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(B) The magnetic field at the center of a circular loop of radius $r$ carrying current $i$ is given by $B = \frac{\mu_0 i}{2r}$.For the two loops,the

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(B) The magnetic field at the center of a circular loop of radius $r$ carrying current $i$ is given by $B = \frac{\mu_0 i}{2r}$.For the two loops,the magnetic fields are $B_1 = \frac{\mu_0 i_1}{2r_1}$ and $B_2 = \frac{\mu_0 i_2}{2r_2}$.Since the currents are in opposite directions,the resultant magnetic field at the center is $B = B_1 - B_2$ (assuming $B_1 > B_2$).Given that $B = \frac{B_1}{2}$,we have:$\frac{B_1}{2} = B_1 - B_2$$B_2 = \frac{B_1}{2}$Substituting the expressions for $B_1$ and $B_2$:$\frac{\mu_0 i_2}{2r_2} = \frac{1}{2} \left( \frac{\mu_0 i_1}{2r_1} \right)$$\frac{i_2}{r_2} = \frac{i_1}{2r_1}$Given $r_2 = 2r_1$,we substitute this into the equation:$\frac{i_2}{2r_1} = \frac{i_1}{2r_1}$$\frac{i_2}{i_1} = 1$.

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