Question

Consider the circuit shown in Fig. 31-40. With the switch S₁ closed and the other two switches open, the circuit has a time constant ${\mathit{\tau }}_{\mathbf{C}}$. With the switch S₂closed and the other two switches open, the circuit has a time constant ${\mathit{\tau }}_{\mathbf{L}}$ . With the switch S₃closed and the other two switches open, the circuit oscillates with a period T. Show that $\mathit{T}\mathbf{=}\mathbf{2}\mathbf{\pi }\sqrt{{\mathbf{\tau }}_{\mathbf{C}}{\mathbf{\tau }}_{\mathbf{L}}}$

Short answer

Hence, proved$T=2\mathrm{\pi }\sqrt{{\mathrm{\tau }}_{\mathrm{C}}{\mathrm{\tau }}_{\mathrm{L}}}$

Step-by-step solution

Given

The circuit diagram is shown in the figure.

Determining the concept

Use the concept of inductive and capacitive time constant and expression of the time period of the LCcircuit.

The formulae are as follows:

$$\begin{gathered} {\tau }_C=RC \\ {\tau }_L=\frac{L}{R} \\ T=2\mathrm{\pi }\sqrt{\mathrm{LC}} \end{gathered}$$

Where, Tis time, Cis capacitance, Ris resistance and $\tau$is torque.

Determining the T=2πτCτL

Proof of $T=2\mathrm{\pi }\sqrt{{\mathrm{\tau }}_{\mathrm{C}}{\mathrm{\tau }}_{\mathrm{L}}}$:

When the switch S₁ is closed and the other two switches are open, the inductor is essentially out of the circuit.

The time constant for the RC circuit is,

$$\begin{gathered} {\tau }_C=RC \\ C=\frac{{\tau }_C}{R}..........\left(1\right) \end{gathered}$$

When the switch S₂ is closed and the other two switches open, the capacitor is essentially out of the circuit.

The time constant for the LR circuit is,

$$\begin{gathered} {\tau }_L=\frac{L}{R} \\ L={\tau }_{L}R..........\left(2\right) \end{gathered}$$

Finally, when the switch S₃ is closed and the other two switches open, the resistor is essentially out of the circuit.

The time period of the oscillation of LC the circuit is,

$T=2\mathrm{\pi }\sqrt{\mathrm{LC}}$

Substitute equations (1) and (2) in this equation as,

$$\begin{gathered} T=2\mathrm{\pi }\sqrt{{\mathrm{\tau }}_{\mathrm{L}}\mathrm{R}\times \frac{{\mathrm{\tau }}_{\mathrm{C}}}{\mathrm{R}}} \\ \mathrm{T}=2\mathrm{\pi }\sqrt{{\mathrm{\tau }}_{\mathrm{C}}{\mathrm{\tau }}_{\mathrm{L}}} \end{gathered}$$

Hence,proved $\mathrm{T}=2\mathrm{\pi }\sqrt{{\mathrm{\tau }}_{\mathrm{C}}{\mathrm{\tau }}_{\mathrm{L}}}$.