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 X -axis. If their planes are parallel to each other and perpendicular to the 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:

Step-by-step solution

The magnetic field B at a distance d along the axis of a circular coil of radius R carrying current I is given by the formula B = ₀ I R² 2(R² + d²)^ 3/2 . For the smaller coil, the radius is R₁ = a and the distance from the origin is d₁ = x . The magnetic field at the origin is B₁ = ₀ I a² 2(a² + x²)^ 3/2 . For the bigger coil, the radius is R₂ = 2a and the distance from the origin is d₂ = 2x . The magnetic field at the origin is B₂ = ₀ I (2a)² 2((2a)² + (2x)²)^ 3/2 . Simplifying B₂ : B₂ = ₀ I (4a²) 2(4a² + 4x²)^ 3