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 12b

Chapter 31, Problem 12b

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?

Verified step by step guidance

First, identify the components of the L-R-C series circuit: a resistor (R = 200 Ω), an inductor (L = 0.400 H), and a capacitor (C = 6.00 μF).

Calculate the angular frequency (ω) of the circuit using the formula for resonance frequency: ω = 1 / √(L * C). Convert the capacitance from microfarads to farads before calculation.

Determine the impedance (Z) of the circuit using the formula: Z = √(R² + (ωL - 1/ωC)²). This accounts for the resistive, inductive, and capacitive components.

Use Ohm's Law to find the current amplitude (I₀) in the circuit. The formula is: I₀ = V₀ / Z, where V₀ is the voltage amplitude across the circuit.

Ensure that all units are consistent when performing calculations, especially converting microfarads to farads and ensuring angular frequency is in radians per second.

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Key Concepts

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

Impedance in an L-R-C Circuit

In an L-R-C series circuit, impedance (Z) is the total opposition to the flow of alternating current, combining resistance (R), inductive reactance (X_L), and capacitive reactance (X_C). It is calculated using the formula Z = √(R² + (X_L - X_C)²), where X_L = ωL and X_C = 1/(ωC), with ω being the angular frequency. Understanding impedance is crucial for determining the current amplitude in the circuit.

Resonance in L-R-C Circuits

Resonance occurs in an L-R-C circuit when the inductive reactance equals the capacitive reactance (X_L = X_C), resulting in the impedance being purely resistive (Z = R). At resonance, the circuit allows maximum current flow, and the current amplitude is at its peak. This concept is essential for analyzing the behavior of the circuit at different frequencies.

Current Amplitude in AC Circuits

The current amplitude in an AC circuit is the maximum value of the current that flows through the circuit. It is determined by the voltage amplitude (V_m) and the impedance (Z) using the formula I_m = V_m/Z. Understanding how to calculate the current amplitude is vital for analyzing the performance of the L-R-C circuit under alternating current conditions.

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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

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.

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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 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

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

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.

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