Capacitor Formulas: Energy and RC Time Constant
Capacitor formulas for charge, stored energy, series and parallel capacitance and the RC time constant, with worked examples and a charging table.
The formulas
| To find | Formula |
|---|---|
| Charge | Q = C × V |
| Stored energy | E = ½ × C × V² |
| Capacitors in parallel | C = C1 + C2 + C3 + … |
| Capacitors in series | 1 / C = 1 / C1 + 1 / C2 + 1 / C3 + … |
| Time constant | τ = R × C |
| Voltage while charging | v(t) = V × (1 − e^(−t/τ)) |
| Voltage while discharging | v(t) = V0 × e^(−t/τ) |
C is in farads, V in volts, Q in coulombs, E in joules, R in ohms and t in seconds. 1 µF is 10⁻⁶ F. For two capacitors in series, C = C1 × C2 / (C1 + C2).
Worked example: a 470 µF capacitor at 24 V
- Q = 470 × 10⁻⁶ × 24 = 11.28 mC
- E = ½ × 470 × 10⁻⁶ × 24² = 0.1354 J
Worked example: 10 µF and 40 µF
- In parallel: C = 10 + 40 = 50 µF
- In series: C = 10 × 40 / (10 + 40) = 8 µF, always less than the smaller capacitor
Worked example: charging through 10 kΩ
A 100 µF capacitor is charged from a 12 V supply through a 10 kΩ resistor.
- τ = 10000 × 0.0001 = 1 s
- After 2 s: v = 12 × (1 − e^(−2/1)) = 10.38 V
Charge and discharge by time constants
| Time | Charged to | Remaining when discharging |
|---|---|---|
| 1 τ | 63.21 % | 36.79 % |
| 2 τ | 86.47 % | 13.53 % |
| 3 τ | 95.02 % | 4.979 % |
| 4 τ | 98.17 % | 1.832 % |
| 5 τ | 99.33 % | 0.6738 % |
Frequently asked questions
How long does a capacitor take to charge fully?
In theory it never does, because the voltage approaches the supply along a curve. After five time constants it is within 1%, which is the usual practical figure.
Why is the series capacitance less than the smallest capacitor?
The voltage splits between the capacitors, so the same charge sits on each plate and the effective capacitance falls.
Why does the energy depend on the square of the voltage?
Doubling the voltage doubles the charge, and the energy is the charge times the average voltage, so it increases four times.
Is a charged capacitor dangerous after the power is off?
It can be. A large capacitor can hold its charge for a long time. Discharge it through a suitable resistor and check with a meter before touching the terminals.
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