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.

Explanation

Magnetic moment of original coil of N turns and radius R carrying a current I
`m_1 = N_1 A_1 I = N.pi.R^2.I`
When the coil is unwound and is rewound to make another coil of radius `R_2 = (R_1)/2 = R/2` , the total number of turns in the coil will change to `N_2 = 2 N_1 = 2N`. As same current I flows through the coil even now, hence new magnetic moment is given by
`m_2 = N_2 A_2 I = 2 N . pi . (R/2)^(2) . I = 1/2 N pi R^2 I = (|vecm_1|)/(2)`