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 12 a

Chapter 31, Problem 12 a

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?

Verified step by step guidance

Step 1: Recall the formula for impedance in an L-R-C series circuit: \( Z = \sqrt{R^2 + (X_L - X_C)^2} \), where \( R \) is the resistance, \( X_L \) is the inductive reactance, and \( X_C \) is the capacitive reactance.

Step 2: Calculate the inductive reactance \( X_L \) using the formula \( X_L = \omega L \), where \( \omega \) is the angular frequency and \( L \) is the inductance. Note that \( \omega = 2\pi f \), where \( f \) is the frequency of the circuit.

Step 3: Calculate the capacitive reactance \( X_C \) using the formula \( X_C = \frac{1}{\omega C} \), where \( C \) is the capacitance and \( \omega \) is the angular frequency.

Step 4: Substitute the values of \( R \), \( X_L \), and \( X_C \) into the impedance formula \( Z = \sqrt{R^2 + (X_L - X_C)^2} \).

Step 5: Simplify the expression to find the impedance \( Z \). Ensure all units are consistent (e.g., convert \( \mu F \) to \( F \) and use SI units for resistance, inductance, and capacitance).

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Key Concepts

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

Impedance in AC Circuits

Impedance is the total opposition that a circuit offers to the flow of alternating current (AC) and is represented as a complex number. It combines resistance (R) and reactance (X), where reactance arises from inductors and capacitors. The formula for impedance in an L-R-C series circuit is Z = √(R² + (X_L - X_C)²), where X_L is the inductive reactance and X_C is the capacitive reactance.

Inductive Reactance

Inductive reactance (X_L) is the opposition to the change of current in an inductor and is given by the formula X_L = 2πfL, where f is the frequency of the AC source and L is the inductance in henries. This reactance increases with frequency, meaning that at higher frequencies, inductors resist changes in current more strongly, affecting the overall impedance of the circuit.

Capacitive Reactance

Capacitive reactance (X_C) is the opposition to the change of voltage across a capacitor and is calculated using the formula X_C = 1/(2πfC), where C is the capacitance in farads. Unlike inductive reactance, capacitive reactance decreases with increasing frequency, which means that capacitors allow more current to pass at higher frequencies, influencing the total impedance in the circuit.

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Related Practice

Textbook Question

(a) Compute the reactance of a 0.450-H inductor at frequencies of 60.0 Hz and 600 Hz. (b) Compute the reactance of a 2.50-μF capacitor at the same frequencies. (c) At what frequency is the reactance of a 0.450-H inductor equal to that of a 2.50-μF capacitor?

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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

A capacitance C and an inductance L are operated at the same angular frequency. (a) At what angular frequency will they have the same reactance? (b) If L = 5 00 mH and C = 3.50 μF, what is the numerical value of the angular frequency in part (a), and what is the reactance of each element?

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Textbook Question

What is the inductance of an inductor whose reactance is 120 Ω at 80.0 Hz?

Textbook Question

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?

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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?