31.47. An L-R-C series circuit consists of a 50.0-Ωresistor, a 10.0-μF capacitor, a 3.50-mH indu

step 1 Hello everyone here it is given an LRC series circuit consists of LRC resistance of 50 o capiste

31.47. An L-R-C series circuit consists of a 50.0-Ωresistor,...

31.47. An $L-R-C$ series circuit consists of a $50.0-\Omega$ resistor, a $10.0-\mu$ F capacitor, a $3.50-\mathrm{mH}$ inductor, and an ac voltage source of voltage anplitude 60.0 $\mathrm{V}$ operating at 1250 $\mathrm{Hz}$ (a) Find the current amplitude and the voltage amplitudes across the inductor, the resistor, and the capacitor. Why can the voltage amplitudes add up to more than 60.0 $\mathrm{V} ?$ (b) If the frequency is now doubled, but nothing else is changed, which of the quantities in part (a) will change? Find the new values for those that do change.

31.47. An $L-R-C$ series circuit consists of a $50.0-\Omega$ resistor, a
$10.0-\mu$ F capacitor, a $3.50-\mathrm{mH}$ inductor, and an ac voltage source of
voltage anplitude 60.0 $\mathrm{V}$ operating at 1250 $\mathrm{Hz}$ (a) Find the current amplitude and the voltage amplitudes across the inductor, the resistor, and the capacitor. Why can the voltage amplitudes add up to more than 60.0 $\mathrm{V} ?$ (b) If the frequency is now doubled, but
nothing else is changed, which of the quantities in part (a) will change? Find the new values for those that do change.

Key Concepts

Frequency Dependence in AC Circuits

The behavior of an LRC circuit changes with frequency. As the frequency varies, both the inductive and capacitive reactances change, which in turn alters the impedance of the circuit and the voltage drops across each component. In particular, doubling the frequency will affect the magnitudes of XL and XC in opposite ways, while the resistor's characteristics remain unchanged. This concept is key in understanding circuit tuning and resonance.

Phasor Representation in AC Circuits

Phasors are used to represent the magnitude and phase relationships of sinusoidal voltages and currents in AC circuits. When analyzing the voltage drops across individual components (resistor, inductor, capacitor), the use of phasors explains why the algebraic sum of individual voltage amplitudes can exceed the source voltage. This is because the voltages are out of phase and must be added vectorially, not algebraically.

Inductive and Capacitive Reactance

Inductive reactance (XL = ?L) and capacitive reactance (XC = 1/(?C)) represent how the inductor and capacitor resist changes in current and voltage respectively at different frequencies. These reactances are frequency dependent; as the frequency increases, the inductive reactance increases, while the capacitive reactance decreases. Their roles are crucial in determining the phase relationship between the voltage and current in the circuit.

Impedance in LRC Circuits

In AC circuits, the total opposition to current flow is described by the impedance, which is a combination of resistance and reactance. The impedance in an LRC series circuit is given by the vector sum of the resistance and the net reactance (the difference between inductive and capacitive reactances). It is used to determine the overall current from the applied voltage via Ohm's law for AC circuits: I = V / |Z|.

Transcript

Please give Ace some feedback