RC Circuits — Question 2

Question 2

The energy half-life is shorter than the charge half-life in a discharging capacitor

A capacitor initially carries charge Q0 and discharges only through a resistor R. Its capacitance is C and tau=RC.

1. Derive the charge and stored-energy functions and their respective half-lives.

2. Find the remaining energy when charge has halved, and the remaining charge when energy has halved.

Question 2 - Solution

The energy half-life is shorter than the charge half-life in a discharging capacitor

(a) Method and calculation

$$R\dot q+q/C=0\Rightarrow q=Q_0e^{-t/\tau}.$$$$U=\frac{q^2}{2C}=U_0e^{-2t/\tau},\quad t_{q,1/2}=\tau\ln2,\quad t_{U,1/2}=\tfrac12\tau\ln2.$$

(b) Interpretation and further calculation

$$q/Q_0=1/2\Rightarrow U/U_0=(1/2)^2=1/4.$$$$U/U_0=1/2\Rightarrow q/Q_0=1/\sqrt2.$$

Physical check

Energy is quadratic in charge, so it decays with twice the exponential rate.