Exponential decay reality check — after the rising edge in a first-order RC pulse network, how long does it take for the voltage across the resistor to “decrease to zero”? (Interpret “zero” as a practical threshold; without τ the value is not unique.)

Introduction / Context:In a first-order RC network, voltages across elements change exponentially. They never reach mathematical zero in finite time; design practice uses thresholds such as 1τ (~63% change) or 5τ (~99% change). If a problem asks for “time to zero” but does not specify τ or a practical threshold, no single numeric answer exists.

Given Data / Assumptions:

• No R, C, or τ given.
• No threshold definition (e.g., 1% or 0.1%) provided.
• Standard first-order behavior assumed.

Concept / Approach:The resistor voltage after a step is an exponential of the form v_R(t) = K * exp(−t/τ). For any finite t, exp(−t/τ) > 0. A practical engineering “zero” requires choosing a percentage (e.g., 5τ for ~99% decay). Without τ and a threshold, the requested time cannot be fixed.

Step-by-Step Solution:

Verification / Alternative check:Assuming τ = 100 µs gives “≈zero” near 500 µs; assuming τ = 1 ms shifts this to ≈5 ms. Different τ yield different times.

Why Other Options Are Wrong:

• 0 s / 1τ / 5τ / half period: each presumes a specific definition or value of τ or period not provided.

Common Pitfalls:Taking 5τ as a universal answer even when τ is unknown; ignoring that “zero” needs a tolerance.

Final Answer:Cannot be determined from the information provided.