A copper wire is wound on a wooden frame, whose shape is that of an equilateral triangle. If the linear dimension of each side of the frame is increased by a factor of 3 , keeping the number of turns of the coil per unit length of the frame the same, then the self inductance of the coil:
Correct Option: , 3
(3) As total length $\mathrm{L}$ of the wire will remain constant
$\mathrm{L}=(3 \mathrm{a}) \mathrm{N}$ $(\mathrm{N}=$ total turns $)$
and length of winding $=($ d $) \mathrm{N}$
$(\mathrm{d}=$ diameter of wire $)$
self inductance $=\mu_{0} \mathrm{n}^{2} \mathrm{~A} \ell$
$=\mu_{0} n^{2}\left(\frac{\sqrt{3} a^{2}}{4}\right) d N$
$\propto \mathrm{a}^{2} \mathrm{~N} \propto \mathrm{a}\left[\mathrm{as} \mathrm{N}=\mathrm{L} / 3 \mathrm{a} \Rightarrow \mathrm{N} \propto \frac{1}{\mathrm{a}}\right]$
Now 'a' increased to ' $3 a$ '
So self inductance will become 3 times
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