For a series RC circuit driven by a sinusoidal source, how does the total impedance magnitude compare with R and the magnitude of Xc?

Introduction / Context:
Series RC circuits exhibit a total impedance formed by combining resistance and capacitive reactance at right angles in the complex plane. Understanding the geometric (phasor) relationship clarifies why the impedance does not simply add arithmetically.

Given Data / Assumptions:

• Components in series: resistor R and capacitor with reactance Xc (magnitude).
• Sinusoidal steady state at angular frequency ω.
• Impedance magnitude is sought, not the complex form.

Concept / Approach:
Total series impedance: Z = R − jXc. Magnitude |Z| = sqrt(R^2 + Xc^2). Since R and Xc combine orthogonally, |Z| is strictly greater than each individual magnitude but strictly less than the arithmetic sum R + Xc.

Step-by-Step Solution:
Write complex form: Z = R + (−jXc).Compute magnitude: |Z| = sqrt(R^2 + Xc^2).Compare: |Z| > max(R, Xc) and |Z| < R + Xc.Therefore, the precise description is “greater than either but less than their sum.”

Verification / Alternative check:
Numerical example: R = 3 Ω, Xc = 4 Ω ⇒ |Z| = 5 Ω which is greater than 4 and 3, but less than 3 + 4 = 7.

Why Other Options Are Wrong:
Greater than sum / equal to sum: would require colinear vectors, not orthogonal reactance.“Less than the sum” alone is true but incomplete; option (c) gives the full correct relationship.None: incorrect since (c) is exact.

Common Pitfalls:
Adding magnitudes arithmetically; ignoring the 90-degree phase between R and Xc; confusing series RC with parallel behavior.

Final Answer:
The impedance is greater than either the resistance or the capacitive reactance but less than their sum