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(D) The magnetic field at the centre of a circular loop of radius $r$ carrying current $i$ is given by $B = \frac{\mu_0 i}{2r}$.For loop $A$,$B_1 = \f

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$\frac{1}{6}$

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(D) The magnetic field at the centre of a circular loop of radius $r$ carrying current $i$ is given by $B = \frac{\mu_0 i}{2r}$.For loop $A$,$B_1 = \frac{\mu_0 i_1}{2r_1}$.For loop $B$,$B_2 = \frac{\mu_0 i_2}{2r_2}$.The ratio of the magnetic flux densities is given as $\frac{B_1}{B_2} = \frac{1}{3}$.Substituting the expressions,we get $\frac{B_1}{B_2} = \frac{i_1}{r_1} 	imes \frac{r_2}{i_2} = \frac{1}{3}$.Given the ratio of radii $\frac{r_1}{r_2} = \frac{1}{2}$,which implies $\frac{r_2}{r_1} = 2$.Substituting this into the ratio equation: $\frac{i_1}{i_2} 	imes 2 = \frac{1}{3}$.Therefore,$\frac{i_1}{i_2} = \frac{1}{3} 	imes \frac{1}{2} = \frac{1}{6}$.

$A$ and $B$ are two concentric circular loops carrying currents $i_1$ and $i_2$ as shown in the figure. If the ratio of their radii is $1:2$ and the ratio of the magnetic flux densities at the centre $O$ due to $A$ and $B$ is $1:3$,then the value of $\frac{i_1}{i_2}$ will be

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