question_answer

How much length of a very thin wire is required to obtain a solenoid of length \[{{l}_{0}}\] and inductance L

A)            \[\sqrt{\frac{2\pi L{{l}_{0}}}{{{\mu }_{0}}}}\]             

B)            \[\sqrt{\frac{4\pi L{{l}_{0}}}{\mu _{0}^{2}}}\]

C)            \[\sqrt{\frac{4\pi L{{l}_{0}}}{{{\mu }_{0}}}}\]             

D)            \[\sqrt{\frac{8\pi L{{l}_{0}}}{{{\mu }_{0}}}}\]

Correct Answer: C

Solution :

                   Suppose solenoid has N turns, each of radius r and length of wire is l. It?s self inductance \[L=\frac{{{\mu }_{0}}{{N}^{2}}A}{{{l}_{0}}}=\frac{{{\mu }_{0}}{{N}^{2}}\pi {{r}^{2}}}{{{l}_{0}}}\] .... (i) Also length of the wire \[l=N\times 2\pi r\]                    \[\Rightarrow {{N}^{2}}{{r}^{2}}=\frac{{{l}^{2}}}{4{{\pi }^{2}}}\]                                                     .... (ii)            From equation (i) and (ii) \[l=\sqrt{\frac{4\pi L{{l}_{o}}}{{{\mu }_{o}}}}\]