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Ch 31: Alternating Current
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 31: Alternating CurrentProblema 6
Capitolo 31, Problema 6

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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1
Understand that reactance is the opposition to the flow of alternating current caused by inductance and capacitance. For inductors, the reactance \( X_L \) is given by \( X_L = \omega L \), and for capacitors, the reactance \( X_C \) is given by \( X_C = \frac{1}{\omega C} \), where \( \omega \) is the angular frequency.
To find the angular frequency at which the reactance of the inductance and capacitance are equal, set \( X_L = X_C \). This gives the equation \( \omega L = \frac{1}{\omega C} \).
Solve the equation \( \omega L = \frac{1}{\omega C} \) for \( \omega \). Multiply both sides by \( \omega \) to get \( \omega^2 = \frac{1}{LC} \). Then, take the square root of both sides to find \( \omega = \frac{1}{\sqrt{LC}} \).
Substitute the given values \( L = 500 \text{ mH} = 0.500 \text{ H} \) and \( C = 3.50 \mu\text{F} = 3.50 \times 10^{-6} \text{ F} \) into the formula \( \omega = \frac{1}{\sqrt{LC}} \) to find the numerical value of the angular frequency.
Once you have the angular frequency, calculate the reactance of each element using \( X_L = \omega L \) and \( X_C = \frac{1}{\omega C} \) with the found \( \omega \).

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Concetti chiave

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Reactance

Reactance is the resistance offered by capacitors and inductors to the flow of alternating current, due to their ability to store and release energy. Capacitive reactance (X_C) decreases with increasing frequency, while inductive reactance (X_L) increases. The reactance is crucial for determining how these components affect AC circuits.
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Inductors and Graphs

Angular Frequency

Angular frequency, denoted by ω, is a measure of how quickly an AC circuit oscillates, expressed in radians per second. It is related to the frequency (f) by the formula ω = 2πf. Angular frequency is essential for calculating reactance and understanding the behavior of AC circuits at different frequencies.
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Resonance in LC Circuits

Resonance occurs in LC circuits when the inductive reactance equals the capacitive reactance, leading to maximum energy transfer. At resonance, the angular frequency ω is given by ω = 1/√(LC). This concept is key to solving the problem, as it determines the frequency at which the reactances of the capacitor and inductor are equal.
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