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. $R_{2}=4 R_{1}$
• B. $R_{2}=2 R_{1}$
• C. $R_{2}=\sqrt{2} R_{1}$
• D. $R_{2}=8 R_{1}$

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}{2 R_{2}} $
Here$ x =\sqrt{3} R_{1}$
$\therefore \frac{R_{1}^{2}}{2\left(R_{1}^{2}+3 R_{1}^{2}\right)^{3 / 2}} =\frac{1}{2 R_{2}} $
$\Rightarrow \frac{R_{1}^{2}}{8 R_{1}^{3}} =\frac{1}{R_{2}} $
$ \Rightarrow R_{2} =8 R_{1} $