Capacitor Charging & Time Constants
When a capacitor charges through a resistor, it doesn't fill up at a steady rate — it charges fast at first, then slower and slower as it gets closer to full voltage. The time constant tells you how quickly this slowdown happens: after one time constant, the capacitor has reached about 63 percent of its final voltage.
The formula
τ = R · C
- τ — time constant (s): the time to reach about 63 percent of full charge
- R — resistance (Ω): how much the resistor limits the charging current
- C — capacitance (F): how much charge the capacitor can store per volt
Worked example
A 1000 Ω resistor charges a 0.001 F capacitor. What is the time constant?
- R = 1000 Ω
- C = 0.001 F
- τ = R · C
- τ = 1000 Ω · 0.001 F
τ = 1 s
Test yourself
If the resistor is changed to 2000 Ω with the same 0.001 F capacitor, what is the new time constant?
- Correct answer: 2 s
- 1 s
- 4 s
Right! τ = R · C = 2000 Ω × 0.001 F = 2 s — doubling the resistance doubles the time constant.
After exactly one time constant has passed, roughly what percent of full voltage has the capacitor reached?
- 37%
- 50%
- Correct answer: 63%
Exactly — after one time constant, the capacitor is at about 63% of its final voltage.
Where you see this
A camera flash can't fire twice in a row: after the burst, the capacitor recharges through a resistor, fast at first then slower, and the ready light comes on only when it has climbed most of the way. Old turn signals and windshield-wiper intervals are the same circuit used as a clock — the blink rhythm is set by how quickly the capacitor fills.
Common mistakes
The reflex error is assuming the capacitor fills at a steady rate — halfway through the time is not halfway to full: after one time constant it has reached about 63 percent, not 50, and it approaches full voltage ever more slowly. The second slip is misreading the knob that sets the pace: τ = R · C, so doubling the resistance from 1000 Ω to 2000 Ω with the same 0.001 F capacitor doubles the time constant from 1 s to 2 s — bigger resistance means slower charging, and ohms times farads genuinely come out in seconds.
How it connects
This closes the circuits module by putting everything together: the loop obeys Kirchhoff's rules at every instant, the resistor obeys Ohm's law, and the capacitor accumulates the difference — which is why the current fades as the gap in voltages closes. The capacitor returns as a headliner in the electromagnetism module, the same device now read as an energy store.