Question

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:

• A. decreases by a factor of 9
• B. decreases by a factor of $9\sqrt{3}$
• C. inreases by a factor of 3
• D. inreases by a factor of 27

Answer 1

The correct option is C inreases by a factor of 3

As the total length of $L$ of the wire will remain constant
$L=\left(3a\right)N(N=\text{total turns})\text{and length of winding}=\left(d\right)N=l$
$(d=\text{diameter of wire})$




Now, self inductance is given by, $L={\mu }_0{n}^{2}Al$
$={\mu }_0{n}^2\left(\frac{\sqrt{3}{a}^2}{4}\right)dN$

$\Rightarrow L\propto a^{2}N$

As $N=\frac{l}{3a}\Rightarrow N\propto \frac{1}{a}$

$\therefore L\propto a$

Now, as $a$ is increased to $3a$, the self inductance will become 3 times its initial value.

Hence, $\left(C\right)$ is the correct answer.