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 of the frame same, then the self inductance of the coil will

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

Answer 1

The correct option is C decrease by a factor of $3$

Since the wire is wounded on the frame, this makes the case of a solenoid.

For a solenoid, the value of self –inductance is given by,

$L={\mu }_0{n}^2\pi R^{2}l$

where, $L=$ coefficient of self-inductance,
$n=\frac{N}{l}=$ number of turns per unit length,
$R=$radius of the wire,
$l=$ length of the solenoid.

$\Rightarrow \frac{L_2}{L_1}=\frac{n_2^2{l}_2}{n_1^2{l}_1}$

$=\frac{{\left(\frac{N_2}{l_2}\right)}^2{l}_2}{{\left(\frac{N_1}{l_1}\right)}^2{l}_1}=\frac{l_1}{l_2}[\because N_1=N_2]$

If $l$ is the length of one side of the triangle then, $l_1=3l$ and $l_2=3\left(3l\right)=9l$

$\Rightarrow L_2=\frac{3l}{9l}=\frac{L_1}{3}$

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