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Chapter 31: Problem 31

(a) At what frequency would a \(6.0 \mathrm{mH}\) inductor and a \(10 \mu \mathrm{F}\) capacitor have the same reactance? (b) What would the reactance be? (c) Show that this frequency would be the natural frequency of an oscillating circuit with the same \(L\) and \(C\).

Short Answer

Expert verified
(a) 650 Hz (b) 24.5 Ohms (c) Confirmed.

Step by step solution

01

Understanding Reactance Equality

The reactance of an inductor is given by \( X_L = \omega L \) and the reactance of a capacitor is \( X_C = \frac{1}{\omega C} \). For these to be equal, \( \omega L = \frac{1}{\omega C} \). We start by setting these formulas equal to each other: \( \omega L = \frac{1}{\omega C} \). This equation relates the angular frequency \( \omega \), which determines how often the inductor and capacitor experience peak reactance.
02

Solving for Angular Frequency

To solve \( \omega L = \frac{1}{\omega C} \), we rearrange it to \( \omega^2 = \frac{1}{LC} \). By taking the square root of both sides, we find the angular frequency \( \omega = \sqrt{\frac{1}{LC}} \). Substitute \( L = 6.0 \times 10^{-3} \) H (henries) and \( C = 10 \times 10^{-6} \) F (farads) into the equation to calculate \( \omega \).
03

Calculating Angular Frequency

Substitute the given values: \( L = 6.0 \times 10^{-3} \) H, \( C = 10 \times 10^{-6} \) F into \( \omega = \sqrt{\frac{1}{LC}} \). We get \( \omega = \sqrt{\frac{1}{6.0 \times 10^{-3} \times 10 \times 10^{-6}}} \). Simplifying gives \( \omega \approx 4082 \) rad/s.
04

Finding Reactance Frequency

Frequency \( f \) is related to angular frequency \( \omega \) by \( \omega = 2 \pi f \). We find \( f \) using \( f = \frac{\omega}{2 \pi} \). Substituting \( \omega\approx 4082 \) rad/s, we calculate \( f \approx 650 \) Hz.
05

Calculating Reactance

To find the reactance \( X \), either \( X = \omega L \) or \( X = \frac{1}{\omega C} \) can be used because both are equal. Using \( X = \omega L = 4082 \times 6.0 \times 10^{-3} \), we find \( X \approx 24.5 \) Ohms.
06

Confirming Resonant Frequency

An oscillating LC circuit鈥檚 natural frequency \( \omega_0 \) is \( \omega_0 = \frac{1}{\sqrt{LC}} \), which simplifies to our previous \( \omega = 4082 \) rad/s when using the given values for \( L \) and \( C \). This confirms that the calculated frequency is indeed the natural frequency of the circuit.

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Key Concepts

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

Inductor
An inductor is a fundamental component in electrical circuits that stores energy in a magnetic field when an electric current flows through it. Inductors consist of wire coils and their ability to store energy is measured in henries (H).
The key property of an inductor is its inductance, denoted as \( L \), which indicates how effective it is at storing this energy.
  • Inductors oppose changes in current through them by generating an internal voltage; this is due to their ability to store energy in magnetic fields.
  • The reactance of an inductor depends on the frequency of the alternating current and is given by the formula \( X_L = \omega L \), where \( \omega \) is the angular frequency.
  • In the context of the exercise, a 6.0 mH inductor is used, which affects the circuit's reactance in relation to changing frequencies.
By understanding inductors, you can better grasp how they function within larger circuits, like oscillating circuits, to influence current flow.
Capacitor
Capacitors are devices that store electrical energy in an electric field between two conducting plates separated by an insulating material. The storage ability or capacitance of a capacitor is measured in farads (F).
In this exercise, a capacitor with a capacitance \( C \) of \( 10 \mu \text{F} \) is used, which plays a key role in determining the circuit's overall reactance.
  • Capacitors oppose changes in voltage across them; when they are charged, they release stored energy when needed by the circuit.
  • The reactance of a capacitor is inversely proportional to the frequency of the alternating current, described by \( X_C = \frac{1}{\omega C} \).
  • This inverse relationship means that as frequency increases, a capacitor's reactance decreases, making it crucial in alternating current circuits.
Understanding capacitors is essential for effectively designing or analyzing circuits that require specific responses to alternating currents.
Oscillating Circuit
An oscillating circuit, often known as a resonant or LC circuit, involves a loop containing both an inductor and a capacitor. This setup can continuously exchange energy between magnetic and electric fields, creating oscillations at a specific natural frequency.
When the inductor and capacitor are at resonance, the energy enters into a state of equilibrium, where the rate of energy transfer between the components is maximized.
  • The resonant frequency \( f_0 \) of an LC circuit is determined by its components and is given by \( f_0 = \frac{1}{2 \pi \sqrt{LC}} \).
  • The focus of the exercise was on finding at which frequency an LC circuit鈥檚 natural frequency comes into play, revealing its unique behavior.
  • Oscillating circuits are widely used in radio transmitters and clocking devices due to their stable frequency characteristics.
By examining oscillating circuits, one learns how to harness this natural tendency to match frequencies across various technologies.
Frequency
Frequency is a measure of how often an event repeats over a specified period of time, usually expressed in hertz (Hz), which implies cycles per second.
In alternating current circuits, frequency derives from how fast the current changes direction, and it plays a crucial role in defining the response of reactive components like inductors and capacitors.
  • The relationship between angular frequency \( \omega \) and frequency \( f \) is \( \omega = 2 \pi f \).
  • High-frequency signals generally have lower wavelengths, affecting how electrical components like inductors and capacitors react within circuits.
  • In the context of this exercise, finding the frequency where elements have the same reactance is key to understanding oscillating circuit behavior.
Mastering frequency concepts allows students to better evaluate and design circuits that are sensitive to changes in charge and discharge cycles.

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Most popular questions from this chapter

An ac generator has \(\operatorname{emf} \mathscr{E}=\mathscr{E}_{m} \sin \left(\omega_{d} t-\pi / 4\right)\), where \(\mathscr{E}_{m}=30.0 \mathrm{~V}\) and \(\omega_{d}=350 \mathrm{rad} / \mathrm{s}\). The current produced in a con- nected circuit is \(i(t)=I \sin \left(\omega_{d} t-3 \pi / 4\right)\), where \(I=620 \mathrm{~mA}\). At what time after \(t=0\) does (a) the generator emf first reach a maximum and (b) the current first reach a maximum? (c) The circuit contains a single element other than the generator. Is it a capacitor, an inductor, or a resistor? Justify your answer. (d) What is the value of the capacitance, inductance, or resistance, as the case may be?

In a certain oscillating \(L C\) circuit, the total energy is converted from electrical energy in the capacitor to magnetic energy in the inductor in \(1.50 \mu \mathrm{s}\). What are (a) the period of oscillation and (b) the frequency of oscillation? (c) How long after the magnetic energy is a maximum will it be a maximum again?

An ac generator has \(\operatorname{emf} \mathscr{E}=\mathscr{E}_{m} \sin \omega_{d} t\), with \(\mathscr{E}_{m}=25.0 \mathrm{~V}\) and \(\omega_{d}=377 \mathrm{rad} / \mathrm{s}\). It is connected to a \(12.7 \mathrm{H}\) inductor. (a) What is the maximum value of the current? (b) When the current is a maximum, what is the emf of the generator? (c) When the emf of the generator is \(-12.5 \mathrm{~V}\) and increasing in magnitude, what is the current?c

To construct an oscillating \(L C\) system, you can choose from a \(10 \mathrm{mH}\) inductor, a \(5.0 \mu \mathrm{F}\) capacitor, and a \(2.0 \mu \mathrm{F}\) capacitor. What are the (a) smallest, (b) second smallest, (c) second largest, and (d) largest oscillation frequency that can be set up by these elements in various combinations?

What is the maximum value of an ac voltage whose rms value is \(100 \mathrm{~V} ?\)
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