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_{1}} (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 $\frac{μ_{0} i}{4} (\frac{r_{1} + r_{2}}{r_{1} r_{2}})$
Magnetic induction due to an arc
$= \frac{μ_{0} I θ}{4 π R} = \frac{μ_{0} I π}{4 π R} {A s θ = π}$
$= \frac{μ_{0} I}{4 R}$
So total induction
$= \frac{μ_{0} I}{4} (\frac{1}{R_{1}} + \frac{1}{R_{2}}) = \frac{μ_{0} I}{4} (\frac{r_{1} + r_{2}}{r_{1} r_{2}})$

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