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Chapter 21: Problem 45

A 0.48 - \(\mu\) F capacitor is connected in series to a $5.00-\mathrm{k} \Omega\( resistor and an ac source of voltage amplitude \)2.0 \mathrm{V}$ (a) At \(f=120 \mathrm{Hz},\) what are the voltage amplitudes across the capacitor and across the resistor? (b) Do the voltage amplitudes add to give the amplitude of the source voltage (i.e., does \(V_{\mathrm{R}}+V_{\mathrm{C}}=2.0 \mathrm{V}\) )? Explain. (c) Draw a phasor diagram to show the addition of the voltages.

Short Answer

Expert verified
Question: Find the voltage amplitudes across the capacitor and the resistor in a series circuit with a capacitor, a resistor, and an AC voltage source at a frequency of 120 Hz. Then, check whether the sum of these voltages is equal to the amplitude of the source voltage and illustrate the voltages in a phasor diagram.

Step by step solution

01

Determine the impedance of the circuit

First, we need to calculate the impedance of the circuit, which combines the resistor's resistance (R) and the capacitor's reactance (X_C). Given that it's a series circuit, we can write the impedance (Z) as: \(Z=\sqrt{R^{2}+X_{C}^{2}}\) The capacitive reactance (\(X_C\)) can be calculated using the formula: \(X_{C}=\frac{1}{2 \pi f C}\) Where f is the frequency (120 Hz) and C is the capacitance (0.48 μF).
02

Calculate the current amplitude

We can find the current amplitude (I) in the circuit by using Ohm's law: \(I=\frac{V_{s}}{Z}\) Where \(V_s\) is the amplitude of the source voltage (2.0 V).
03

Find the voltage amplitudes across the resistor and the capacitor

Now, we can determine the voltage amplitudes across the resistor and the capacitor using Ohm's law. For the resistor, we have: \(V_{R}=I \times R\) And for the capacitor, we have: \(V_{C}=I \times X_C\)
04

Check if the voltage amplitudes add up to the amplitude of the source voltage

To verify if the voltage amplitudes across the resistor and the capacitor add up to the amplitude of the source voltage, we can add \(V_R\) and \(V_C\): \(V_R + V_C = I \times R + I \times X_C\) We can then check if this sum is equal to the amplitude of the source voltage (\(V_s = 2.0 V\)).
05

Draw a phasor diagram

Finally, we can draw a phasor diagram to show the addition of the voltages. On the diagram, the resistor voltage phasor should be on the horizontal axis and the capacitor voltage phasor should be on the vertical axis. The source voltage phasor would be the hypotenuse of the right-angled triangle formed by the resistor and capacitor voltage phasors. The angle between the source voltage phasor and the resistor voltage phasor would be the phase angle of the circuit.

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Most popular questions from this chapter

A variable capacitor is connected in series to an inductor with negligible internal resistance and of inductance \(2.4 \times 10^{-4} \mathrm{H} .\) The combination is used as a tuner for a radio. If the lowest frequency to be tuned in is \(0.52 \mathrm{MHz}\). what is the maximum capacitance required?
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