(II) ( a ) What is the rms current in a series R C circuit if R = 3.8 k Ω, C = 0.80 μ F , a

(II) ( a ) What is the rms current in a series R C circuit...

(II) $( a )$ What is the rms current in a series $R C$ circuit if $R = 3.8 \mathrm { k } \Omega , \quad C = 0.80 \mu \mathrm { F } ,$ and the rms applied voltage is 120$\mathrm { V }$ at 60.0$\mathrm { Hz }$ ? (b) What is the phase angle between voltage and current? (c) What is the power dissipated by the circuit? (d) What are the voltmeter readings across $R$ and $C$ ?

Key Concepts

Voltage Division in AC Circuits

Voltage division in AC circuits involves the distribution of the applied voltage among the circuit's components based on their impedances. This concept allows for calculating individual voltage drops across elements such as resistors and capacitors, providing insight into circuit behavior and design.

Power Dissipation in AC Circuits

Power dissipation in AC circuits refers to the conversion of electrical energy into heat, primarily occurring in resistive components. The average power is calculated using the RMS values of voltage and current along with the cosine of the phase angle, illustrating how both amplitude and phase difference contribute to energy consumption.

Phase Angle

The phase angle in AC circuits indicates the time difference between the peaks of voltage and current waveforms. In circuits with reactive components like capacitors and inductors, the phase angle shows how much the current leads or lags the voltage, which has implications for reactive power and circuit behavior.

Capacitive Reactance

Capacitive reactance is the opposition that a capacitor offers to AC current, depending inversely on both the frequency of the AC signal and the capacitance. It introduces a phase shift between voltage and current and plays a critical role in determining the overall impedance of circuits containing capacitors.

Complex Impedance

Complex impedance generalizes resistance to AC circuits by combining actual resistance with reactance (due to capacitors and inductors) into a complex quantity. It is essential in determining the circuit's total opposition to AC current, both in magnitude and phase, and is fundamental for analyzing how AC signals behave in series and parallel circuits.

Root Mean Square (RMS) Values

RMS values are the effective values of alternating current or voltage, calculated as the square root of the mean of the squares of instantaneous values over a cycle. They represent the equivalent DC value that would produce the same heating effect in a resistor, making them central to AC circuit analysis.

Transcript

00:01 We're told that a series rc circuit has a resistance of 3 .8 kilo -ooms, a capacitance of 0 .8 microferids, and we're applying a voltage of 120 volts at 60 hertz.

00:13 We want to find what the rms current is in this circuit in part a.

00:18 So we want to find the rms current.

00:23 So we can start by finding the impedance for the circuit, which is given by the resistance squared, plus the reactants of the capacitor squared, all square rooted.

00:37 And since the reactance of the capacitor is given by 1 over 2 pi times the frequency times the capacitance, our impedance is r squared plus 1 over 4 pi squared, frequency squared, capacity, capacitance squared, all square rooted.

01:01 So now substituting in the values we have, we find that this has a value of, 5 ,043 .2 oms.

01:10 So now that we know the impedance of the circuit, we can calculate the rms current, which of course is just going to be given by the voltage applied over the impedance of the circuit.

01:26 Since we're given the rms voltage, and we just calculated the impedance, we know all these values.

01:31 So we plug them in to find that the current has a value of 0 .024 amps.

01:39 So that's our answer to part a.

01:40 In part b, we want to find the phase angle.

01:48 So the phase angle of this circuit is given by the inverse tan of negative the reactants of the capacitor over the resistance.

02:05 Writing this out explicitly is the inverse tangent of minus 1 over 2 pi times the frequency, times the capacitance, times the resistance...

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