Question

Two concentric co-planar circular loops of radii $r_1$ and $r_2$ carry currents of respectively $i_1$ and $i_2$ in opposite directions (one clockwise and the other anticlockwise.) The magnetic induction at the centre of loops is half that due to $i_1$ alone at the centre. If $r_2=2r_1$. the value of $i_2/i_1$ is

• A. $2$
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $1$

Answer 1

The correct option is D $1$

$\begin{array}{c}B=\frac{{\mu }_o{i}_1}{2r_1}-\frac{{\mu }_o{i}_2}{2\left(2r_1\right)}\\ =\frac{{\mu }_o{i}_1}{2r_1}-\frac{{\mu }_o{i}_2}{4r_2}\\ B^{{}^{\prime }}=\frac{{\mu }_o{i}_1}{2r_1}\\ \Rightarrow B=\frac{B^{{}^{\prime }}}{2}\\ \Rightarrow \frac{{\mu }_o{i}_1}{2r_1}-\frac{{\mu }_o{i}_2}{4r_1}=\frac{{\mu }_o{i}_1}{4r_1}\\ \Rightarrow \frac{{\mu }_o{i}_1}{4r_1}=\frac{{\mu }_o{i}_2}{4r_1}\\ \Rightarrow \frac{i_1}{i_2}=1\\ Hence,\\ optionDiscorrectanswer.\end{array}$