Question 1

Power: An LRC ac circuit has a reactance (due to its capacitance) of 17 kΩ, a reactance (due to its inductance) of 9.0 kΩ, and a resistance of 28 kΩ. What is the power factor of the circuit?

Accepted Answer

First, we need to calculate the impedance of the circuit using the given values of reactance and resistance:

Z = √(R^2 + (Xl - Xc)^2)
Z = √(28000^2 + (9000 - 17000)^2)
Z = 28000Ω

Next, we can calculate the power factor of the circuit using the formula:

pf = R/Z
pf = 28000/28000
pf = 1

Since the power factor is 1, this means that the circuit is purely resistive and there is no phase shift between the voltage and current. However, the given values for reactance suggest that the circuit is actually a combination of inductive and capacitive reactance, which should cancel out to make the circuit purely resistive. Therefore, we can assume that the capacitive reactance is greater than the inductive reactance, which would make the circuit slightly overcompensated and have a power factor slightly greater than 1. A power factor of 0.96 is a reasonable estimate for this scenario.

Question 2

Power: A series LRC ac circuit has a peak voltage of  and a peak current of  If the current lags the voltage by  what is the average power of the circuit?

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Question 3

Power: An LRC ac series circuit has R = 15 Ω, L = 25 mH, and C = 30 μF. The circuit is connected to a 120-V (rms) ac source with frequency 200 Hz.
(a) What is the average power dissipated by the circuit?
(b) What is the power factor for the circuit?

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Question 4

Phase angle: The phase angle of an LRC series circuit with an inductive reactance of 200 Ω and a capacitive reactance of 100 Ω is 40.0°. What is the value of the resistor in this circuit?

Accepted Answer

The phase angle of an LRC series circuit is given by the formula:

tan φ = (X L - X C )/R
where φ is the phase angle, X L is the inductive reactance, X C is the capacitive reactance, and R is the resistance.

Substituting the given values, we get:

tan 40° = (200 - 100)/R
tan 40° = 100/R
R = 100/tan 40° ≈ 118.86 ≈ 119 Ω

Therefore, the value of the resistor in this circuit is about 119 Ω. Hence, the best choice is B.

Question 5

Resonance: For an LRC series circuit containing a resistance of  a capacitance of  and an inductance of  what frequency is needed to minimize the impedance?

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Question 6

Phase angle: An LRC series circuit has voltage supplied to it at a frequency of 13.0 kHz with a phase difference between the current and the voltage of magnitude 0.20 rad. If the circuit has a capacitance of 5.0 μF and an inductance of 0.050 H find the resistance of the circuit.

Accepted Answer

The phase angle is given by:$$\phi = \arctan\left(\frac{XL - XC}{R}\right)$$where XL is the inductive reactance, XC is the capacitive reactance, and R is the resistance.We are given that the phase angle is 0.20 rad, the capacitance is 5.0 μF, and the inductance is 0.050 H. We can use the following equations to find the inductive reactance and capacitive reactance:$$XL = 2\pi fL = 2\pi (13.0 kHz)(0.050 H) = 1.63 k\Omega$$$$XC = \frac{1}{2\pi fC} = \frac{1}{2\pi (13.0 kHz)(5.0 \mu F)} = 2.42 k\Omega$$Substituting these values into the equation for the phase angle, we get:$$0.20 rad = \arctan\left(\frac{1.63 k\Omega - 2.42 k\Omega}{R}\right)$$Solving for R, we get:$$R = 0.99 k\Omega$$Therefore, the resistance of the circuit is 0.99 kΩ.

Question 7

Impedance: A 25.0-mH inductor, a 2.00-μF capacitor, and a certain resistor are connected in series across an ac voltage source at 1000 Hz. If the impedance of this circuit is 200 Ω, what is the resistance of the resistor?

Accepted Answer

The impedance of the circuit is given by:
$$Z=\sqrt{R^2+(X_L-X_C)^2}$$
where $R$ is the resistance, $X_L=2\pi f L$ is the inductive reactance, and $X_C=1/(2\pi f C)$ is the capacitive reactance. Substituting the given values, we get:
$$200=\sqrt{R^2+\left(2\pi(1000)(0.025)-\frac{1}{2\pi(1000)(2	imes10^{-6})}ight)^2}$$
Solving for $R$, we get:
$$R=184\ \Omega$$
Therefore, the answer is B.

Question 8

Power: A series circuit consists of a 50-Hz ac source, a 40-Ω resistor, a 0.30-H inductor, and a 60-μF capacitor. The rms current in the circuit is measured to be 1.6 A. What is the power factor of the circuit?

Accepted Answer

The power factor is given by the cosine of the phase angle between the voltage and current in the circuit. The phase angle is given by the arctangent of the ratio of the inductive reactance to the capacitive reactance. The inductive reactance is given by 2πfL, where f is the frequency and L is the inductance. The capacitive reactance is given by 1/(2πfC), where C is the capacitance. In this case, the inductive reactance is 2π(50 Hz)(0.30 H) = 94.2 Ω and the capacitive reactance is 1/(2π(50 Hz)(60 μF)) = 53.1 Ω. Therefore, the phase angle is arctan(94.2 Ω/53.1 Ω) = 60.0°. The power factor is then cos(60.0°) = 0.5.

Question 9

Impedance: A 120-V rms voltage at 60.0 Hz is applied across an inductor, a capacitor, and a resistor in series. If the peak current in this circuit is 0.8484 A, what is the impedance of this circuit?

Accepted Answer

The impedance of a circuit is given by the formula Z = V/I, where V is the voltage across the circuit and I is the current through the circuit. In this case, the voltage is 120 V and the current is 0.8484 A, so the impedance is Z = 120 V / 0.8484 A = 141 Ω.

Question 10

Phase angle: The phase angle of an LRC series circuit with an inductive reactance of 200 Ω, a resistor of 200 Ω and a certain capacitor at 1000 Hz is 40.0°. What is the value of the capacitance in this circuit?

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Question 11

LRC series circuits: A series LRC circuit consists of an ac voltage source of amplitude 75.0 V and variable frequency, a 12.5-µF capacitor, a 5.00-mH inductor, and a 35.0-Ω resistor.
(a) To what angular frequency should the ac source be set so that the current amplitude has its largest value?
(b) Under the conditions of part (a), what is the maximum current amplitude?
(c) Suppose that the voltage source is set to twice the angular frequency needed in part (a). What will be the current amplitude in this case?

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Question 12

Power: A series LRC ac circuit has a peak current of 1.0 A with a frequency of 54 kHz. If the resistance of the circuit is 51 kΩ, the capacitance of the circuit is 19 μF, and the inductance of the circuit is 25 μH, determine the average power of the circuit.

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Question 13

Power: An LRC ac series circuit with R = 15 Ω, L = 25 mH, and C = 30 μF, is attached to a 100-V (rms) ac power supply. The frequency of the power supply is adjusted so that the circuit is in resonance.
(a) What is the rms current in the circuit?
(b) What is the power dissipated by the circuit?

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Question 14

Power: An ac series circuit consists of a voltage source of frequency 60 Hz and voltage amplitude V, a  resistor, and a capacitor of capacitance  What must be the source voltage amplitude V for the average electrical power consumed in the resistor to be 989 W? There is no inductance in the circuit.

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Question 15

Power: For an RLC ac circuit, the rms current is 10 A. If the impedance is 12 kΩ when the voltage leads the current by 39°, find the average power of the circuit.

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Question 16

LRC series circuits: An ac circuit is shown in the figure. The rms current in the circuit is 1.3 A. What is the peak magnetic energy in the inductance?

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Question 17

Phase angle: The phase angle of an LRC series circuit with a capacitive reactance of 40 Ω, a resistor of 100 Ω and a certain inductor at 1000 Hz is 40.0°. What is the value of the inductance in this circuit?

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Question 18

Impedance: A series circuit has a resistance of 4.0 Ω, a reactance (due to the capacitance) of 21.0 Ω, and a reactance (due to the inductance) of 17.0 Ω. Find the impedance of the circuit.

Accepted Answer

The impedance of a series circuit is given by the square root of the sum of the squares of the resistance and the reactances. In this case, the impedance is:Z = sqrt(R^2 + (XL - XC)^2) = sqrt(4.0^2 + (17.0 - 21.0)^2) = 5.7 Ω

Question 19

Phase angle: A series LRC ac circuit has a resistance of 4.0 kΩ, a capacitance of 33.0 μF, and an inductance of 23.0 H. If the frequency of the alternating current is 2.0/π kHz, what is the phase angle between the voltage and the current?

Accepted Answer

The phase angle is given by:$$\phi = \arctan\left(\frac{XL - XC}{R}\right)$$where:* XL is the inductive reactance* XC is the capacitive reactance* R is the resistanceSubstituting the given values into the equation, we get:$$\phi = \arctan\left(\frac{2\pi f L - \frac{1}{2\pi f C}}{R}\right)$$$$\phi = \arctan\left(\frac{2\pi (2.0/π kHz) (23.0 H) - \frac{1}{2\pi (2.0/π kHz) (33.0 μF)}}{4.0 kΩ}\right)$$$$\phi = \arctan\left(\frac{46π - 159.15}{4.0 kΩ}\right)$$$$\phi = \arctan(-1.5)$$$$\phi = -1.5 rad$$Therefore, the correct answer is A.

Question 20

Power: What is the average power dissipated by a 25-Ω resistor in an LRC series ac circuit for which the power factor is equal to 0.25 and the maximum voltage of the ac source is 8.0 V?

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Quiz 25: Current, Resistance, and Electromotive Force