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 12a

Chapter 31, Problem 12a

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?

Verified step by step guidance

Start by understanding the concept of impedance in an RLC circuit. Impedance is the total opposition that a circuit presents to the flow of alternating current (AC) and is a combination of resistance (R), inductive reactance (X_L), and capacitive reactance (X_C). In a series RLC circuit, impedance (Z) is given by the formula:

Z = \sqrt{R^{2} + (X^{L} - X^{C}) 2}

Calculate the inductive reactance (X_L) using the formula:

X L_{=} ω L

, where

ω

is the angular frequency and

L

is the inductance. Substitute the given values:

ω = 250

rad/s and

L = 0.400

Calculate the capacitive reactance (X_C) using the formula:

X C_{=} \frac{1}{ω C}

, where

C

is the capacitance. Substitute the given values:

C = 6.00 \times 10 - 6

Substitute the calculated values of X_L and X_C into the impedance formula:

Z = \sqrt{200^{2} + (X L_{-} X C) 2}

Finally, simplify the expression to find the impedance Z. Remember that impedance is a complex quantity, but in this case, you are finding its magnitude, which is a real number.

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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 a circuit offers to the flow of alternating current (AC) and is a combination of resistance and reactance. In a series RLC circuit, impedance (Z) is calculated using Z = √(R² + (X_L - X_C)²), where R is resistance, X_L is inductive reactance, and X_C is capacitive reactance.

Inductive Reactance

Inductive reactance (X_L) is the opposition to the change in current by an inductor in an AC circuit. It is given by X_L = ωL, where ω is the angular frequency and L is the inductance. It increases with frequency, causing a phase shift between voltage and current.

Angular Frequency

Angular frequency (ω) is a measure of how quickly the AC voltage oscillates, expressed in radians per second. It is related to the frequency (f) by ω = 2πf. Angular frequency is crucial for calculating reactance and impedance in AC circuits, affecting how components like inductors and capacitors behave.

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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. They are connected to form an L-R-C series circuit. 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

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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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?