Two similar coils of radius $R$ are lying concentrically with their planes at right angles to each other. The currents flowing in them are $I$ and $2I$, respectively. The resultant magnetic field induction at the centre will be
AIPMT 2012, Medium
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Magnetic field induction due to vertical loop at the centre $O$ is
$B_{1}=\frac{\mu_{0} I}{2 R}$
It acts in horizontal direction.
Magnetic field induction due to horizontal loop at the centre $O$ is
$B_{2}=\frac{\mu_{0} 2 I}{2 R}$
It acts in vertically upward direction.
As $B_{1}$ and $B_{2}$ are perpendicular to each other, therefore the resultant magnetic field induction at the centre $O$ is
${B_{\text {net }}=\sqrt{B_{1}^{2}+B_{2}^{2}}=\sqrt{\left(\frac{\mu_{0} I}{2 R}\right)^{2}+\left(\frac{\mu_{0} 2 I}{2 R}\right)^{2}}}$
${B_{\text {net }}=\frac{\mu_{0} I}{2 R} \sqrt{(1)^{2}+(2)^{2}}=\frac{\sqrt{5} \mu_{0} I}{2 R}}$
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