A 60.0-Hz ac generator with a peak voltage of 110 V drives a series RC circuit with R = 10.0 and C = 10.0 mF. The impedance is
Recall that
Z^2 = R^2 + 1/(ωC)^2
and ω = 2πf
Xc = 1/(2pi*F*C) = 1/(377*10^-5) = 265.3 ohms.
Z = 10 - j265.3 = 265.5 ohms.[-88o].
To find the impedance (Z) of the series RC circuit, we need to calculate the total resistance (R) and the total reactance (X), and then find the magnitude of the impedance using the Pythagorean theorem.
The total resistance (R) includes the resistance (10.0 Ω) of the resistor (R) in the circuit.
The total reactance (X) includes the reactance (Xc) of the capacitor (C) in the circuit, which is given by the equation Xc = 1 / (2πfC), where f is the frequency (60.0 Hz) and C is the capacitance (10.0 mF).
Let's calculate R and Xc:
R = 10.0 Ω (given)
Xc = 1 / (2πfC)
= 1 / (2π * 60.0 Hz * 10.0 mF)
= 1 / (0.012π F)
≈ 26.53 Ω
Now, let's find the impedance (Z) using the equation Z = √(R^2 + X^2):
Z = √(R^2 + Xc^2)
= √(10.0 Ω^2 + 26.53 Ω^2)
= √(100.0 Ω^2 + 703.96 Ω^2)
= √(803.96 Ω^2)
≈ 28.36 Ω
Therefore, the impedance (Z) of the series RC circuit is approximately 28.36 Ω.