Impedance of an RC series circuit: Is it correct that the magnitude of impedance is (R^2 + Xc^2)^(1/2), where Xc = 1 / (2 * pi * f * C), rather than (R^2 + C^2)^(1/2)?

Introduction / Context:Impedance generalizes resistance to AC by incorporating phase and frequency dependence. For a series RC, the impedance vector is Z = R − j * Xc. Confusing the capacitance C (in farads) with the capacitive reactance Xc (in ohms) leads to dimensional errors and wrong results.

Given Data / Assumptions:

• Sinusoidal steady state at frequency f.
• Ideal R and C elements.
• Magnitude operation on complex impedance.

Concept / Approach:Capacitive reactance is Xc = 1 / (2 * pi * f * C). The series impedance is Z = R − j * Xc. The magnitude is |Z| = sqrt(R^2 + Xc^2). Substituting C directly into sqrt(R^2 + C^2) is incorrect because C has units of farads, not ohms. Proper dimensional analysis reinforces the correct formula.

Step-by-Step Solution:

Verification / Alternative check:Phasor diagrams and impedance triangles used in laboratory measurements match the sqrt(R^2 + Xc^2) expression and produce correct current and phase predictions.

Why Other Options Are Wrong:

Common Pitfalls:Plugging capacitance directly where reactance belongs; forgetting the frequency dependence of Xc.

Final Answer:Correct