When the response is due to the initial energy stored and the physical characteristics of the circuit and not due to some external voltage or current source, it is called the natural response of the circuit.
The natural response of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitation.
The rapidity with which the voltage (or current) decreases is expressed in terms of the time constant, denoted by \(\tau\) (tau).
The time constant \(\tau\) of a circuit is the time required for the response to decay to a factor of \(e^{-1}\) or \(36.8\) percent of its initial value.
This implies that at \(t = \tau\)
\(\displaystyle V_0\,e^{\displaystyle -t / \tau} = V_0\,e^{-1} = 0.368 V_0\)
A circuit with a small time constant gives a fast response in that it reaches the steady state (or final state) quickly due to quick dissipation of energy stored, whereas a circuit with a large time constant gives a slow response because it takes longer to reach steady state.
At any rate, whether the time constant is small or large, the circuit approximately reaches the steady state in five time constants, \(t = 5\tau\).