RC series behavior — interpreting zero voltage across the resistor In a series R–C circuit connected to a DC source, if the measured voltage across the resistor is zero at steady observation, the capacitor is:

Introduction / Context:Reading voltages in an RC circuit is a quick way to infer the state of charge of the capacitor. For DC sources, steady-state behavior is simple and governed by exponential transients that settle over several time constants.

Given Data / Assumptions:

• A single resistor and capacitor in series with a DC source.
• The observed voltage across the resistor is zero (within measurement resolution).
• Ideal components and steady observation (after transients).

Concept / Approach:In a series RC with DC, the transient current i(t) flows while the capacitor voltage changes. Once the capacitor reaches its final DC voltage, dv/dt = 0, so i = C * dv/dt = 0. With i = 0, the resistor drop V_R = i * R = 0 V. Therefore, zero across R indicates the capacitor has reached its final (charged) value.

Step-by-Step Solution:Write the capacitor differential relation: i = C * dvC/dt.At steady DC, dvC/dt = 0 ⇒ i = 0.Resistor voltage: V_R = i * R = 0 V.Hence the capacitor must be fully charged to the source DC level (minus any series drops).

Verification / Alternative check:Transient expression: vC(t) = V * (1 - e^(-t/(RC))). As t → ∞, vC → V and i(t) = (V/R) * e^(-t/(RC)) → 0. Thus V_R = i*R → 0, consistent with the observation.

Why Other Options Are Wrong:

• Charging/discharging: during either, current is nonzero, so the resistor would show a nonzero drop.
• Fully discharged: would produce maximum current initially and a large resistor drop, not zero.

Common Pitfalls:

• Confusing the instant just after switching (when V_R is largest) with steady state.
• Assuming zero across R means an open circuit elsewhere; here it simply means i = 0 in steady DC.

Final Answer:fully charged