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A circular loop of radius $R$ carrying a current $I$ is unwound and rewound into a loop of half the radius. The current in the loop remains the same. The ratio of magnetic moments of the new loop to that of the old loop is

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1 : 2

1 : 4

1 : 3

2 : 1

answer is A.

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Detailed Solution

The ratio of magnetic moments of the new loop to that of the old loop is

1 : 2

.
Let Area of the coil initially,

A = π R^{2}

Initial magnetic moment

μ = NIA = NI (π R^{2})

Length of the coil initially

L = N (2 πR)

Radius of new coil formed

R^{'} = \frac{R}{2}

and the number of turns in new coil be

N

.
But the length of the new coil

(L')

formed must be equal to that of original coil

L' = L

∴ N^{'} = 2 N

Thus, area of new coil

A' = \frac{{πR}^{2}}{4}

Magnetic moment of new coil

μ' = N' IA'

\Rightarrow μ' = 2 NI \times \frac{π R^{2}}{4}

\Rightarrow μ' = \frac{NIπ R^{2}}{2}

Hence, the ratio between the magnetic moments is given by

\Rightarrow \frac{μ}{μ'} = \frac{1}{2}

Hence, the ratio of magnetic moments of new loop to that of old loop is

1 : 2

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