Question

Two circular loops $L_1$ and $L_2$ of wire carrying equal and opposite currents are placed parallel to each other with a common axis. The radius of loop $L_1$ is $R_1$ and that of $L_2$ is $R_2$. The distance between the centres of the loops is $\sqrt{3}{R}_1.$ The magnetic field at the centre of $L_2$ shall be zero, if

• A. $R_2=4R_1$
• B. $R_2=2R_1$
• C. $R_2=\sqrt{2}{R}_1$
• D. $R_2=8R_1$

Answer 1

Answer: (D)

Magnetic field at centre of $L_2$ will be zero when

$B_{L_1}=B_{L_2}$ (magnitude)
$\frac{{\mu }_{0}I{R}_1^2}{2{\left(R_1^2+x^2\right)}^{3/2}}=\frac{{\mu }_{0}I}{2R_2}$
Here$x=\sqrt{3}{R}_1$
$\therefore \frac{R_1^2}{2{\left(R_1^2+3R_1^2\right)}^{3/2}}=\frac{1}{2R_2}$
$\Rightarrow \frac{R_1^2}{8R_1^3}=\frac{1}{R_2}$
$\Rightarrow R_2=8R_1$