In a series RC circuit, impedance is:
Group of answer choices
none of the above.
always smaller than R.
the same as R.
larger than R.
The correct answer and explanation is:
Correct Answer: larger than R.
Explanation:
In a series RC circuit, the total impedance ZZ is the combination of the resistance RR and the capacitive reactance XCX_C. Because resistance and reactance are perpendicular in the complex plane, impedance is calculated using the Pythagorean theorem: Z=R2+XC2Z = \sqrt{R^2 + X_C^2}
where
- RR is the resistance (measured in ohms),
- XCX_C is the capacitive reactance (also measured in ohms), defined as:
XC=12πfCX_C = \frac{1}{2 \pi f C}
where ff is the frequency in hertz and CC is the capacitance in farads.
Why is impedance larger than R?
- Reactance adds magnitude:
The resistance RR represents the real part of impedance, while the capacitive reactance XCX_C represents the imaginary part. Since impedance ZZ is the vector sum of RR and XCX_C, its magnitude is always at least as large as RR, and typically larger unless XC=0X_C = 0 (which happens only at infinite frequency). - Impedance is never smaller than resistance alone:
Because of the Pythagorean sum, if XC≠0X_C \neq 0, then Z=R2+XC2>RZ = \sqrt{R^2 + X_C^2} > R - Special cases:
- At very high frequency, the capacitive reactance XC→0X_C \to 0, so Z→RZ \to R.
- At very low frequency, XC→∞X_C \to \infty, so Z→∞Z \to \infty.
- Physical interpretation:
The capacitor opposes changes in voltage by introducing reactance, which increases the total opposition to current flow beyond the resistance alone. This means the circuit impedes current more than just the resistor would by itself.
Summary:
- Impedance ZZ in a series RC circuit is the combination of resistance and capacitive reactance.
- Because of the perpendicular (vector) nature of resistance and reactance, impedance magnitude is always larger than the resistance alone unless the reactance is zero.
- Thus, the correct choice is “larger than R.”