Two concentric coplanar circular loops of radii ' $r_1$ ' and ' $r_2$ ' respectively carry currents '…
Two concentric coplanar circular loops of radii ' $r_1$ ' and ' $r_2$ ' respectively carry currents ' $\mathrm{i}_1$ ' and ' $\mathrm{i}_2$ ' in opposite directions (one clockwise and other anticlockwise). The magnetic induction at the center of the loops is half that due to ' $\mathrm{i}_1$ ' alone at the center. If $r_2=2 r_1$, the value of $\frac{i_2}{i_1}$
$\frac{1}{4}$
1
2
$\frac{1}{2}$
Solution
Magnetic field due to current $i_1$ is given by
$B_1=\frac{\mu_0 i_1}{2 r_1}$
Similarly, $B_2=\frac{\mu_0 i_2}{2 r_2}$
The resultant field at the center, $\mathrm{B}=\mathrm{B}_1-\mathrm{B}_2$
It is given that $B=\frac{B_1}{2}$
$\begin{aligned}
& \therefore \frac{\mathrm{B}_1}{2}=\mathrm{B}_1-\mathrm{B}_2 \\
& \therefore \mathrm{B}_2=\frac{\mathrm{B}_1}{2} \\
& \therefore \frac{\mu_0 \mathrm{i}_2}{2 \mathrm{r}_2}=\frac{1}{2}\left(\frac{\mu_0 \mathrm{i}_1}{2 \mathrm{r}_1}\right) \\
& \therefore \frac{\mathrm{i}_2}{\mathrm{i}_1}=\frac{1}{2} \cdot \frac{\mathrm{r}_2}{\mathrm{r}_1}=\frac{1}{2} \cdot 2=1
\end{aligned}$