RC series phase relationship (magnitude vs. sign): For a series R–C circuit, is the phase angle magnitude given by |φ| = arctan(Xc / R) = arctan(Vc / Vr), with the understanding that the capacitor causes a negative (lagging) angle for the total voltage relative to current?

Introduction / Context:
Phase angle in AC circuits relates voltages and currents through impedance. For a series R–C circuit, engineers often use impedance triangles and phasors to express the relationship between resistive and reactive drops. This question checks whether the common magnitude formula for the phase angle in a series R–C circuit is correctly stated, and clarifies the sign convention for capacitive behavior.

Given Data / Assumptions:

• Series R (ohms) and C (farads) driven by a sinusoidal source.
• Capacitive reactance Xc = 1 / (2π f C).
• Phasor notation with current as the reference; ideal components; steady state.

Concept / Approach:
In series R–C, the impedance is Z = R − jXc. The phase angle φ between total voltage and current satisfies tan(φ) = −Xc / R. The magnitude is |φ| = arctan(Xc/R). Because the drop across the capacitor lags the current by 90°, the overall angle is negative (capacitive). The ratio of drops also reflects this: Vc / Vr = Xc / R, consistent with the triangle of voltages in a series path.

Step-by-Step Solution:

Verification / Alternative check:
Construct the right triangle with sides R (adjacent) and Xc (opposite). The angle at the series node has tangent Xc/R (magnitude). Sign is assigned by the reactive sign (capacitive negative). Measurements with a scope confirm Vc leads Vr in quadrature, matching the geometry.

Why Other Options Are Wrong:
Incorrect: conflicts with standard phasor analysis.
Only true at resonance: R–C series has no resonance without an L; the relation holds at all f.
Cannot be used unless R >> Xc: no such restriction exists for the identity.

Common Pitfalls:
Forgetting the negative sign for capacitive circuits; mixing up magnitude and signed angle; confusing Vc/Vr with Vr/Vc.

Final Answer:
Correct