Sinusoidal AC phase relationships For an ideal capacitor driven by a sinusoidal source, the source voltage across the capacitor will _____ the current by 90°.

Introduction / Context:
In AC circuit analysis, understanding phase relationships between voltage and current for reactive elements (capacitors and inductors) is essential for designing filters, compensating power factor, and interpreting impedance. For a capacitor, current and voltage are out of phase by a quarter cycle (90°).

Given Data / Assumptions:

• Ideal capacitor with capacitance C.
• Sinusoidal steady-state excitation at angular frequency ω.
• No series resistance or parasitic effects.

Concept / Approach:
The capacitor current–voltage relation is i(t) = C * dv(t)/dt. For a sinusoidal voltage v(t) = Vp * sin(ωt), differentiation produces a cosine, which is a sine shifted by +90°. Therefore current leads voltage by 90°, or equivalently, voltage lags current by 90°.

Step-by-Step Solution:
Start with i(t) = C * dv/dt.Let v(t) = Vp * sin(ωt) → dv/dt = ω * Vp * cos(ωt).Since cos(ωt) = sin(ωt + 90°), current i(t) leads v(t) by 90°.Thus, phrased from the perspective of voltage: voltage lags current by 90°.

Verification / Alternative check:
Using phasors, capacitor impedance is Zc = 1/(jωC) = −j/(ωC). The −j indicates a −90° voltage phase relative to current, confirming that voltage lags current by 90°.

Why Other Options Are Wrong:
Lead by 90°: reversed relationship (true for current, not for voltage).Lag/lead by 180°: would imply inversion without reactance timing; not applicable to ideal capacitive behavior.

Common Pitfalls:
Memorizing incorrectly that “voltage leads current” for capacitors; it is the current that leads.Mixing up capacitor and inductor phase rules (inductor voltage leads current by 90°).

Final Answer:
lag the current by 90 degrees