AC phase relationships: in a purely inductive circuit, how does the current waveform relate in phase to the applied voltage waveform?

Introduction / Context:
Understanding phase relationships is essential for analyzing reactive AC circuits, power factor correction, and resonance. Inductors and capacitors shift current relative to voltage in opposite ways, which affects real and reactive power flow.

Given Data / Assumptions:

• Purely inductive load (ideal inductor with no resistance).
• Sinusoidal steady-state excitation.
• Focus on phase, not magnitude.

Concept / Approach:
For an ideal inductor, v_L = L * di/dt. A sinusoidal voltage leads the resulting sinusoidal current by 90 degrees; equivalently, current lags voltage by 90 degrees. This stems from the inductor opposing changes in current, causing a delay in current relative to the applied voltage.

Step-by-Step Solution:
Assume v(t) = V_m * sin(ωt).Then i(t) = (1/ωL) * ∫ v(t) dt = (V_m/(ωL)) * (-cos(ωt)) = (V_m/(ωL)) * sin(ωt − 90°).Therefore, current lags voltage by 90° in an ideal inductor.Select “Current lags the voltage through an inductor.”

Verification / Alternative check:
Phasor diagrams place the inductor’s impedance at +jX_L; current phasor is 90° behind the voltage across the inductor, confirming the lag.

Why Other Options Are Wrong:
A: In phase applies to pure resistors. B: “Current leads” describes capacitors, not inductors. D/E: Not applicable because one specific relationship holds for ideal inductors.

Common Pitfalls:
Forgetting the effect of series resistance (which reduces but does not reverse the lag); confusing capacitor and inductor phase behaviors.

Final Answer:
Current lags the voltage through an inductor