Question

A uniformly wound long solenoid of inductance $L$ and resistance $R$ is cut into two parts in the ratio $\eta :1$, which are then connected in parallel. The combination is then connected to a cell of emf $E$. The time constant of the circuit is :

• A. $\frac{L}{R}$
• B. $\frac{L}{(\eta +1)R}$
• C. $\left(\frac{\eta }{\eta +1}\right)\frac{L}{R}$
• D. $\left(\frac{\eta +1}{\eta }\right)\frac{L}{R}$

Answer 1

The correct option is A $\frac{L}{R}$

$L_1=\left(\frac{\eta }{\eta +1}\right)L,R_1=\left(\frac{\eta }{\eta +1}\right)R$
$L_2=\left(\frac{1}{\eta +1}\right)L,R_2=\left(\frac{1}{\eta +1}\right)R$
$L_{net}=\frac{L_1{L}_2}{L_1+L_2}=\frac{\eta LL}{(\eta +1)(\eta L+L)}$
Similarly, $R_{net}=\frac{R_1{R}_2}{R_1+R_2}=\frac{\eta RR}{(\eta +1)(\eta R+R)}$
${\tau }_L=\frac{L_{net}}{R_{net}}=\frac{L}{R}$