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Chapter 21: Problem 26

(a) What inductance has the same reactance in a 120 \(\mathrm{V}, 60-\mathrm{Hz}\) circuit as a capacitance of \(10 \mu \mathrm{F} ?\) (b) What would be the ratio of inductive reactance to capacitive reactance if the frequency were changed to \(120 \mathrm{~Hz} ?\)

Short Answer

Expert verified
The inductance is 0.702 H, and the reactance ratio is about 3.99 at 120 Hz.

Step by step solution

01

Calculate Capacitive Reactance at 60 Hz

The capacitive reactance \(X_C\) is given by the formula\[ X_C = \frac{1}{2\pi f C} \]where \(f = 60\, \text{Hz}\) and \(C = 10\.\mu\text{F} = 10 \times 10^{-6} \text{F}\).Plugging in the values, \[ X_C = \frac{1}{2\pi \times 60 \times 10\times 10^{-6}} = \frac{1}{0.0037699} \approx 265.26 \text{ ohms}. \]
02

Find Inductance for Equivalent Inductive Reactance at 60 Hz

To find the inductance \(L\) that has the same reactance as the capacitance, we first have the inductive reactance formula:\[X_L = 2\pi f L.\]Set \(X_L = X_C = 265.26 \text{ ohms} \) from Step 1 and solve for \(L\):\[265.26 = 2\pi \times 60 \times L\]Solve for \(L\): \[L = \frac{265.26}{2\pi \times 60} = 0.702 \text{H}\].
03

Calculate New Capacitive Reactance at 120 Hz

If the frequency is doubled to \(120 \text{ Hz}\), calculate the new capacitive reactance:\[X_C' = \frac{1}{2\pi \times 120 \times 10 \times 10^{-6}} = \frac{1}{0.0075398} \approx 132.63 \text{ ohms}\].
04

Calculate New Inductive Reactance at 120 Hz

Calculate the new inductive reactance \(X_L'\) at \(120 \text{ Hz}\) using \(L\) from Step 2:\[X_L' = 2\pi \times 120 \times 0.702 = 529.68 \text{ ohms}\].
05

Calculate the Ratio of Inductive to Capacitive Reactance at 120 Hz

Compute the ratio of inductive to capacitive reactance at \(120 \text{ Hz}\):\[\text{Ratio} = \frac{529.68}{132.63} \approx 3.99\].

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Key Concepts

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

Inductance
Inductance is a property of a circuit or component that opposes changes in current. It is often associated with coils and inductors.
Inductance is measured in henrys (H). In electronics, inductance can be thought of as a temporary storage of energy in a magnetic field.
  • When alternating current (AC) flows through an inductor, it generates an electromagnetic field around it.
  • This field creates a reactance, which is a form of resistance to changes in current flowing through the circuit.
  • The formula for inductive reactance (\[XL\]) is \(X_L = 2\pi f L\).
Understanding inductance is important because it plays a key role in tuning circuits and filtering signals. In applications like radios or TVs, inductance is used to select specific frequencies.
Capacitance
Capacitance is another fundamental property of electric circuits. It refers to the ability of a component, known as a capacitor, to store and release energy in the form of an electric field.
Capacitance is measured in farads (F). A capacitor works by accumulating electric charge on its plates when voltage is applied.
  • The formula for capacitive reactance (\[XC\]) is \(X_C = \frac{1}{2\pi f C}\).
  • Capacitive reactance decreases with an increase in frequency, unlike inductive reactance.
  • The opposition of capacitors to changing voltage allows them to filter and smooth signals in electronic circuits.
In practical terms, capacitors are used to manage power supply and ensure stable voltage levels in a circuit, making them vital for daily gadgets like smartphones and computers.
Frequency
Frequency in an electrical circuit refers to how many times the current changes direction in one second. It is measured in hertz (Hz) and is crucial in determining how electrical components like inductors and capacitors behave.
When frequency increases:
  • Inductive reactance increases because the inductor resists rapid changes in current.
  • Conversely, capacitive reactance decreases since the capacitor finds it easier to react to fast changes in voltage.
Frequency's impact on reactance means that by adjusting the frequency, you can control the behavior of the circuit. This makes frequency adjustments key in applications like radio broadcasting, where different stations use different frequencies to transmit signals.

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Most popular questions from this chapter

The current in a resistor is given by \(I=(8.0 \mathrm{~A}) \sin (40 \pi t)\) when a voltage given by \(V=(60 \mathrm{~V}) \sin (40 \pi t)\) is applied to it. (a) What is the resistance value? (b) What are the frequency and period of the voltage source? (c) What is the average power delivered to the resistor?

The current in a \(60-\Omega\) resistor is given by \(I=(2.0 \mathrm{~A}) \sin (380 t) .\) (a) What is the frequency of the current? (b) What is the rms current? (c) How much average power is delivered to the resistor? (d) Write an equation for the voltage across the resistor as a function of time. (e) Write an equation for the power delivered to the resistor as a function of time. (f) Show that the rms power obtained in part (e) is the same as your answer to part (c).

How much current is in a circuit containing only a 50- \(\mu \mathrm{F}\) capacitor connected to an ac generator with an output of \(120 \mathrm{~V}\) and \(60 \mathrm{~Hz} ?\)

A series RLC circuit has a resistance of \(25 \Omega\), a capacitance of \(0.80 \mu \mathrm{F},\) and an inductance of \(250 \mathrm{mH}\). The circuit is connected to a variable-frequency source with a fixed rms voltage output of \(12 \mathrm{~V}\). If the frequency that is supplied is set at the circuit's resonance frequency, what is the rms voltage across each of the circuit elements?

A \(1.0-\mu \mathrm{F}\) capacitor is connected to \(\mathrm{a} 120-\mathrm{V}, 60\) - \(\mathrm{Hz}\) source. (a) What is the capacitive reactance of the circuit? (b) How much current is in the circuit? (c) What is the phase angle between the current and the applied voltage? (d) What is the maximum energy stored in the capacitor? (f) What is the power dissipated by this circuit?
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