In an L-R-C series circuit, L=0.280 H and C= 4.00 μF. The voltage amplitude of the source is 120

step 1 For this problem, we have this LRC series circuit, and we know the parameters for R, L, and C. A

In an L-R-C series circuit, L=0.280 H and C= 4.00 μF. The...

In an $L-R-C$ series circuit, $L=0.280 \mathrm{H}$ and $C=$ $4.00 \mu \mathrm{F}$. The voltage amplitude of the source is $120 \mathrm{~V}$. (a) What is the resonance angular frequency of the circuit? (b) When the source operates at the resonance angular frequency, the current amplitude in the circuit is 1.70 A. What is the resistance $R$ of the resistor? (c) At the resonance angular frequency, what are the peak voltages across the inductor, the capacitor, and the resistor?

Key Concepts

Resonance in RLC Circuits

This concept refers to the condition in an RLC series circuit when the inductive and capacitive reactances are equal in magnitude but opposite in phase, causing them to cancel each other out. At resonance, the circuit behaves as a purely resistive circuit, and the overall impedance is minimized.

Resonant Angular Frequency Calculation

The resonant angular frequency is determined by the circuit's inductance and capacitance through the formula ?? = 1/?(LC). This frequency marks the point at which the circuit’s reactive components cancel, maximizing the current for a given voltage.

Reactance in AC Circuits

Reactance is the opposition offered by capacitors and inductors to alternating currents. Inductive reactance is given by X_L = ?L while capacitive reactance is X_C = 1/(?C). At resonance, these two reactances become equal in magnitude, leading to the cancellation that defines resonance.

Impedance at Resonance

In a series RLC circuit at resonance, the impedance reduces to just the resistance R because the inductive and capacitive reactances cancel each other. This results in the current amplitude being determined solely by the resistor according to Ohm's law.

Voltage Distribution in Series Circuits

Despite the supply voltage being fixed, the peak voltages across the inductor and capacitor can be much higher due to their reactive properties (given by V = I × X). At resonance, although the net reactive voltage drop is zero, individual voltages across L and C can be significantly large and are equal in magnitude but opposite in phase, with the resistor's voltage drop equal to the supply voltage when considering the current through R.

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