In a series RC circuit, both the source frequency f and resistance R are halved. Without numerical values of R, C, and f, how does the total impedance |Z| change?
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Acannot be determined without values
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Bdoubles
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Cis halved
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Dis reduced to one-fourth
Answer
Correct Answer: cannot be determined without values
Explanation
Introduction / Context:The magnitude of impedance in a series RC network depends on both resistance and frequency-dependent reactance. When two parameters change simultaneously, the combined effect on |Z| cannot be predicted uniquely without specific values.
Given Data / Assumptions:
- Series RC with R and C constant initially.
- Change: R → R/2 and f → f/2.
- Capacitance C unchanged.
Concept / Approach:
Impedance magnitude is |Z| = √(R^2 + Xc^2), where Xc = 1 / (2 * π * f * C). Halving f doubles Xc; halving R halves R. The new magnitude depends on the relative sizes of R and Xc before the change.
Step-by-Step Solution:
Original: |Z| = √(R^2 + Xc^2).After change: R′ = R/2, Xc′ = 2 * Xc.New |Z′| = √((R/2)^2 + (2Xc)^2) = √(R^2/4 + 4Xc^2).The ratio |Z′| / |Z| = √((R^2/4 + 4Xc^2) / (R^2 + Xc^2)) depends on the R/Xc balance and has no single fixed value.Verification / Alternative check:
If R ≫ Xc, halving R may dominate and |Z′| decreases; if Xc ≫ R, doubling Xc may dominate and |Z′| increases. Different starting conditions give different outcomes.
Why Other Options Are Wrong:
'Doubles', 'halved', and 'one-fourth' assert universal factors that ignore the relative sizes of R and Xc.
Common Pitfalls:
Assuming proportional changes carry over to the square-root sum; forgetting Xc depends on f.
Final Answer:
cannot be determined without values