the time constant
/ tau / taw /
How long does a process 'take'? An exponentially decaying quantity never quite reaches its final value, so there is no single moment it is 'done.' Engineers and scientists handle this with the time constant — a single number, usually written tau (the Greek letter), that sets the natural timescale of the approach. It answers 'how fast does this settle?' in one tidy figure.
For a decay y(t) = y0 e^(-t/tau), the time constant tau is the time it takes to fall to 1/e ≈ 0.368 of the starting value — that is, to lose about 63% of the way toward zero. (Equivalently, for an approach to a steady value, tau is the time to cover 63% of the remaining gap.) It is simply the reciprocal of the rate constant: if dy/dt = -k y, then tau = 1/k. A small tau means a fast, snappy response; a large tau means a slow, sluggish one. A useful rule of thumb: after about 5 time constants the process is essentially complete (within 1%).
Every first-order linear model has one — Newton's cooling has tau = 1/k, the RC circuit has tau = RC, the RL circuit has tau = L/R — and it is the quickest way to compare how fast different systems respond. It is a close relative of half-life and doubling time, differing only by the constant ln 2: half-life = tau · ln 2 ≈ 0.693 tau. The time constant is the universal yardstick of first-order dynamics.
An RC circuit with R = 2000 ohm and C = 5 microfarad has time constant tau = RC = 0.01 s. The capacitor reaches about 63% of its final charge in 0.01 s and is essentially fully charged after 5 tau = 0.05 s.
tau = 1/k: time to fall to 1/e of the start (about 63% of the way).
The time constant is a property of first-order (single-exponential) behaviour. A system with two decay modes, or an oscillating second-order system, is not captured by a single tau.