Charging and discharging a capacitor

What is it about?

A capacitor stores charge, $Q = C \cdot U$ . Charging through a resistor $R$ from $U_0$ :

U_C(t) = U_0 \cdot \left(1 - e^{-t/\tau}\right), \qquad I(t) = \frac{U_0}{R} \cdot e^{-t/\tau}, \qquad \tau = R \cdot C .

Discharging: $U_C(t) = U_0 \cdot e^{-t/\tau}$ , and the current reverses (negative in the graph).

After $\tau$ the voltage has reached 63 % (charging) or dropped to 37 % (discharging).
The tangent at $t = 0$ meets the final value exactly at $t = \tau$ .
Half-life: $t_{1/2} = \tau \cdot \ln 2$ .
After about $5\,\tau$ the process is practically complete.
Stored energy: $W = \tfrac12 C U^2$ .

The dots are electrons, moving against the conventional current (red arrow) with a speed proportional to $I$ . Very small or very large $\tau$ are shown in time lapse or slow motion; the graphs always show real time.

Try it

Watch the charging process. When is the current largest? Charge
Tap the switch while charging. Charge
Double $R$ or $C$ . R = 20 kΩ · C = 200 µF
Discharge with half-life markers. Half-life

Tasks

Task 1: Time constant

$C = 100\,\mu\text{F}$ , $R = 10\,\text{k}\Omega$ , $U_0 = 10\,\text{V}$ . Find $\tau$ , $I_0$ and $U_C(\tau)$ . Load task

Show solution

$\tau = 1.0\,\text{s}$ , $I_0 = 1.0\,\text{mA}$ , $U_C(\tau) \approx 6.32\,\text{V}$ . Show in the simulation

Task 2: Half-life

When has the discharging capacitor dropped to $5\,\text{V}$ , and to $2.5\,\text{V}$ ? Load task

Show solution

$t_{1/2} \approx 0.693\,\text{s}$ , two half-lives $\approx 1.39\,\text{s}$ . Show in the simulation

Task 3: Energy

Energy stored at $10\,\text{V}$ ? Load task

Show solution

$W = \tfrac12 \cdot 10^{-4}\,\text{F} \cdot (10\,\text{V})^2 = 5\,\text{mJ}$ . Show in the simulation

Ideas for the classroom

Compare with a real measurement using a large electrolytic capacitor.
Common stumbling blocks: “Current flows through the capacitor.” – Only in the leads. “The current keeps its direction when discharging.” – It reverses.