Two similar coils of radius R and number of turns N are lying concentrically with their planes at right angles to each other. The currents flowing in them are i and $\sqrt{3} i$ respectively. The resultant magnetic induction at the centre will be (in Wb/m²)

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$\frac{μ_{0} Ni}{\sqrt{3} R}$

$\frac{μ_{0} Ni}{R}$

$\frac{\sqrt{3} μ_{0} N}{iR}$

$\frac{\sqrt{3} μ_{0} Ni}{R}$

answer is A.

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Detailed Solution

See fig The magnetic induction at the centre of a circular coil of N turns, radius R and carrying a current i is given by
$B = \frac{μ_{0} Ni}{2 R}$
Here, $B_{1} = \frac{μ_{0} Ni}{2 R} and B_{2} = \frac{μ_{0} N (\sqrt{3} i)}{2 R}$
Now, $B = \sqrt{{(B_{1})}^{2} + {(B_{2})}^{2}}$
$= \frac{μ_{0} Ni}{2 R} \sqrt{(1 + 3)} = \frac{2 μ_{0} Ni}{2 R} = \frac{μ_{0} Ni}{R}$