Question

A uniformly wound long solenoid of self inductance $L$ is cut into two parts having lengths in the ratio $1:2$, which are then connected in parallel. The effective inductance of the combination will be -

• A. $\frac{2L}{9}$
• B. $\frac{L}{9}$
• C. $\frac{4L}{9}$
• D. $\frac{5L}{9}$

Answer 1

The correct option is A $\frac{2L}{9}$

We know that the self inductance of a long solenoid is,

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

$\Rightarrow L\propto l$

So,

$L_1=\frac{1}{1+2}\times L=\frac{L}{3}$ and $L_2=\frac{2L}{3}$

The long solenoid is first cut into two parts and these two parts are joined in parallel.

The effective inductance of the combination,

$L_{\text{eq}}=\frac{L_1{L}_2}{L_1+L_2}=\frac{L/3\times 2L/3}{L/3+2L/3}=\frac{2L}{9}$

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