Foundations — what do we call the characteristic time measure for first-order charging/discharging (e.g., in RC circuits) that sets the exponential response speed?

Introduction / Context:
In first-order systems such as RC and RL circuits, the output follows an exponential trajectory toward a new value after a step input. The pace of this response is governed by a single parameter that engineers use constantly for design, quick estimates, and troubleshooting. Recognizing and naming this parameter correctly is essential for time-domain intuition and for converting between time and frequency viewpoints.

Given Data / Assumptions:

• We consider linear first-order systems (e.g., RC, RL).
• The response to a step is exponential with a single scale factor.
• Terminology must be standard and unambiguous.

Concept / Approach:
The exponential law for an RC step is V(t) = V_final + (V_initial − V_final) * e^(−t/τ). The parameter τ (tau) is called the time constant. It equals R * C for RC circuits and L / R for RL circuits. After 1τ, the response has moved about 63.2% toward its final value; after 5τ, it is within about 0.7% of final, which is often treated as practically settled or “steady state” for many applications.

Step-by-Step Solution:

Verification / Alternative check:
Frequency domain linkage: the cutoff frequency f_c of a first-order filter is 1 / (2π τ). Thus time constant also determines bandwidth, confirming its centrality.

Why Other Options Are Wrong:

Common Pitfalls:
Equating “steady state” with a fixed time like 5τ; in precision work, required settling might be 7–10τ depending on tolerance.

Final Answer:
time constant