47. A circuit consists of a 220.0Ω
resistor and a 0.5II inductor
connected across a generator that
has a frequency of 120Hz and a
voltage if 230V. Determine the
phase angle between the current
and the voltage
A. 59.73°
B. 68.38°
C. 26.75°
D. 30.27°
E. 38.21°
To find the phase angle between the current and the voltage, we can use the formula:
tan(φ) = (wL - 1/wC) / R
Where:
φ = phase angle
w = angular frequency = 2πf
L = inductance
C = capacitance
R = resistance
In this case, there is no capacitor in the circuit, so wC = 0. Thus, the formula simplifies to:
tan(φ) = wL / R
Given:
f = 120 Hz
w = 2πf = 2π(120) = 240π rad/s
L = 0.5 H
R = 220.0 Ω
Plugging in these values, we get:
tan(φ) = (240π)(0.5) / 220.0
Using a calculator, we find:
tan(φ) ≈ 3.456
To find the phase angle φ, we take the arctan of 3.456:
φ ≈ arctan(3.456)
Using a calculator, we find:
φ ≈ 68.38°
Therefore, the correct answer is B. 68.38°.
To find the phase angle between the current and the voltage in this circuit, we need to first calculate the reactance of the inductor, then find the impedance of the circuit.
1. Calculate the reactance of the inductor (XL):
- XL = 2πfL
- Given frequency (f) = 120 Hz
- Given inductance (L) = 0.5 H
- XL = 2π(120)(0.5)
- XL = 377 Ω
2. Calculate the impedance of the circuit (Z):
- Z = √(R^2 + XL^2)
- Given resistor value (R) = 220 Ω
- Z = √(220^2 + 377^2)
- Z ≈ 430 Ω
3. Calculate the phase angle (θ):
- θ = arctan(XL/R)
- θ = arctan(377/220)
- θ ≈ 59.73° (rounded to two decimal places)
Therefore, the correct answer is option A) 59.73°.