Two concentric and coplanar circular loops have their radii in the ratio 1:3 . They carry currents in opposite directions such that the net magnetic field at their common centre is exactly zero. The ratio of the currents ( I₁ : I₂ ) flowing in the inner and outer loops respectively is:

Step-by-step solution

Let the radii of the inner and outer loops be R₁ and R₂ respectively, and their currents be I₁ and I₂ . The magnetic field at the centre of a circular loop is given by B = ₀ I 2R . For the net magnetic field at the common centre to be zero, the magnitudes of the magnetic fields produced by the two loops must be equal, because their currents are in opposite directions and thus their fields oppose each other. Therefore, B₁ = B₂ ₀ I₁ 2R₁ = ₀ I₂ 2R₂ I₁ I₂ = R₁ R₂ Given that the ratio of the radii is R₁ R₂ = 1 3 , we ha