RC Time Constant

Ohms multiplied by farads give seconds, and that is not a coincidence. A farad is a coulomb per volt, how much charge a volt buys, and an ohm is a volt per ampere, an ampere being a coulomb per second. Multiply and the coulombs and volts cancel, leaving time. The physical reading is just as direct: $C$ says how much charge the capacitor must accumulate to reach a given voltage, $R$ says how slowly that charge is allowed to arrive, and $\tau = RC$ is the ratio of the job to the rate at which it can be done.

A 10 kΩ resistor with a 100 µF capacitor gives $\tau = 10\,000 \times 100 \times 10^{-6} = 1\ \text{s}$. Charging from a supply, the capacitor covers 63.2% of the remaining gap in the first second, 86.5% by two seconds, 95% by three, and 99.3% by five, which is why 5τ is the working rule for "finished". The step sizes shrink because the driving voltage is the difference between the supply and what the capacitor has already reached, so as the gap closes the current that closes it falls in proportion. That self-limiting behaviour is what makes the curve exponential rather than a straight ramp.

The exponential is not asserted, it falls out of one line of calculus: the current through the resistor is $(V_s - V_C)/R$ and it must equal $C\,dV_C/dt$, giving $RC\,dV_C/dt = V_s - V_C$, whose solution is $V_C = V_s(1 - e^{-t/RC})$. Set $t = \tau$ and the bracket is $1 - 1/e = 0.632$. The inductive version on this site, $\tau = L/R$, is the same equation with the roles of the storage element and the resistor exchanged, and the RC discharge page is this one running downhill. Timing circuits, debounce networks, RC snubbers and the anti-aliasing filter in front of an ADC are all this constant chosen deliberately.

The unit slip is the most common failure by a wide margin. Capacitance is almost never quoted in farads (microfarads, nanofarads and picofarads are what appear on parts) and 10 kΩ with "100" entered as farads rather than microfarads produces a time constant of $10^6$ seconds, or eleven days. If an answer looks absurd, check the prefix first. The second mistake is misreading what τ measures: it is not the time to charge, and the capacitor never mathematically arrives at all. The third is using the wrong resistance. What matters is the total resistance in the charging path, including the source's internal resistance and anything you have connected to watch it: a 10 MΩ oscilloscope probe across a 1 µF capacitor imposes its own 10-second discharge whether you wanted one or not. And a circuit's charge and discharge paths are often different, through a diode or a separate bleeder, in which case it has two time constants and only one of them is $RC$ as drawn.