/*! 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 42 Calculate the reactance of, and ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 30: Problem 42

Calculate the reactance of, and rms current in, a \(36.0-\mathrm{mH}\) radio coil connected to a \(250-\mathrm{V}(\mathrm{rms}) 33.3-\mathrm{kHz}\) ac line. Ignore resistance.

Short Answer

Expert verified
Inductive reactance is approx. 7.54 Ohms; RMS current is approx. 33.16 A.

Step by step solution

01

Understand the Problem

We need to calculate the inductive reactance of an inductor and the RMS current flowing through it. We are given the inductance of the coil as 36.0 mH, the RMS voltage as 250 V, and the frequency of the AC supply as 33.3 kHz.
02

Calculate Inductive Reactance

Inductive reactance can be calculated using the formula \[ X_L = 2\pi f L \]where \( f \) is the frequency of the AC supply and \( L \) is the inductance. Substituting the values,\[ X_L = 2\pi \times 33,300 \text{ Hz} \times 36.0 \times 10^{-3} \text{ H} \]\[ X_L = 2\pi \times 33,300 \times 0.036 \]\[ X_L \approx 7.54 \text{ Ohms} \]
03

Calculate RMS Current

The RMS current, \( I_{ ext{rms}} \), can be found using Ohm鈥檚 Law adapted for AC circuits:\[ I_{ ext{rms}} = \frac{V_{ ext{rms}}}{X_L} \]Substituting the given values,\[ I_{ ext{rms}} = \frac{250 \text{ V}}{7.54 \text{ ohms}} \]\[ I_{ ext{rms}} \approx 33.16 \text{ A} \]

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.

RMS Current
Root Mean Square (RMS) Current is a way to express the average current value in AC circuits. Unlike DC circuits where current value is steady, AC current varies with time, routinely crossing zero. The RMS current value gives us a constant value that represents the AC current's equivalent power handling as a DC current would. To calculate RMS Current, the formula used is:
  • For AC, it's expressed as
    \[ I_{\text{rms}} = \frac{V_{\text{rms}}}{X_L} \]
  • Where \( V_{\text{rms}} \) represents the RMS voltage.
  • \( X_L \) is the inductive reactance, which acts as resistance in an AC circuit.
In our sample problem, the RMS current was found to be approximately 33.16 Amperes. This demonstrates how the current, in this specific AC setup, would behave in terms of power, as if it were in a DC setting.
AC Frequency
Alternating Current (AC) Frequency is a measure of how many cycles a waveform completes in one second. It's expressed in hertz (Hz). This frequency is a critical concept in AC circuits, influencing how capacitors and inductors like our radio coil behave. For the given problem, we have an AC supply with a frequency of 33.3 kHz. That's 33,300 cycles per second! The frequency impacts the inductive reactance significantly, as seen in the formula \( X_L = 2\pi f L \).
  • The higher the frequency, the greater the inductive reactance, \( X_L \).
  • This implies current will be less for a given voltage when frequency is higher.
Thus, frequency directly affects the circuit's impedance and overall performance.
Ohm's Law
Ohm鈥檚 Law is a fundamental principle used to relate voltage, current, and resistance in electrical circuits. In its traditional form for DC circuits, it is expressed as:
  • \( V = IR \)
  • Where \( V \) is voltage, \( I \) is current, and \( R \) is resistance.
However, with AC circuits containing inductors, we adapt Ohm's Law to account for reactance (\( X_L \)) rather than resistance:
  • In this context: \( V_{\text{rms}} = I_{\text{rms}} \times X_L \)
  • Resistance (\( R \)) from Ohm鈥檚 Law becomes inductive reactance (\( X_L \)).
This adaptation demonstrates how current and voltage are influenced by inductive reactance in AC circuits, allowing us to solve for the RMS current with the help of known voltage and reactance.
Inductance
Inductance is a property of an electrical component, such as a coil or inductor, that describes its ability to store energy in a magnetic field. Measured in henrys (H), inductance determines how much the coil will oppose changes in current flow.For the exercise, the coil's inductance is 36.0 mH (millihenrys). This inductance is used in the formula for inductive reactance:
  • \( X_L = 2\pi f L \)
  • Where \( f \) is frequency, \( L \) is the coil's inductance.
A higher inductance means the coil offers more reactance to AC at a given frequency, impacting how the current reacts within the circuit. Understanding inductance helps in designing circuits, especially in ensuring that components operate effectively at the desired frequencies and under various loads.

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

At \(t=0\), let \(Q=Q_{0}\), and \(I=0\) in an \(L C\) circuit. (a) At the first moment when the energy is shared equally by the inductor and the capacitor, what is the charge on the capacitor? (b) How much time has elapsed (in terms of the period \(T\) )?

(II) Capacitors made from piezoelectric materials are commonly used as sound transducers ("speakers"). They often require a large operating voltage. One method for providing the required voltage is to include the speaker as part of an \(L R C\) circuit as shown in Fig. \(29 ,\) where the speaker is modeled electrically as the capacitance \(C = 1.0 \mathrm { nF } .\) Take \(R = 35 \Omega\) and \(L = 55 \mathrm { mH } .\) (a) What is the resonant frequency \(f _ { 0 }\) for this circuit? (b) If the voltage source has peak amplitude \(V _ { 0 } = 2.0 \mathrm { V }\) at frequency \(f = f _ { 0 } ,\) find the peak voltage \(V _ { c 0 }\) across the speaker (i.e., the capacitor \(C\) ). (c) Determine the ratio \(V _ { C 0 } / V _ { 0 . }\)

(II) ( \(a\) ) Determine a formula for the average power \(\bar{P}\) dissipated in an \(L R C\) circuit in terms of \(L, R, C, \omega,\) and \(V_{0}\) (b) At what frequency is the power a maximum? (c) Find an approximate formula for the width of the resonance peak in average power, \(\Delta \omega\), which is the difference in the two (angular) frequencies where \(\bar{P}\) has half its maximum value. Assume a sharp peak.

A voltage \(V=0.95 \sin 754 t\) is applied to an \(L R C\) circuit ( \(I\) is in amperes, \(t\) is in seconds, \(V\) is in volts, and the "angle" is in radians) which has \(L=22.0 \mathrm{mH}, R=23.2 \mathrm{k} \Omega\), and \(C=0.42 \mu \mathrm{F}\). (a) What is the impedance and phase angle? (b) How much power is dissipated in the circuit? (c) What is the rms current and voltage across each element?

A 2200 -pF capacitor is charged to 120\(\mathrm { V }\) and then quickly connected to an inductor. The frequency of oscillation is observed to be 17\(\mathrm { kHz }\) . Determine \(( a )\) the inductance, (b) the peak value of the current, and (c) the maximum energy stored in the magnetic field of the inductor.
See all solutions

Recommended explanations on Physics Textbooks

Modern Physics

Read Explanation

Electric Charge Field and Potential

Read Explanation

Turning Points in Physics

Read Explanation

Energy Physics

Read Explanation

Quantum Physics

Read Explanation

Further Mechanics and Thermal 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.