/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 13 31.13. A \(150-\Omega\) resistor... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 31: Problem 13

31.13. A \(150-\Omega\) resistor is connected in series with a \(0.250-\mathrm{H}\) inductor. The voltage across the resistor is \(v_{R}=(3.80 \mathrm{V}) \cos [(720 \mathrm{rad} / \mathrm{s}) t]\) (a) Derive an expression for the circuit current. (b) Determine the inductive reactance of the inductor. (c) Derive an expression for the voltage \(v_{L}\) across the inductor.

Short Answer

Expert verified
(a) \(I(t) = 0.0253 \cos(720t)\) A, (b) \(X_L = 180 \, \Omega\), (c) \(v_L = -4.554 \sin(720t)\) V.

Step by step solution

01

Identify the Given Parameters

Start by identifying all given parameters in the problem. We have a resistor with resistance \(R = 150 \, \Omega\), and an inductor with inductance \(L = 0.250 \, \mathrm{H}\). The voltage across the resistor is given by \(v_R = (3.80 \, \mathrm{V}) \cos[(720 \, \mathrm{rad/s}) t]\).
02

Derive Circuit Current Expression

For a resistor, the voltage \(v_R = I \cdot R\), where \(I\) is the current. Given \(v_R = (3.80 \, \mathrm{V}) \cos[(720 \, \mathrm{rad/s}) t]\), we find the current using \(I = \frac{v_R}{R}\).\[I(t) = \frac{3.80}{150} \cos[(720 \, \mathrm{rad/s}) t] = 0.0253 \cos[(720 \, \mathrm{rad/s}) t] \, \mathrm{A}\]
03

Calculate Inductive Reactance

Inductive reactance is given by \(X_L = \omega L\), where \(\omega = 720 \, \mathrm{rad/s}\). So the inductive reactance is:\[X_L = 720 \times 0.250 = 180 \, \Omega\]
04

Derive Voltage Expression across the Inductor

The voltage across the inductor is given by \(v_L = L \frac{dI}{dt}\). Using the expression for \(I(t)\), find the derivative:\[I(t) = 0.0253 \cos(720t)\]\[\frac{dI}{dt} = -0.0253 \times 720 \sin(720t) = -18.216 \sin(720t)\]So the voltage across the inductor is:\[v_L = 0.250 \times (-18.216 \sin(720t)) = -4.554 \sin(720t) \, \mathrm{V}\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Inductive Reactance
Inductive reactance (X_L) is a key concept when studying AC circuits. It describes the opposition that an inductor presents to alternating current (AC) due to its inductance. The inductor resists changes in current, and this resistance is frequency-dependent, meaning it changes with the frequency of the AC signal.
In mathematical terms, inductive reactance is calculated using the formula:
  • \[X_L = \omega L\]
  • where \(\omega\) is the angular frequency in radians per second and \(L\) is the inductance in henries.
The higher the frequency of the current, the greater the inductive reactance. This is because a more rapidly changing current creates a larger back voltage due to Faraday's law of induction. Inductive reactance is expressed in ohms (\(\Omega\)), similar to resistance, but it specifically accounts for the frequency-dependent behavior of inductors in AC circuits.
Series R-L Circuit
In a series R-L circuit, a resistor and an inductor are connected in a single loop, so the same current flows through both components. This simple arrangement is widely used in electrical engineering to model the behavior of many systems.
A series R-L circuit is characterized by:
  • A resistor, which restricts the flow of current according to Ohm's law.
  • An inductor, which opposes changes in the current flow due to its inductive properties.
In such circuits, the total opposition to the current flow is not just the resistance (R), but also the inductive reactance (X_L). The total opposition, known as impedance (Z), combines these two:
  • \[Z = \sqrt{R^2 + X_L^2}\]
This impedance affects how voltage and current are related over time, introducing phase shifts between them. When analyzing these circuits, it's crucial to consider both the resistive and reactive components to understand the overall behavior.
Derivative in AC Circuits
The concept of derivatives is essential in understanding AC circuits, especially when dealing with inductors. In the context of an AC circuit, the current and voltage across an inductor are related through differentiation.
The voltage across an inductor (V_L) is given by the formula:
  • \[v_L = L \frac{dI}{dt}\]
  • where \(L\) is the inductance and \(\frac{dI}{dt}\) is the derivative of the current with respect to time.
This equation illustrates how the inductor resists changes in current. The derivative \(\frac{dI}{dt}\) indicates how quickly the current changes. A faster rate of change results in a higher induced voltage across the inductor, demonstrating the inductor's ability to "store" energy temporarily in its magnetic field. This relationship is fundamental for AC circuit analysis, as it highlights how voltage and current are out of phase, with voltage leading current by 90掳, due to the inherent properties of the inductance.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

31.31. In an \(L-R-C\) series circuit, \(R=300 \Omega, L=0.400 \mathrm{H},\) and \(C=6.00 \times 10^{-8} \mathrm{F}\) . When the ac source operates at the resonance frequency of the circuit, the current amplitude is 0.500 \(\mathrm{A}\) . (a) What is the voltage amplitude of the source? (b) What is the amplitude of the voltage across the resistor, across the inductor, and across the capacitor? (c) What is the average power supplied by the source?

31.32. An \(L-R-C\) series circuit consists of a source with voltage amplitude 120 \(\mathrm{V}\) and angular frequency 50.0 \(\mathrm{rad} / \mathrm{s}\) , a resistor with \(R=400 \Omega\) an inductor with \(L=9.00 \mathrm{H}\) , and a capacitor with capacitance \(C .\) (a) For what value of \(C\) will the current amplitude in the circuit be a maximum? (b) When \(C\) has the value calculated in part (a), what is the amplitude of the voltage across the inductor?

31.45. A series circuit has an impedance of 60.0\(\Omega\) and a power factor of 0.720 at 50.0 \(\mathrm{Hz}\) . The source voltage lags the current. (a) What circuit element, an inductor or a capacitor, should be placed in series with the circuit to raise its power factor? (b) What size element will raise the power factor to unity?

31.67. You want to double the resonance angular frequency of a series \(R-L-C\) circuit by changing only the pertinent circuit elements all by the same factor. (a) Which ones should you change? (b) By what factor should you change them?

31\. A8. At a frequency \(\omega_{1}\) the reactance of a certain capacitor equals that of a certain inductor. (a) If the frequency is changed to \(\omega_{2}=2 \omega_{1},\) what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger? (b) If the frequency is changed to \(\omega_{3}=\omega_{1} / 3,\) what is the ratio of the reactance of the inductor to that of the capacitor? Which reactance is larger? (c) If the capacitor and inductor are placed in series with a resistor of resistance \(R\) to form a series \(L-R-C\) circuit? what will be the resonance angular frequency of the circuit?
See all solutions

Recommended explanations on Physics Textbooks

Astrophysics

Read Explanation

Force

Read Explanation

Magnetism

Read Explanation

Fluids

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Fields in Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.