Two circular coils of radius '\(a\)' and '\(2 a\)' are placed coaxially at a distance ' \(x\) and '\(2 x\)' respectively from the origin along the \(\mathrm{X}\)-axis. If their planes are parallel to each other and perpendicular to the \(\mathrm{X}\) - axis and both carry the same current in the same direction, then the ratio of the magnetic field induction at the origin due to the smaller coil to that of the bigger one is:

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.