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 . . .

Question :  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. 

Doubt by Nevaeh

Solution : 

In all such questions, length of the wire will be same as the same coil is unwound and rewound. 

R1 = R 

R2 = R/2 (Given) 

R1/R2 = 2 — (1)

Length of the wire = N1 × 2πR = N2 × 2π (R/2)

N1 = N2/2

N1/N2 = 1/2 — (2)

Now, The ratio of magnetic moments :

M1/M2 = N1IA1/N2IA2

M1/M2 = N1A1/N2A2

             = N1×πR1/N2×πR2 

             = (N1/N2)×(R1/R2)2

             = 1/2 × (2)2

             = 2 

M1  : M= 2 : 1 

Similar Question : A circular coil of N turns and diameter d carries a current I. It is unwounded and rewound to make another coil of diameter 2d, current I remaining the same. Calculate the ratio of the magnetic moments of the new coil and the original coil.
Ans : M1 : M2 = 1 : 2