Impedance magnitude of a series RC at 3 kHz: A 0.22 µF capacitor is in series with a 200 Ω resistor and connected to a 3 kHz sinusoidal source. What is the magnitude of the total impedance |Z|?

Introduction:In a series RC circuit, impedance is the vector sum of resistance and capacitive reactance. The magnitude determines current for a given applied voltage and frequency, which is key in filter and timing applications.

Given Data / Assumptions:

• R = 200 Ω
• C = 0.22 µF
• f = 3 kHz
• Series connection, ideal components

Concept / Approach:

Compute capacitive reactance Xc = 1 / (2 * π * f * C). Then the impedance magnitude is |Z| = sqrt(R^2 + Xc^2).

Step-by-Step Solution:

Verification / Alternative check:

Since R and Xc are comparable (200 Ω vs 241 Ω), |Z| must exceed each but be less than their sum; 313 Ω satisfies this bound (200 < 313 < 441).

Why Other Options Are Wrong:

• 214 Ω: Less than R; impossible for magnitude with additional reactance.
• 414 Ω and 880 Ω: Overestimates; closer to linear addition, not vector.
• 241 Ω: This is Xc alone, not the total magnitude.

Common Pitfalls:

• Adding R and Xc arithmetically instead of vectorially.
• Using wrong units for C (µF vs F).

Final Answer:

313 Ω