Ch 31: Alternating Current

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 31: Alternating Current

Problem 18

Chapter 31, Problem 18

A resistor with R = 300 Ω and an inductor are connected in series across an ac source that has voltage amplitude 500 V. The rate at which electrical energy is dissipated in the resistor is 286 W. What is (a) the impedance Z of the circuit; (b) the amplitude of the voltage across the inductor; (c) the power factor?

Verified step by step guidance

Step 1: Start by understanding the relationship between power, voltage, and resistance in an AC circuit. The power dissipated in the resistor can be calculated using the formula: \( P = \frac{V^2}{R} \), where \( P \) is the power, \( V \) is the voltage across the resistor, and \( R \) is the resistance.

Step 2: Given that the power dissipated in the resistor is 286 W and the resistance \( R \) is 300 Ω, use the formula \( P = \frac{V^2}{R} \) to find the voltage across the resistor. Rearrange the formula to solve for \( V \): \( V = \sqrt{P \times R} \).

Step 3: To find the impedance \( Z \) of the circuit, use the formula for the total voltage in an AC circuit: \( V_{total} = I \times Z \), where \( I \) is the current. First, find the current using \( I = \frac{V_{resistor}}{R} \), then use \( Z = \frac{V_{total}}{I} \) to find the impedance.

Step 4: To find the amplitude of the voltage across the inductor, use the relationship \( V_{inductor} = I \times X_L \), where \( X_L \) is the inductive reactance. Since \( Z = \sqrt{R^2 + X_L^2} \), you can solve for \( X_L \) using the impedance found in Step 3.

Step 5: The power factor is the ratio of the real power to the apparent power in the circuit. It can be calculated using \( \text{Power Factor} = \frac{R}{Z} \). Use the values of \( R \) and \( Z \) obtained in previous steps to find the power factor.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

13m

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Impedance in AC Circuits

Impedance (Z) is the total opposition a circuit presents to the flow of alternating current (AC) and is a combination of resistance (R) and reactance (X). In a series R-L circuit, impedance is calculated using Z = √(R² + (ωL)²), where ω is the angular frequency and L is the inductance. Impedance affects both the current flow and the phase relationship between voltage and current.

Voltage Across an Inductor

In an AC circuit, the voltage across an inductor is determined by the inductive reactance (X_L = ωL) and the current flowing through it. The voltage amplitude across the inductor can be found using V_L = I * X_L, where I is the current amplitude. This voltage is out of phase with the current by 90 degrees, leading the current in time.

Power Factor in AC Circuits

The power factor in an AC circuit is the cosine of the phase angle (φ) between the voltage and current, given by cos(φ) = R/Z. It indicates how effectively the circuit converts electrical power into useful work. A power factor of 1 means all the power is used effectively, while a lower power factor indicates more power is wasted in the form of reactive power.

Recommended video:

Guided course

05:37

Power in AC Circuits

Related Practice

Textbook Question

In an L-R-C series circuit, the components have the following values: L = 20.0 mH, C = 140 nF, and R = 350 Ω.The generator has an rms voltage of 120 V and a frequency of 1.25 kHz. Determine (a) the power supplied by the generator and (b) the power dissipated in the resistor.

views

Textbook Question

You have a 200-Ω resistor, a 0.400-H inductor, and a 6.00-μF capacitor. They are connected to form an L-R-C series circuit. What is the impedance of the circuit?

Textbook Question

You have a 200-Ω resistor, a 0.400-H inductor, and a 6.00-μF capacitor. Suppose you take the resistor and inductor and make a series circuit with a voltage source that has voltage amplitude 30.0 V and an angular frequency of 250 rad/s. What is the impedance of the circuit?

Textbook Question

An L-R-C series circuit with L = 0.120 H, R = 240 Ω, and C = 7.30 μF carries an rms current of 0.450 A with a frequency of 400 Hz. What are the phase angle and power factor for this circuit?

views

Textbook Question

The power of a certain CD player operating at 120 V rms is 20.0 W. Assuming that the CD player behaves like a pure resistor, find the maximum instantaneous power.

views

Textbook Question

You have a 200-Ω resistor, a 0.400-H inductor, and a 6.00-μF capacitor. They are connected to form an L-R-C series circuit. What is the current amplitude?