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(C) Suppose the solenoid has $N$ turns,each of radius $r$,and the total length of the wire is $l$.The self-inductance $L$ of the solenoid is given by

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$\sqrt{\frac{4\pi L l_0}{\mu_0}}$

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(C) Suppose the solenoid has $N$ turns,each of radius $r$,and the total length of the wire is $l$.The self-inductance $L$ of the solenoid is given by $L = \frac{\mu_0 N^2 A}{l_0} = \frac{\mu_0 N^2 \pi r^2}{l_0}$ .... $(i)$The total length of the wire $l$ is given by $l = N 	imes 2\pi r$.Squaring both sides,we get $l^2 = N^2 	imes 4\pi^2 r^2$,which implies $N^2 r^2 = \frac{l^2}{4\pi^2}$ .... $(ii)$Substituting the value of $N^2 r^2$ from equation $(ii)$ into equation $(i)$:$L = \frac{\mu_0 \pi}{l_0} 	imes \frac{l^2}{4\pi^2} = \frac{\mu_0 l^2}{4\pi l_0}$Rearranging for $l$,we get $l^2 = \frac{4\pi L l_0}{\mu_0}$,so $l = \sqrt{\frac{4\pi L l_0}{\mu_0}}$.

What length of a very thin wire is required to obtain a solenoid of length $l_0$ and inductance $L$?

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