Series RC magnitude: A 47 Ω resistor is in series with a capacitor having reactance Xc = 120 Ω across an AC source. What is the magnitude of the circuit impedance Z?

Introduction / Context:
The magnitude of impedance in a series RC circuit is found using the Pythagorean relationship between resistance and reactance. This is a basic but critical tool for determining current and voltage drops in AC circuits.

Given Data / Assumptions:

• R = 47 Ω (purely resistive).
• Xc = 120 Ω (purely capacitive reactance).
• Series connection; ignore parasitics.

Concept / Approach:

Impedance magnitude for series RC: |Z| = √(R^2 + Xc^2). This comes from vector addition of orthogonal components on the complex plane (R along real axis, −Xc along imaginary axis).

Step-by-Step Solution:

Verification / Alternative check:

Because Xc ≫ R, the impedance should be closer to Xc than to R; 129 Ω is just slightly above 120 Ω, which is reasonable.

Why Other Options Are Wrong:

12.9 Ω is off by a factor of 10 (decimal misplacement). 167 Ω suggests adding magnitudes linearly. 73 Ω is smaller than both R and Xc, impossible for a series combination.

Common Pitfalls:

Adding R and Xc directly; forgetting to square before adding; ignoring that Xc is orthogonal to R.

Final Answer:

129 Ω