Two concentric circular loops,one of radius R | vedclass

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(B) Step $1$: Analyze the direction of the magnetic field. According to the Biot-Savart Law,$d\vec{B} = \frac{\mu_0 I}{4\pi} \frac{d\vec{\ell} 	imes

Vedclass · Accepted Answer

$A, B$

Vedclass · Answer

(B) Step $1$: Analyze the direction of the magnetic field. According to the Biot-Savart Law,$d\vec{B} = \frac{\mu_0 I}{4\pi} \frac{d\vec{\ell} 	imes \vec{r}}{r^3}$. Since the current elements $d\vec{\ell}$ lie in the $xy$-plane and the position vector $\vec{r}$ also lies in the $xy$-plane,the cross product $d\vec{\ell} 	imes \vec{r}$ is always perpendicular to the $xy$-plane (i.e.,along the $z$-axis). Thus,statement $(A)$ is correct.Step $2$: Analyze the dependence on coordinates. Due to the circular symmetry of the loops,the magnetic field magnitude at any point $(x, y)$ depends only on the radial distance $r = \sqrt{x^2 + y^2}$ from the origin. Thus,statement $(B)$ is correct.Step $3$: Analyze the magnitude at different points. At the center $(r=0)$,the magnetic field due to the smaller loop is $B_1 = \frac{\mu_0 I_1}{2R}$ (outward) and due to the larger loop is $B_2 = \frac{\mu_0 I_2}{4R}$ (inward). Since $I_2 > 2I_1$,$B_2 > B_1$,so the net field is inward. As we move towards the smaller loop,$B_1$ increases and approaches infinity at $r=R$. Since $B_1$ and $B_2$ have opposite directions,there must be a point where the net magnetic field is zero. Thus,statement $(C)$ is incorrect.Step $4$: Analyze the direction between the loops. Between the loops,the field from the inner loop is inward and the field from the outer loop is also inward. Thus,the net field points inward,not outward. Statement $(D)$ is incorrect.Therefore,the correct statements are $(A)$ and $(B)$.

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