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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 original coil.
Solution
The magnetic moment m of a current loop is the vector whose magnitude is equal to the area of the loop times the magnitude of the current flowing in it i.e.
m =N IA
Where, N = number of turns
Let m1 and m2 be the magnetic moments of circular original coil of radius ‘R’ and new coil of radius ‘R/2’.
Length of wire remains same. Thus,
`N (2piR) = N'(2pi(R/2))`
N' = 2N
Now,
`m_1NIA_1 =NI (pir_1^2) =NIpiR^2`
`m_2 = N'IA_2 = 2NI(pir_2^2) = 2NI(pi(R/2)^2) = 1/2NIpiR^2`
`m_2/m_1 = (1/2)/1 =1/2`
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