A straight conducting wire of a certain length is bent into a single-turn circular loop. When a steady current I is passed through it, the magnetic field produced at its centre is B . If the same wire is now rewound into a closely packed circular coil of n turns and the identical current I is passed through it, what will be the new magnetic field at the centre of the coil?

Step-by-step solution

Let the length of the wire be L . For the single-turn loop, the radius R₁ is given by: L = 2 R₁ R₁ = L 2 The magnetic field at the centre of this single-turn loop is: B = ₀ I 2R₁ When the same wire is rewound into a coil of n turns, let the new radius be R_n . The total length of the wire remains constant: L = n(2 R_n) R_n = L n(2 ) = R₁ n The new magnetic field at the centre of the n -turn coil is: B_n = ₀ n I 2R_n Substituting R_n = R₁ n into the equation: B_n = ₀ n I 2 ( R₁ n ) = n^2 ( ₀ I 2R₁ ) Since B = ₀ I 2R