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

\frac{μ_{0} i}{4} (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_{1} - r_{2}}{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.
$∴ B = B_{1} + B_{2}$

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

The direction of the magnetic field will be inside the plane of the rings.

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