A coil of N turns and radius R carries I . It is unwound and rewound to make a square coil of side 'a' having same number of turns N. Keeping the curretn I same, find the ratio of the magnetic moments of the square coil and the circular coil.

Explanation

Magnetic moment of circular coil of N turns and radius R carrying a current I
`m_1 = N A_1 I = N pi R^2 I" ".........(i) `
When the coil is unwound and is rewound to make another square coil of side .a. and number of turns N, as total length of wire remains constant , we have
`N . 2 pi R = N 4 a implies a= (pi R)/(2)`
`:.` Magnetic moment of square coil for same current I
`m_2 = NA_2 I = N a^2 I = N ((pi R^2)/(2)) I = N(pi^2 R^2)/(4) I" " .......(ii)`
`implies (m_2)/(m_1) = pi/4`.