One number describes the whole curve
τ = R × C
With R in ohms and C in farads, τ comes out in seconds. It is the time a capacitor takes to charge through that resistor to 63.2% of the supply, and that one number sets everything else about the curve.
10 kΩ with 100 µF gives τ = 10,000 × 0.0001 = 1 second.
How full after each time constant
- 1τ: 63.2%
- 2τ: 86.5%
- 3τ: 95.0%
- 4τ: 98.2%
- 5τ: 99.3%, which every datasheet treats as fully charged
Discharging follows the same curve downwards: after 1τ the capacitor is down to 36.8% of where it started.
Why 63.2%
The charge follows V = Vs(1 − e−t/RC). At t = RC the bracket is 1 − 1/e = 0.632. The curve never quite arrives, which is why "fully charged" is a convention rather than a moment.
The same number as a filter
An RC pair is also a low-pass filter, and its corner is
f = 1 ÷ (2πRC)
That 10 kΩ and 100 µF gives 0.159 Hz: slow enough to smooth a reading, far too slow for audio. Swap the capacitor for 100 nF and the corner moves to 159 Hz. A long time constant and a low cutoff are the same fact seen from two directions.
Stored energy
E = ½CV²
100 µF charged to 5 V holds 1.25 mJ, which is nothing. The same capacitance at 400 V in a power supply holds 8 J, which is enough to hurt, and is why those capacitors have bleed resistors across them.
Where 1.1 RC comes from
A 555 timer in one-shot mode charges its capacitor until it reaches two thirds of the supply, not 63.2%. Solving the same exponential for 2/3 gives t = RC × ln(3) = 1.0986 RC, which every 555 datasheet rounds to 1.1 RC. It is the same curve, read at a different level.
Picking the parts
For a given τ you can use a big resistor with a small capacitor or the other way round. Prefer the smaller capacitor: large electrolytics leak, drift with temperature and age, and their tolerance is often ±20%, so a 10% resistor with a small film capacitor is far more repeatable than the other way round.
Tools
The RC Time Constant Calculator gives τ, the charge times, the cutoff frequency and the stored energy. The 555 Timer Calculator uses the 1.1 RC above, Ohm's Law Calculator covers the steady state, and the LED Resistor Calculator sizes the resistor beside it.