Ch 32: AC Circuits

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 32: AC Circuits

Problem 32a

Chapter 32, Problem 32a

For the circuit of FIGURE EX32.32, What is the resonance frequency, in both rad/s and Hz?

Verified step by step guidance

Step 1: Identify the components in the circuit. The circuit consists of a resistor (R = 10 Ω), an inductor (L = 10 mH), and a capacitor (C = 10 μF) connected in series with an AC voltage source.

Step 2: Recall the formula for resonance frequency in a series RLC circuit. The resonance frequency occurs when the inductive reactance and capacitive reactance cancel each other out. The formula for angular resonance frequency (ω₀) is: ω₀ = 1 / √(L * C).

Step 3: Substitute the given values into the formula for angular resonance frequency. Use L = 10 mH = 10 × 10⁻³ H and C = 10 μF = 10 × 10⁻⁶ F. The formula becomes: ω₀ = 1 / √((10 × 10⁻³) * (10 × 10⁻⁶)).

Step 4: Convert the angular resonance frequency (ω₀) to the resonance frequency in Hz (f₀). The relationship between angular frequency and frequency is: f₀ = ω₀ / (2π).

Step 5: After calculating ω₀ and f₀, you will have the resonance frequency in both rad/s and Hz. Ensure units are consistent throughout the calculation.

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Key Concepts

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

Resonance Frequency

Resonance frequency is the frequency at which a system naturally oscillates with maximum amplitude. In electrical circuits, it occurs when the inductive reactance equals the capacitive reactance, leading to a condition where the impedance is minimized. This frequency can be calculated using the formula f₀ = 1 / (2π√(LC)), where L is inductance and C is capacitance.

Inductive Reactance

Inductive reactance (X_L) is the opposition that an inductor presents to alternating current (AC) due to its inductance. It is calculated using the formula X_L = ωL, where ω is the angular frequency in rad/s and L is the inductance in henries. This concept is crucial for determining the behavior of circuits at different frequencies, especially in resonance conditions.

Capacitive Reactance

Capacitive reactance (X_C) is the opposition that a capacitor presents to AC, which decreases with increasing frequency. It is given by the formula X_C = 1 / (ωC), where C is the capacitance in farads. Understanding capacitive reactance is essential for analyzing how capacitors interact with inductors in resonant circuits, affecting the overall impedance and resonance frequency.

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Textbook Question

For the circuit of FIGURE EX32.32, Find V_R and V_L at resonance.

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A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the capacitor value is doubled?

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A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the capacitor value is doubled and, at the same time, the inductor value is halved?

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For the circuit of FIGURE EX32.33, What is the resonance frequency, in both rad/s and Hz?

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A series RLC circuit has a 200 kHz resonance frequency. What is the resonance frequency if the resistor value is doubled?

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