According to the sources, the magnitude of the magnetic field $B$ at the centre of a circular loop of radius $R$ carrying current $I$ is given by the formula $B = \frac{\mu_0 I}{2R}$ .

1. Field from the first coil ($B_1$): For the coil carrying current $I$, the magnetic field at the centre is $B_1 = \frac{\mu_0 I}{2R}$.
2. Field from the second coil ($B_2$): For the coil carrying current $2I$, the magnetic field at the centre is $B_2 = \frac{\mu_0 (2I)}{2R} = \frac{2\mu_0 I}{2R}$.
3. Resultant Field ($B_{net}$): Since the planes of the two coils are at right angles, the magnetic field vectors they produce at the common centre are also perpendicular to each other. The resultant magnetic field is calculated using the Pythagorean theorem for perpendicular vectors: $B_{net} = \sqrt{B_1^2 + B_2^2}$ .

Substituting the values: $B_{net} = \sqrt{(\frac{\mu_0 I}{2R})^2 + (\frac{2\mu_0 I}{2R})^2} = \sqrt{\frac{\mu_0^2 I^2}{4R^2} (1 + 4)} = \frac{\sqrt{5}\mu_0 I}{2R}$.