Magnetic field induction due to vertical loop at the centre O is

${B_1} = {{{\mu _0}I} \over {2R}}$

It acts in horizontal direction.

Magnetic field induction due to horizontal loop at the centre O is

${B_2} = {{{\mu _0}2I} \over {2R}}$

It acts in vertically upward direction.

As B₁ and B₂ are perpendicular to each other, therefore the resultant magnetic field induction at the centre O is

${B_{net}} = \sqrt {B_1^2 + B_2^2}$

$= \sqrt {{{\left( {{{{\mu _0}I} \over {2R}}} \right)}^2} + {{\left( {{{{\mu _0}2I} \over {2R}}} \right)}^2}}$

${B_{net}} = {{{\mu _0}I} \over {2R}}\sqrt {{{\left( 1 \right)}^2} + {{\left( 2 \right)}^2}} = {{\sqrt 5 {\mu _0}I} \over {2R}}$