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RC: $\tau =RC$ — RL: $\tau =\frac{L}{R}$ — one time constant is how long the circuit takes to cover 63.2 % of the remaining gap to its final value.

RC circuit

Example τ = 100 ms

How long until the capacitor crosses a given voltage — the question a bare τ never answers on its own.

Example Charging: reaches 2 V after 51.08 ms (51.1% of τ)

RL circuit

Example τ = 100 µs

How long until the current crosses a given value, driven by a supply Vs through R.

Example Rising: reaches 30 mA after 91.63 µs (91.6% of τ), Iss = 50 mA