When does current flow in a capacitive branch? For an ideal capacitor in a circuit, current is present during which time intervals?

Introduction / Context:
Capacitive current is tied to how the capacitor's voltage changes with time. Recognizing when current flows (and when it does not) is key for waveform analysis, timing circuits, and filter design.

Given Data / Assumptions:

• Ideal capacitor with i = C * dv/dt.
• Charging and discharging refer to periods when the capacitor voltage is increasing or decreasing.
• Steady DC conditions imply no change in capacitor voltage.

Concept / Approach:
If dv/dt ≠ 0 (voltage is changing), current flows. That occurs both while the capacitor is charging (dv/dt > 0 for a positive-going change) and while it is discharging (dv/dt < 0). When the voltage is constant, dv/dt = 0 and current is zero.

Step-by-Step Solution:
Start from i = C * dv/dt.Charging: dv/dt is positive ⇒ i is positive (for chosen polarity).Discharging: dv/dt is negative ⇒ i is negative; magnitude is nonzero.Steady DC: dv/dt = 0 ⇒ i = 0 (no current in an ideal capacitor at steady state).

Verification / Alternative check:
Consider an RC step response. Current i(t) = (V/R) * e^(-t/(RC)) during charging and i(t) = -(V/R) * e^(-t/(RC)) during discharging. In both cases, current exists until it decays to zero.

Why Other Options Are Wrong:

• Charging periods only / discharging periods only: incomplete; current flows in both phases.
• Neither: incorrect; only at steady state is current zero.

Common Pitfalls:

• Thinking only “charging” causes current; any time the voltage changes, current flows.
• Ignoring polarity: the sign of current flips between charge and discharge, but magnitude is what matters for conduction.

Final Answer:
both charging periods and discharging periods