Two similar coils are kept mutually perpendicular such that their...

Suppose the magnetic field produced due to each coil is $B$. The two coils are kept perpendicular hence, the angle between these is $90^{\circ}$ therefore, the resultant magnetic field is given by $$\begin{aligned} & =\sqrt{B^2+B^2+2 B \cdot B \cdot \cos 90^{\circ}} \\ & =\sqrt{2 B^2+2 B^2 \times 0} \\ & =\sqrt{2 B^2}=B \sqrt{2}\end{aligned}$$ Hence, the ratio of magnetic field due to one coil and the resultant magnetic field is given by $=\frac{B}{\sqrt{2} B}=1: \sqrt{2}$