a
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

\({B_{\text {net }}=\sqrt{B_{1}^{2}+B_{2}^{2}}=\sqrt{\left(\frac{\mu_{0} I}{2 R}\right)^{2}+\left(\frac{\mu_{0} 2 I}{2 R}\right)^{2}}}\)

\({B_{\text {net }}=\frac{\mu_{0} I}{2 R} \sqrt{(1)^{2}+(2)^{2}}=\frac{\sqrt{5} \mu_{0} I}{2 R}}\)