A uniform conducting wire of a given length is bent into a single-turn circular loop. When a current I is passed through it, the magnetic field at its centre is B and its magnetic dipole moment is M . If the same wire is now rewound to form a circular coil of 4 turns and the same current I is passed through it, what will be the new magnetic field at the centre and the new magnetic dipole moment, respectively?

Step-by-step solution

Let the total length of the uniform wire be L . For the single-turn loop (initial state): L = 2 R R = L 2 Magnetic field at the centre, B = ₀ I 2R Magnetic dipole moment, M = I R^2 When the same wire is rewound into n = 4 turns (final state): L = 4 2 R' R' = R 4 The new magnetic field at the centre is: B' = ₀ n I 2R' = ₀ (4) I 2(R/4) = 16 ( ₀ I 2R ) = 16B The new magnetic dipole moment is: M' = n I (R')^2 = 4 I ( R 4 )^2 = 4 16 I R^2 = 1 4 (I R^2) = M 4 Therefore, the new magnetic field is 16B and the new magnetic