An R C circuit consists of a resistor R=32 Ω, a capacitor C=25 μF, and an ac generator with an r

step 1 In the first item, we can use HOM's Law to relate the voltage with the current as follows. VRMS

An R C circuit consists of a resistor R=32 Ω, a capacitor...

An $R C$ circuit consists of a resistor $R=32 \Omega,$ a capacitor $C=25 \mu \mathrm{F},$ and an ac generator with an rms voltage of 120 $\mathrm{V}$ . (a) At what frequency will the rms current in this circuit be 2.9 $\mathrm{A}$ ? For this frequency, what are (b) the rms voltage across the resistor, $V_{\mathrm{rms}, R},$ and $(\mathrm{c})$ the rms voltage across the capacitor, $V_{\mathrm{mss}} c^{2}(\mathrm{d})$ Show that $V_{\operatorname{mss}, R}+V_{\text { rms }, C}>120 \mathrm{V}$ but that $\sqrt{V_{\mathrm{rms}, R}^{2}+V_{\mathrm{rms}, c}^{2}}=120 \mathrm{V}$

An $R C$ circuit consists of a resistor $R=32 \Omega,$ a capacitor $C=25 \mu \mathrm{F},$ and an ac generator with an rms voltage of 120 $\mathrm{V}$ .
(a) At what frequency will the rms current in this circuit be 2.9 $\mathrm{A}$ ?
For this frequency, what are (b) the rms voltage across the resistor, $V_{\mathrm{rms}, R},$ and $(\mathrm{c})$ the rms voltage across the capacitor, $V_{\mathrm{mss}} c^{2}(\mathrm{d})$ Show that $V_{\operatorname{mss}, R}+V_{\text { rms }, C}>120 \mathrm{V}$ but that $\sqrt{V_{\mathrm{rms}, R}^{2}+V_{\mathrm{rms}, c}^{2}}=120 \mathrm{V}$

Key Concepts

Phasor Addition and Phase Relationships

In AC circuits, voltages across resistive and reactive components are not in phase. Phasor representation is used to account for both magnitude and phase. The voltages add vectorially, which explains why the algebraic sum of individual rms voltages may exceed the rms input voltage, while their vector (or phasor) addition correctly reconstructs the original voltage magnitude.

Voltage Division in AC Circuits

Voltage division in AC circuits involves determining how the total voltage is distributed across the various circuit components based on their impedance. Unlike resistive voltage division, in AC circuits the division must take into account both magnitude and phase differences, making the analysis more complex.

Root Mean Square (RMS) Values in AC Circuits

RMS values provide a measure of the effective or equivalent DC value of an AC signal. In AC circuit analysis, both current and voltage are often expressed in RMS terms to accurately relate the delivered power and the circuit’s behavior under sinusoidal excitation.

Impedance

Impedance is the total opposition a circuit offers to alternating current and is a combination of resistance and reactance. In an RC circuit, the impedance is given by the square root of the sum of the squares of the resistor value and the capacitive reactance, highlighting how both the resistor and the capacitor influence the overall current flow.

RC Circuit Analysis

An RC circuit combines resistors and capacitors, which interact to affect both the amplitude and the phase of an AC signal. Understanding how these two components work together is fundamental for analyzing circuit behavior, determining impedance, and predicting voltage and current distributions across the circuit.

Capacitive Reactance

Capacitive reactance represents the opposition to AC current provided by a capacitor and depends on the frequency of the applied signal and the capacitance. It is calculated as the reciprocal of the product of the angular frequency and the capacitance, which means that as frequency increases, the reactance decreases, influencing the circuit’s impedance.

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