An L-R-C series circuit is connected to a 120-Hz ac source that has V_rms = 80.0 V. The circuit has a resistance of 75.0 Ω and an impedance at this frequency of 105 Ω. What average power is delivered to the circuit by the source?

Verified step by step guidance

1. Start by understanding the concept of average power in an AC circuit, which is given by the formula: \( P_{avg} = I_{rms}^2 \times R \), where \( I_{rms} \) is the root mean square current and \( R \) is the resistance.
2. To find \( I_{rms} \), use the relationship between voltage, current, and impedance in an AC circuit: \( V_{rms} = I_{rms} \times Z \), where \( Z \) is the impedance. Rearrange this formula to solve for \( I_{rms} \): \( I_{rms} = \frac{V_{rms}}{Z} \).
3. Substitute the given values into the formula: \( V_{rms} = 80.0 \) V and \( Z = 105 \) Ω to calculate \( I_{rms} \).
4. Once \( I_{rms} \) is calculated, substitute it into the average power formula: \( P_{avg} = I_{rms}^2 \times R \), using \( R = 75.0 \) Ω.
5. Calculate \( P_{avg} \) using the values obtained from the previous steps to find the average power delivered to the circuit.

Key Concepts

Impedance in AC Circuits

Impedance is the total opposition a circuit presents to the flow of alternating current (AC) and is a combination of resistance, inductive reactance, and capacitive reactance. In an L-R-C circuit, impedance determines how much voltage is required to drive a certain current through the circuit. It is crucial for calculating power in AC circuits.

RMS Voltage

RMS (Root Mean Square) voltage is a measure of the effective value of an alternating voltage, which is equivalent to a DC voltage that would deliver the same power to a load. It is used in AC circuits to calculate power because it accounts for the varying nature of AC voltage over time.

Average Power in AC Circuits

Average power in AC circuits is calculated using the formula P_avg = V_rms * I_rms * cos(φ), where φ is the phase angle between the voltage and current. This formula accounts for the fact that not all the power in an AC circuit is used effectively due to phase differences, making it essential for determining the actual power delivered to the circuit.