Two concentric coplanar 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 the loops is half that due to ${i_1}$ alone at the centre. If ${r_2} = 2{r_1}.$ the value of ${I_2}/{I_1}$ is....
Diffcult
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(d) Magnetic field at centre due to smaller loop
${B_1} = \frac{{{\mu _0}}}{{4\pi }}.\frac{{2\pi {i_1}}}{{{r_1}}}$..... $(i)$
Due to Bigger loop ${B_2} = \frac{{{\mu _0}}}{{4\pi }}.\frac{{2\pi {i_2}}}{{{r_2}}}$ So net magnetic field at centre
$B = {B_1} - {B_2} = \frac{{{\mu _0}}}{{4\pi }} \times 2\pi \left( {\frac{{{i_1}}}{{{r_1}}} - \frac{{{i_2}}}{{{r_2}}}} \right)$
According to question $B = \frac{1}{2} \times {B_1}$
$ \Rightarrow \frac{{{\mu _0}}}{{4\pi }}.2\pi \left( {\frac{{{i_1}}}{{{r_1}}} - \frac{{{i_2}}}{{{r_2}}}} \right) = \frac{1}{2} \times \frac{{{\mu _0}}}{{4\pi }}.\frac{{2\pi {i_1}}}{{{r_1}}}$ $\frac{{{i_1}}}{{{r_1}}} - \frac{{{i_2}}}{{{r_2}}} = \frac{{{i_1}}}{{2{r_1}}} \Rightarrow \frac{{{i_1}}}{{2{r_1}}} = \frac{{{i_2}}}{{{r_2}}} \Rightarrow \frac{{{i_1}}}{{{i_2}}} = 1$ $\{ {r_2} = 2{r_1}\} $
${B_1} = \frac{{{\mu _0}}}{{4\pi }}.\frac{{2\pi {i_1}}}{{{r_1}}}$..... $(i)$
Due to Bigger loop ${B_2} = \frac{{{\mu _0}}}{{4\pi }}.\frac{{2\pi {i_2}}}{{{r_2}}}$ So net magnetic field at centre
$B = {B_1} - {B_2} = \frac{{{\mu _0}}}{{4\pi }} \times 2\pi \left( {\frac{{{i_1}}}{{{r_1}}} - \frac{{{i_2}}}{{{r_2}}}} \right)$
According to question $B = \frac{1}{2} \times {B_1}$
$ \Rightarrow \frac{{{\mu _0}}}{{4\pi }}.2\pi \left( {\frac{{{i_1}}}{{{r_1}}} - \frac{{{i_2}}}{{{r_2}}}} \right) = \frac{1}{2} \times \frac{{{\mu _0}}}{{4\pi }}.\frac{{2\pi {i_1}}}{{{r_1}}}$ $\frac{{{i_1}}}{{{r_1}}} - \frac{{{i_2}}}{{{r_2}}} = \frac{{{i_1}}}{{2{r_1}}} \Rightarrow \frac{{{i_1}}}{{2{r_1}}} = \frac{{{i_2}}}{{{r_2}}} \Rightarrow \frac{{{i_1}}}{{{i_2}}} = 1$ $\{ {r_2} = 2{r_1}\} $
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