The magnetic field at the centre of a wire loop formed by two semicircular wires of radii $R_1=2 \pi\ \mathrm{m}$ and $R_2=4 \pi\ \mathrm{m}$ carrying current $I=4 \mathrm{~A}$ as per figure given below is $\alpha \times 10^{-7} \mathrm{~T}$. The value of $\alpha$ is___________ (Centre $\mathrm{O}$ is common for all segments)
JEE MAIN 2024, Diffcult
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$ \frac{\mu_0 \mathrm{i}}{2 \mathrm{R}_2}\left(\frac{\pi}{2 \pi}\right) \otimes+\frac{\mu_0 \mathrm{i}}{2 \mathrm{R}_1}\left(\frac{\pi}{2 \pi}\right) \otimes $
$ \left(\frac{\mu_0 \mathrm{i}}{4 \mathrm{R}_2}+\frac{\mu_0 \mathrm{i}}{4 \mathrm{R}_1}\right) \otimes $
$ \frac{4 \pi \times 10^{-7} \times 4}{4 \times 4 \pi}+\frac{4 \pi \times 10^{-7} \times 4}{4 \times 2 \pi} $
$ =3 \times 10^{-7}=\alpha \times 10^{-7} $
$ \alpha=3$
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