A resistor and capacitor are connected in series across an...

A resistor and capacitor are connected in series across an ac generator. The emf of the generator is given by $v(t)=V_{0} \cos \omega t, \quad$ where $\quad V_{0}=120 \mathrm{V}$ $\omega=120 \pi \mathrm{rad} / \mathrm{s}, \quad R=400 \Omega,$ and $C=4.0 \mu \mathrm{F} .(\mathrm{a}) \mathrm{What}$ is the impedance of the circuit? (b) What is the amplitude of the current through the resistor? (c) Write an expression for the current through the resistor. (d) Write expressions representing the voltages across the resistor and across the capacitor.

A resistor and capacitor are connected in series across an ac generator. The emf of the generator is given by $v(t)=V_{0} \cos \omega t, \quad$ where $\quad V_{0}=120 \mathrm{V}$ $\omega=120 \pi \mathrm{rad} / \mathrm{s}, \quad R=400 \Omega,$ and $C=4.0 \mu \mathrm{F} .(\mathrm{a}) \mathrm{What}$
is the impedance of the circuit? (b) What is the amplitude of the current through the resistor? (c) Write an expression for the current through the resistor. (d) Write expressions representing the voltages across the resistor and across the capacitor.

Key Concepts

Voltage and Current Phase Relationships

In an RC circuit, the phase relationships between the voltage across components and the circuit current are crucial. The resistor causes no phase shift between its voltage and the current, while the capacitor causes the voltage to lag behind the current by 90 degrees. This difference is central when writing the expressions for instantaneous voltages across each component, as their phase shifts must be accurately incorporated to reflect the physics of the circuit.

Ohm's Law in AC Circuits

Ohm's Law in AC circuits relates the phasor voltage and the phasor current through the complex impedance, written as V = I·Z. This relationship is used to determine quantities such as the amplitude of the current when given the amplitude of the voltage and the impedance of the circuit. The approach involves dividing the voltage phasor by the total impedance to find the current phasor, from which both magnitude and phase are derived.

Phasor Representation

Phasor representation is a mathematical technique used to analyze AC circuits by converting sinusoidally varying signals into complex numbers. This allows phasor algebra to simplify the handling of phase relationships between current and voltage. In circuits with resistors and capacitors, phasors help in converting differential equations into algebraic equations, making it more straightforward to solve for amplitudes and phase shifts.

Complex Impedance in RC Circuits

In an RC series circuit driven by an AC voltage source, the resistor and capacitor contribute to the total impedance in different ways. The resistor has a real impedance R, while the capacitor's impedance is given by 1/(j?C), which is purely imaginary. The combination results in a complex impedance of the form Z = R - j/(?C), and the magnitude of this impedance is calculated using the Pythagorean theorem applied to its real and imaginary parts.

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