The figure shows an experimental plot for discharging of a capacitor in an $R-C$ circuit. The time constant…
The figure shows an experimental plot for discharging of a capacitor in an $R-C$ circuit. The time constant $\tau$ of this circuit lies between:
$150 \mathrm{~sec}$ and $200 \mathrm{~sec}$
$0$ and $50 \mathrm{~sec}$
$50 \mathrm{~sec}$ and $100 \mathrm{~sec}$
$100 \mathrm{~sec}$ and $150 \mathrm{~sec}$
Solution
For discharging of an RC circuit,
$V=V_0 e^{-t / \tau}$
So, when
$V=\frac{V_0}{2}$
$\frac{\mathrm{V}_0}{2}=\mathrm{V}_0 \mathrm{e}^{-\mathrm{t} / \tau}$
$\ln \frac{1}{2}=-\frac{\mathrm{t}}{\tau} \Rightarrow \tau=\frac{\mathrm{t}}{\ln 2}$
From graph when $\mathrm{V}=\frac{\mathrm{V}_0}{2}, \mathrm{t}=100 \mathrm{~s} \quad \therefore \tau=\frac{100}{\ln 2}=144.3 \mathrm{~sec}$
