If a single-turn circular coil ( of radius R₁ ) carrying a current I is rewound into a coil of N turns ( of radius R₂ ) using the same total length of wire and carrying the same current I , the ratio of the magnetic field at the centre of the new coil (B₂) to that of the original coil (B₁) is

Step-by-step solution

Let the total length of the wire be L . For the single-turn coil of radius R₁ , the length is L = 2 R₁ . For the N -turn coil of radius R₂ , the length is L = N(2 R₂) . Equating the lengths, 2 R₁ = N(2 R₂) R₂ = R₁ N . The magnetic field at the centre of the original coil is B₁ = ₀ I 2 R₁ . The magnetic field at the centre of the new coil is B₂ = ₀ N I 2 R₂ . Substituting R₂ = R₁ N into the expression for B₂ : B₂ = ₀ N I 2 ( R₁ N ) = N^2 ( ₀ I 2 R₁ ) = N^2 B₁ . The ratio of the magnetic field at the centre of the ne