Compute capacitive reactance: a 0.47 µF capacitor across a 2 kHz sine-wave source—what is Xc?

Introduction / Context:Capacitive reactance indicates how a capacitor impedes AC at a given frequency. It is critical for sizing coupling capacitors, setting filter cutoffs, and calculating impedance in signal paths.

Given Data / Assumptions:

• Capacitance C = 0.47 µF = 0.47 × 10^-6 F.
• Frequency f = 2 kHz.
• Ideal capacitor (neglect ESR).

Concept / Approach:The formula for capacitive reactance is Xc = 1 / (2 * π * f * C). Plug in values carefully, keeping units consistent, to avoid power-of-ten mistakes.

Step-by-Step Solution:

Verification / Alternative check:Sanity check: At audio kHz and sub-microfarad capacitance, Xc should be on the order of hundreds of ohms; result is reasonable.

Why Other Options Are Wrong:

• 17 Ω or 1.7 Ω: Off by decade(s), likely unit conversion errors.
• 0.000169 Ω: Physically unrealistic; would imply a near-short.
• 340 Ω: Would correspond to half the frequency or half the capacitance; not our case.

Common Pitfalls:

• Losing micro (10^-6) in calculations.
• Rounding early instead of at the end.

Final Answer:170 Ω