Question

A long solenoid has an inductance $L$ and resistance $R$. It is broken into two equal parts and then joined in parallel. The combination is joined to a cell of emf $E$. The time constant of the circuit will be

• A. L/R
• B. 2L/R
• C. L/2R
• D. R/L

Answer 1

Answer: (A)

The time constant of the circuit will be $\frac{L}{R}$.

Answer 2

A solenoid with inductance $L$ and resistance $R$ is broken into two equal parts. Each part will have half the inductance and half the resistance, i.e., $L/2$ and $R/2$.

When these two parts are connected in parallel, the equivalent inductance $L_{eq}$ is given by: $L_{eq} = \frac{(L/2) \times (L/2)}{(L/2) + (L/2)} = \frac{L^2/4}{L} = L/4$.

The equivalent resistance $R_{eq}$ is given by: $R_{eq} = \frac{(R/2) \times (R/2)}{(R/2) + (R/2)} = \frac{R^2/4}{R} = R/4$.

The time constant ($\tau$) of an RL circuit is given by the ratio of inductance to resistance: $\tau = \frac{L_{eq}}{R_{eq}} = \frac{L/4}{R/4} = \frac{L}{R}$.