How much length of a very thin wire is required to obtain a solenoid of length `l_(0)` and inductance `L`

Options

A. `sqrt((2piLl_(0))/(mu_(0)))`
B. `sqrt((4piLl_(0))/(mu_(0)^(2)))`
C. `sqrt((4piLl_(0))/(mu_(0)))`
D. `sqrt((8piLl_(0))/(mu_(0)))`

Answer

C

Explanation

Suppose solenoid has `N` turns, each of radius `r` and length of wire is `l`.
its self-inductance `L=(mu_(0)N^(2)A)/(l_(0))=(mu_(0)N^(2)pir^(2))/(l_(0))`…..`(i)`
Also length of the wire `l=Nxx2pir`
`impliesN^(2)r^(2)=(l^(2))/(4pi^(2))`
from equation (i) and (ii) `l=sqrt((4piLl_(0))/(mu_(0)))`