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An LRC series circuit is constructed using a 175-Ω resistor, a 12.5-μF capacitor, and an 8.00-mH inductor, all connected across an ac source having a variable frequency and a voltage amplitude of 25.0 V. (a) At what angular frequency will the impedance be smallest, and what is the impedance at this frequency? (b) At the angular frequency in part (a), what is the maximum current through the inductor? (c) At the angular frequency in part (a), find the potential difference across the ac source, the resistor, the capacitor, and the inductor at the instant that the current is equal to one-half its greatest positive value. (d) In part (c), how are the potential differences across the resistor, inductor, and capacitor related to the potential difference across the ac source?

Short Answer

Expert verified
(a) 3162 rad/s, 175 Ω. (b) 0.143 A. (c) Resistor: 12.51 V. Capacitor: -12.51 V. Inductor: +12.51 V. (d) V_ac = V_R.

Step by step solution

01

Identify Resonance Frequency

The impedance of an L-R-C circuit is minimal at its resonance angular frequency. The formula for the resonance frequency ω0 in a series L-R-C circuit is given by:ω0=1LCWhere:- L=8.00×103 H- C=12.5×106 FSubstitute in the values:ω0=18.00×10312.5×1063162 rad/s
02

Calculate Minimum Impedance

At resonance, the impedance Z of the circuit is smallest and equals the resistance R, because the inductive and capacitive reactances cancel each other out.Thus, the minimum impedance Zmin is:Zmin=R=175 Ω
03

Compute Maximum Current

The maximum current Imax through the inductor at resonance is obtained using Ohm's law:Imax=V0ZGiven V0=25.0 V and Z=175 Ω:Imax=25.01750.143 A
04

Find Voltage Relationships at Specific Current

The potential differences at the instant when the current is half its maximum, I=0.5Imax, are analyzed. Since:- I=0.5×0.1430.0715 AThe voltage across each component is:- Resistor: VR=I×R=0.0715×17512.51 V- Capacitor: At resonance, voltage across the capacitor VC is VR, and hence is 12.51 V.- Inductor: At resonance, voltage across the inductor VL is +12.51 V.
05

Relate Potential Differences Across Elements

At resonance, potential differences across inductor and capacitor directly oppose each other, and their magnitudes are equal, thus canceling each other out. Consequently, the potential difference across the ac source equals the potential across the resistor:Vac=VR12.51 V

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Resonance Frequency
In an RLC circuit, the resonance frequency holds incredible importance due to its capability to minimize impedance. It is the frequency at which the inductive and capacitive reactances cancel each other out, making the net reactive part of the impedance zero. At this resonance frequency, only the resistive component contributes to the impedance, meaning it is at its minimum.The formula for finding the resonance angular frequency ω0 is:ω0=1LCwhere L is the inductance and C is the capacitance. For our circuit with an inductor of 8.00 mH and a capacitor of 12.5 µF, substituting these values into the formula gives a resonance angular frequency of approximately 3162 rad/s. This frequency is crucial for achieving minimal impedance and optimal functioning of the RLC circuit.
Impedance
In an RLC circuit, impedance is the total opposition to the flow of alternating current and is symbolized as Z. This impedance is frequency-dependent, composed of resistance R, inductive reactance XL=ωL, and capacitive reactance XC=1ωC.At resonance frequency, the impedance in a series RLC circuit is simplified to purely resistive, as the inductive and capacitive reactances cancel each other out, making the reactive component zero. Therefore, the impedance Z at resonance is equal to the resistance R alone. In our specific example, this results in a minimum impedance value of 175 Ω. Recognizing this resonating condition in circuits can significantly impact the design and functionality in real-world applications, such as tuning radio frequencies.
Maximum Current
Maximum current in an RLC circuit occurs when the impedance is at its minimum, which is at the resonance frequency. At this point, the circuit behaves almost like a resistive circuit, allowing for maximum current flow. The relationship can be defined by Ohm's Law:Imax=V0Zwhere Imax is the maximum current, V0 is the voltage amplitude, and Z is the circuit's impedance. For our circuit with a voltage amplitude of 25.0 V and impedance of 175 Ω, the maximum current is approximately 0.143 A. Understanding this peak current is integral for safe and efficient circuit design, preventing component overload and maximizing performance.
Voltage Relationship
Inside an RLC circuit, the voltage relationships vary significantly among different components due to the presence of reactance. When the current is at half its maximum value, the potential differences can be uniquely distributed.- **Resistor:** The voltage across a resistor VR follows Ohm's Law, calculated as I×R. For a half maximum current, this results in approximately 12.51 V.- **Inductor and Capacitor:** At resonance, the voltages across the inductor VL and capacitor VC are equal in magnitude but opposite in phase, thus directly counteracting each other. Therefore, VC=VR=12.51V and VL=+12.51V.This equality results in their contributions canceling each other out. The primary voltage across the AC source directly matches the resistor's voltage at resonance, simplifying analysis by equating Vac to VR. This fundamental concept helps engineers design effective filtering and energy resonance systems.

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