An AC source operating at 60. Hz with a maximum voltage of...

An AC source operating at 60. Hz with a maximum voltage of 170 V is connected in series with a resistor $(R=1.2 \mathrm{k} \Omega)$ and a capacitor $(C=2.5 \mu \mathrm{F})$ (a) What is the maximum value of the current in the circuit? (b) What are the maximum values of the potential difference across the resistor and the capacitor? (c) When the current is zero, what are the magnitudes of the potential difference across the resistor, the capacitor, and the AC source? How much charge is on the capacitor at this instant? (d) When the current is at a maximum, what are the magnitudes of the potential differences across the resistor, the capacitor, and the AC source? How much charge is on the capacitor at this instant?

An AC source operating at 60. Hz with a maximum voltage of 170 V is connected in series with a resistor $(R=1.2 \mathrm{k} \Omega)$ and a capacitor $(C=2.5 \mu \mathrm{F})$ (a) What is the maximum value of the current in the circuit? (b) What are the maximum values of the potential difference across the resistor and the capacitor? (c) When the current is zero, what are the magnitudes of the potential difference across the resistor, the capacitor, and the AC source? How much charge is on the capacitor at
this instant? (d) When the current is at a maximum, what are the magnitudes of the potential differences across the resistor, the capacitor, and the AC source? How much charge is on the capacitor at this instant?

Key Concepts

Alternating Current (AC) Circuits

AC circuits involve voltages and currents that oscillate sinusoidally with time, characterized by parameters such as amplitude (or maximum value), frequency, and phase. Understanding these oscillations and their peak and root-mean-square (RMS) values is fundamental when analyzing how circuit elements respond over different parts of the cycle.

Series RC Circuit

A series RC circuit consists of a resistor and a capacitor connected in a single loop, which means that the same current flows through both components. However, the voltage across each component can differ in both amplitude and phase, making it necessary to consider both resistive and reactive properties when analyzing the circuit behavior.

Impedance

Impedance extends the concept of resistance to AC circuits by incorporating both the resistive (real) and reactive (imaginary) components of a circuit element. In a series RC circuit, the resistor contributes a real impedance while the capacitor contributes an imaginary impedance (via capacitive reactance). The overall impedance, calculated as the vector sum of these components, determines the magnitude and phase of the current relative to the applied voltage.

Capacitive Reactance

Capacitive reactance is the opposition offered by a capacitor to the flow of AC current and is inversely proportional to both the frequency of the AC signal and the capacitance of the capacitor. This concept is critical in AC circuit analysis because it plays a key role in determining the voltage drop across the capacitor as compared to that across the resistor.

Phasor Analysis

Phasor analysis is a mathematical technique used to simplify the analysis of AC circuits by representing sinusoidal voltages and currents as complex numbers or rotating vectors. This method allows for easy handling of amplitude and phase relationships between different circuit variables, such as the phase difference between the voltage and current in resistor and capacitor elements.

Voltage Division in AC Circuits

In an AC circuit, voltage division refers to the way the total voltage is shared among the circuit components based on their impedances. In a series RC circuit, the voltage drop across each component is determined by its impedance relative to the total impedance of the circuit, which also introduces phase differences between the voltage drops.

Charge on a Capacitor

The charge on a capacitor is directly related to the voltage across its plates by the relationship Q = C × V. This concept is especially important in AC circuits for determining how much charge is stored at any given instant, which can vary depending on the phase of the AC cycle, such as when the current is zero.

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