnetic Field between the Loops ($R < r < 2R$) (Option D)

For any point in the region $R < r < 2R$:

1. The point lies outside the inner loop of radius $R$. The anti-clockwise current $I_1$ produces a magnetic field directed into the plane ($-\hat{k}$) in the region outside its radius ($r > R$).
2. The point lies inside the outer loop of radius $2R$. The clockwise current $I_2$ produces a magnetic field directed into the plane ($-\hat{k}$) in the region inside its radius ($r < 2R$).

Since both magnetic field contributions are directed into the plane ($-\hat{k}$), the net field $\vec{B}(x, y)$ points normally inwards (into the plane), not outwards.

Option D is INCORRECT.

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Conclusion

The correct options are A and B.