Question

A circular coil of ${}^{\prime }{N}^{\prime }$ turns and radius ${}^{\prime }{R}^{\prime }$ carries a current ${}^{\prime }{I}^{\prime }$. It is unwound and rewound to make another coil of radius ${}^{\prime }R/2^{\prime }$, current ${}^{\prime }{I}^{\prime }$ remaining the same. Calculate the ratio of the magnetic moments of the new coil and original coil.

Answer 1

Area of the coil initially $A=\pi R^2$

$\therefore$ Initial magnetic moment $\mu =NIA=NI\left(\pi R^2\right)$

Length of the coil initially $L=N\left(2\pi R\right)$

Radius of new coil formed $R^{\prime }=\frac{R}{2}$

Let the number of turns in new coil be $N^{\prime }$.

But the length of the new coil formed must be equal to that of original coil i.e $L^{\prime }=L$

$\therefore$ $N^{\prime }\times 2\pi \frac{R}{2}=N\times 2\pi R$ $⟹N^{\prime }=2N$

Thus area of new coil $A^{\prime }=\frac{\pi R^2}{4}$

$\therefore$ Magnetic moment of new coil ${\mu }^{\prime }=N^{\prime }I{A}^{\prime }=2NI\times \frac{\pi R^2}{4}=\frac{NI\pi R^2}{2}$

$⟹$ $\frac{{\mu }^{\prime }}{\mu }=\frac{1}{2}$