In the figure shown there are two semicircles of radii $r_{1}$ and $r_{2}$ in which a current i is flowing. The magnetic induction at the centre O will be

$\frac{μ_{0} i}{r} (r_{1} + r_{2})$

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$\frac{μ_{0} i}{4} (r_{1} - r_{2})$

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$\frac{μ_{0} i}{4} (\frac{r_{1} + r_{2}}{r_{1} r_{2}})$

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$\frac{μ_{0} i}{4} (\frac{r_{2} - r_{1}}{r_{1} r_{2}})$

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Solution

The correct option is C
$\frac{μ_{0} i}{4} (\frac{r_{1} + r_{2}}{r_{1} r_{2}})$

The magnetic induction due to both semicircular parts will be in the same direction perpendicular to the paper inwards.

$∴ B = B_{1} + B_{2} = \frac{μ_{0} i}{4 r_{1}} + \frac{μ_{0} i}{4 r_{2}} = \frac{μ_{0} i}{4} (\frac{r_{1} + r_{2}}{r_{1} r_{2}}) \otimes$

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