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?

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:

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