Question

The equation of discharge of a capacitor is given by Q = Qoe^(-t/RC) , define the terms half life and time constant of a discharging capacitor and derive the expression that links them.

Answer 1

• The half-time $T$ is the time during which the charge stored is decreased by two times : $Q_0 e^{-\frac{T}{RC}} = \frac{1}{2} Q_0$. Thus $\frac{T}{RC}=\ln 2$ and so $T = RC\ln2$.
• The characteristic time $\tau$ is defined such that $Q = Q_0 e^{-\frac{t}{\tau}}$ and thus $\tau = RC$.

We see from these two relations that $T = \tau \ln 2$. We can also verify that $RC$ have the dimensions of a time.