Question

A solenoid would over a rectangular frame. If all the linear dimensions of the frame are increased by a factor $3$ and the number of turns per unit length remains the same, the inductance increased by a factor of :-

• A. $3$
• B. $9$
• C. $27$
• D. $63$

Answer 1

The correct option is(D) $9$

Firstly, the relation between inductance and length is calculated

$emf=\frac{-Nd\varphi }{dt}=-NA\frac{dB}{dt}$

$B=\mu \frac{N}{l}I$

$emf=-\frac{\mu N^{2}A}{l}\frac{dI}{dt}$

$emf=-L\frac{dI}{dt}$

$L=\frac{\mu N^{2}A}{l}$

So, L is inversely proportional to length of the solenoid.

If all the linear dimensions of the frame are increased by a factor of 3 and the number of turns per unit length remains the same

Then area $\begin{align}

$A=3l\times 3b$

$=9lb$

$L^{{}^{\prime }}=\frac{9{\mu }_0{N}^{2}A}{l}$

$L^{\prime }=9L$

So new inductance becomes 9 times.