Whenever the alternating current frequency in a series RLC circuit is halved,
A) the inductive reactance is doubled and the capacitive reactance is halved.
B) the inductive reactance is doubled and the capacitive reactance is doubled.
C) the inductive reactance is halved and the capacitive reactance is halved.
D) the inductive reactance is halved and the capacitive reactance is doubled.
E) the reactance of the circuit remains the same.

Question

Quizplus · Accepted Answer

In a series RLC circuit, the total impedance at any frequency is given by:

Z = √(R² + (Xl - Xc)²)

Where R is the resistance, Xl is the inductive reactance and Xc is the capacitive reactance.

At resonance, Xl = Xc and the total impedance is purely resistive, i.e. Z = R.

When the frequency is halved, the inductive reactance (Xl) and capacitive reactance (Xc) both change according to the formulas:

Xl = 2πfL

Xc = 1/(2πfC)

Where L is the inductance and C is the capacitance.

Halving the frequency means dividing f by 2, so we get:

Xl' = 2π(f/2)L = πL

Xc' = 1/(2π(f/2)C) = 2/(πC)

Thus, the inductive reactance is halved (Xl' = 0.5Xl) and the capacitive reactance is doubled (Xc' = 2Xc).

Therefore, the total impedance of the circuit is altered since it's no longer purely resistive. This change in impedance affects the behavior of the circuit, causing it to behave differently at different frequencies. So the correct choice is D.

Whenever the Alternating Current Frequency in a Series RLC Circuit