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$V_{C}$ is the ideal voltmeter in the figure. Resistance of the resistor shown is R. Initially the switch is in position 2 when charging of the capacitor starts. Initially the capacitor was uncharged. The switch in circuit shifts automatically from 2 to 1 when $V_{C} > 2 V / 3$ and goes back to 2 from 1 when $V_{C} < V / 3.$ The ideal voltmeter reads voltage across capacitor as plotted. What is the period T of the wave form in terms of R and C?

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$3 R C \ln 3$

$2 R C \ln 2$

$(R C / 2) \ln 2$

none of these

answer is B.

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Detailed Solution

$t_{1}$ is the time of capacitor to discharge from 2V/3 to V/3. For this, $\frac{V}{3} - \frac{2 V}{3} e^{- t_{1} / τ}$

$o r t_{1} = τ \ln 2 = R C \ln 2$

$t_{2}$ is time for charging of capacitor from V/3 to 2V/3.

So $\int_{V C / 3}^{2 V C / 3} \frac{d q}{V C - q} = \int_{0}^{t_{2}} \frac{d t}{R C}$

$o r t_{2} = R C \ln 2$

$T = t_{1} + t_{2} = 2 R C \ln 2$

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