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Two wires A and B are of lengths 40 cm and 30 cm. A is bent into a circle of radius r and B into an arc of radius r. A current $i_{1}$ is passed through A and $i_{2}$ through B. To have the same magnetic induction at the centre, the ratio of $i_{1} : i_{2}$ is

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a

$3 : 4$

b

$3 : 5$

c

$2 : 3$

d

$4 : 3$

answer is A.

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Detailed Solution

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$\begin{array}{l} B_{1} = \frac{μ_{0} i_{1}}{2 r}; B_{2} = \frac{μ_{0} i_{2}}{2 r} \frac{θ}{2 π} \\ \Rightarrow \frac{μ_{0} i_{1}}{2 r} = \frac{μ_{0} i_{2}}{2 r} \frac{θ}{2 π} \\ \Rightarrow \frac{i_{1}}{i_{2}} = \frac{l_{2}}{2 πr} \\ \Rightarrow \frac{i_{1}}{i_{2}} = \frac{l_{2}}{l_{1}} \\ \Rightarrow \frac{i_{1}}{i_{2}} = \frac{3}{4} \end{array}$

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