A circular coil of N turns and radius R carries a current I. It is unwound and rewound to make another coil of radius R/2, current I remaining the same. Calculate the ratio of the magnetic moments of the new coil and original coil.

Area of the coil initially A = πR²
∴Initial magnetic moment μ = NIA = NI(πR²)
Length of the coil initially L = N(2πR)
Radius of new coil formed R' = R/2
Let the number of turns in new coil be N'.
But the length of the new coil formed must be equal to that of original coil i.e L' = L
∴N' × 2π(R/2) = N × 2πR ⇒ N' = 2N
Thus area of new coil A' = π(R²/4)
∴Magnetic moment of new coil μ' = N'IA' = 2NI × π(R²/4) = NIπR²/2
⇒ μ'/μ = 1/2