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 the original coil.

Solution

Length of the wire remains same,
$straight N subscript 1 space straight x space 2 πR space equals space straight N subscript 2 space straight x space space 2 straight pi space straight R over 2 therefore space straight N subscript 2 space equals space 2 space straight N$

Magnetic moment of a coil, m = NAI

For the coil of radius R, magnetic moment,
$straight m subscript 1 space equals space straight N subscript 1 IA subscript 1 space equals space straight N subscript 1 Iπ space straight R squared$
Magnetic moment for coil of radius R/2,
$straight m subscript 2 space equals space straight N subscript 2 space IA subscript 2 space equals space fraction numerator 2 straight N subscript 1 IπR squared over denominator 4 end fraction equals fraction numerator straight N subscript 1 IπR squared over denominator 2 end fraction$
Therefore,
$straight m subscript 2 over straight m subscript 1 space equals space 1 colon thin space 2$