Area of the coil initially, \( A = \pi R^{2} \)

Initial magnetic moment, \( \mu = NIA = NI(\pi R^{2}) \)

Length of the coil initially, L = L= \(N(2\pi R) \)

Radius of new coil formed,\({R}’ = \frac{R}{2} \)

Let the number of turns in new coil be\( {N}’ \)

But the length of the new coil formed must be equal to that of the original coil \({L}’ = L \)

Therefore, \({N}’ * 2\pi \frac{R}{2} = N * 2\pi R \) \(\Rightarrow {N}’ = 2N \)

Thus area of new coil \({A}’ = {A}’ = \frac{\pi R^{2}}{4} \)

Therefore, the magnetic moment of new coil, \({\mu }’ = {N}’I{A}’ \) \(\Rightarrow 2NI * \frac{\pi R^{2}}{4} \) \(\Rightarrow \frac{NI\pi R^{2}}{2} \) \(\Rightarrow \frac{\mu }{{\mu }’} = \frac{1}{2} \)

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