What length of wire is required to construct | vedclass

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(C) Let the solenoid have $N$ turns,each of radius $r$,and let the total length of the wire be $l$.The inductance $L$ of a solenoid is given by:$L = \

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$\sqrt {\frac{{4\pi L{l_0}}}{{{\mu _0}}}} $

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(C) Let the solenoid have $N$ turns,each of radius $r$,and let the total length of the wire be $l$.The inductance $L$ of a 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 the circumference of one turn multiplied by the number of turns:$l = N 	imes 2\pi r \Rightarrow N^2 r^2 = \frac{{{l^2}}}{{4{\pi ^2}}}$ .... $(ii)$Substituting $(ii)$ into $(i)$:$L = \frac{{{\mu _0} \pi}}{{{l_0}}} 	imes \frac{{{l^2}}}{{4{\pi ^2}}} = \frac{{{\mu _0}{l^2}}}{{4\pi {l_0}}}$Rearranging for $l$:$l^2 = \frac{{4\pi L{l_0}}}{{{\mu _0}}}$$l = \sqrt {\frac{{4\pi L{l_0}}}{{{\mu _0}}}} $

What length of wire is required to construct a solenoid of length $l_0$ and inductance $L$?

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