the RC time constant
/ ar-see TIME KON-stant /
The RC time constant is the single number that tells you how fast an RC circuit charges or discharges, its natural sense of pace. A big time constant means a lazy, slow response; a small one means a snappy, quick one. Think of it as the circuit's 'reaction time', set once and for all by the size of the resistor and the capacitor. It answers the question: roughly how long before this capacitor is meaningfully charged or drained?
Precisely, the time constant, written with the Greek letter tau, is the product of resistance and capacitance, tau = R C. With R in ohms and C in farads, tau comes out in seconds. It is the time in which a charging capacitor reaches a fraction 1 - 1/e, about 63 percent, of its final charge, and the time in which a discharging one falls to 1/e, about 37 percent, of its starting value. In the exponential formulas e^(-t/RC), tau = RC is exactly the timescale in the exponent.
A useful rule of thumb is that after about 5 time constants the circuit is more than 99 percent of the way to its final state, so engineers treat '5 tau' as effectively settled. Choosing R and C therefore lets you dial in any timing you like, from microseconds in fast electronics to many seconds in a slow blinker. One honest subtlety: because the change is exponential, there is no sharp moment when charging 'finishes', so tau describes the characteristic pace rather than a hard deadline.
With R = 1 megaohm (1 x 10^6 ohms) and C = 1 microfarad (1 x 10^-6 F), the time constant is tau = R C = 1e6 x 1e-6 = 1 second. After 1 s a charging capacitor sits at about 63 percent of full; after roughly 5 s it is over 99 percent charged.
tau = RC sets the pace: about 63 percent charged after one tau, essentially done after five.
tau = RC has units of seconds when R is in ohms and C in farads. Because charging is exponential, there is no exact 'finish' time; 5 tau (over 99 percent) is the practical rule.