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Chapter 24: Problem 19

Find the impedance of a \(60.0-\mathrm{Hz}\) circuit with a \(45.5-\Omega\) resistor connected in series with a \(95.0-\mu \mathrm{F}\) capacitor.

Short Answer

Expert verified
The impedance of the circuit is approximately 53.39 惟.

Step by step solution

01

Understand Impedance in Series Circuit

Impedance (\(Z\)) in a series circuit with a resistor and a capacitor is calculated using the formula \(Z = \sqrt{R^2 + (X_C)^2}\), where \(R\) is the resistance and \(X_C\) is the capacitive reactance.
02

Calculate the Capacitive Reactance

The capacitive reactance \(X_C\) can be found using the formula \(X_C = \frac{1}{2\pi f C}\), where \(f = 60.0\, \text{Hz}\) and \(C = 95.0\, \mu\text{F} = 95.0 \times 10^{-6} \text{F}\). Substitute the values into the formula: \[ X_C = \frac{1}{2\pi \times 60 \times 95.0 \times 10^{-6}} \approx 27.93\, \Omega \]
03

Calculate the Impedance

Substitute \(R = 45.5\, \Omega\) and \(X_C \approx 27.93\, \Omega\) into the impedance formula:\[ Z = \sqrt{(45.5)^2 + (27.93)^2} \] Calculate the inside terms first:\[ Z = \sqrt{2070.25 + 779.09} \approx \sqrt{2849.34} \approx 53.39\, \Omega \]
04

Conclusion

The total impedance \(Z\) of the circuit is approximately \(53.39\, \Omega\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Series Circuit
A series circuit is a type of electrical circuit in which components are connected one after another in a single path. In a series configuration, the same current flows through each component. This makes it easier to analyze when understanding concepts like impedance.

Series circuits can include a combination of resistors, capacitors, and other electrical components. In calculations involving series circuits, it's vital to remember that the total impedance is the sum of individual impedances of these components.
  • Uniform Current Flow: The same current passes through all components in the series.
  • Voltage Division: The total voltage of the power source is divided among the components.
Understanding series circuits is crucial when working with AC circuits where resistors, inductors, and capacitors might be present together.
Capacitive Reactance
Capacitive reactance (\(X_C\)) is a measure of a capacitor's opposition to alternating current (AC). Unlike resistance, which opposes both direct current (DC) and AC, capacitance primarily affects AC. Capacitive reactance decreases with increasing frequency and capacitance:

\[X_C = \frac{1}{2\pi f C}\]Here, \(f\) is the frequency of the AC source and \(C\) is the capacitance. Notice how as the frequency or capacitance goes up, \(X_C\) decreases.
  • Frequency Dependence: Higher frequencies result in lower reactance.
  • Capacitance Influence: Larger capacitors offer less reactance.
By understanding capacitive reactance, we can see how capacitors affect the power flow in AC circuits.
Resistance in AC Circuits
In AC circuits, resistance is the property that limits the flow of current, similar to its role in DC circuits. However, in AC circuits, resistance combines with other factors like inductive and capacitive reactance to form impedance.

The resistance component in AC circuits is straightforward; it doesn't change with frequency and behaves similarly in both AC and DC conditions. It's the real part of impedance and can be calculated using Ohm鈥檚 Law:
\[R = \frac{V}{I}\]where \(V\) is the voltage across the resistor and \(I\) is the current flowing through it.
  • Constant Behavior: Resistance remains constant regardless of frequency in AC conditions.
  • Heat Generation: Resistors dissipate power in the form of heat in circuits.
Understanding resistance is crucial when determining the total impedance in complex AC circuits.

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Most popular questions from this chapter

C E An inductor and a capacitor are to be connected to a generator. Will the generator supply more current at low frequency if the inductor and capacitor are connected in series or in parallel? Explain.

IP A circuit is constructed by connecting a \(1.00-\mathrm{k} \Omega\) resistor, a \(252-\mu \mathrm{F}\) capacitor, and a \(515-\mathrm{mH}\) inductor in series. (a) What is the highest frequency at which the impedance of this circuit is equal to \(2.00 \mathrm{k} \Omega ?\) (b) To reduce the impedance of this circuit, should the frequency be increased or decreased from its value in part (a)? Explain.

IP An RLC circuit has a resistance of \(105 \Omega\), an inductance of \(85.0 \mathrm{mH},\) and a capacitance of \(13.2 \mu \mathrm{F} .\) (a) What is the power factor for this circuit when it is connected to a \(125-\mathrm{Hz}\) ac generator? (b) Will the power factor increase, decrease, or stay the same if the resistance is increased? Explain. (c) Calculate the power factor for a resistance of \(525 \Omega\)

(a) Sketch the phasor diagram for an ac circuit with a \(105-\Omega\) resistor in series with a \(32.2-\mu \mathrm{F}\) capacitor. The frequency of the generator is \(60.0 \mathrm{Hz}\). (b) If the rms voltage of the generator is \(120 \mathrm{V},\) what is the average power consumed by the circuit?

The rms current in an \(R C\) circuit is 0.72 A. The capacitor in this circuit has a capacitance of \(13 \mu \mathrm{F}\) and the ac generator has a frequency of \(150 \mathrm{Hz}\) and an rms voltage of \(95 \mathrm{V}\). What is the resistance in this circuit?
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