Final answer:
The resonant frequency of the RLC circuit is approximately 1.80 Hz. The complex impedance Z is approximately 33.3 kΩ with a phase angle of 0.452 radians.
Step-by-step explanation:
The resonant frequency of an RLC circuit can be determined using the formula:
f_res = 1 / (2π√(LC))
Where f_res is the resonant frequency, L is the inductance, and C is the capacitance.
In this case, the given values are L = 900 H and C = 4.94 µF (or 4.94 x 10^-6 F).
Substituting these values into the formula, we can calculate:
f_res = 1 / (2π√(900 H x 4.94 x 10^-6 F))
f_res ≈ 1.80 Hz
The resonant frequency of this RLC circuit is approximately 1.80 Hz.
To find the complex impedance Z, we can use the formula:
Z = √(R^2 + (Xl - Xc)^2)
Where Z is the impedance, R is the resistance, Xl is the inductive reactance, and Xc is the capacitive reactance.
The given values are R = 30.0 kΩ, Xl = 2πfL, and Xc = 1 / (2πfC).
Substituting these values and the given driving frequency of 4.00 Hz, we can calculate:
Xl = 2π x 4.00 Hz x 900 H ≈ 2261 Ω
Xc = 1 / (2π x 4.00 Hz x 4.94 x 10^-6 F) ≈ 8073 Ω
Z = √((30.0 kΩ)^2 + (2261 Ω - 8073 Ω)^2)
Z ≈ 33.3 kΩ (magnitude) and Ø ≈ 0.452 rad (phase angle)
The magnitude |Z| of the impedance is approximately 33.3 kΩ and the phase angle Ø is approximately 0.452 radians.