First slide

Inductance

Question

How much length of a very thin wire is required to obtain a solenoid of length $l_{0}$ and inductance L?

Difficult

A

$\sqrt{\frac{2 π L l_{0}}{μ_{0}}}$
B

$\sqrt{\frac{4 π L l_{0}}{μ_{0}^{2}}}$
C

$\sqrt{\frac{4 π L l_{0}}{μ_{0}}}$
D

$\sqrt{\frac{8 π L l_{0}}{μ_{0}}}$

Solution

Suppose solenoid has N turns, each of radius r and length of wire is l.
Its self-inductance, $L = \frac{μ_{0} N^{2} A}{l_{0}} = \frac{μ_{0} N^{2} π r^{2}}{l_{0}}$ ….(i)
Also, length of the wire, $l = N \times 2 πr$
$\Rightarrow N^{2} r^{2} = \frac{l^{2}}{4 π^{2}}$ … (ii)
From equations (i) and (ii), $l = \sqrt{\frac{4 {πLl}_{0}}{μ_{0}}}$

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