Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 21: Problem 25

What is the reactance of an air core solenoid of length \(8.0 \mathrm{cm},\) radius \(1.0 \mathrm{cm},\) and 240 turns at a frequency of $15.0 \mathrm{kHz} ?$

Short Answer

Expert verified
Short answer: To find the reactance of the air core solenoid, first calculate the cross-sectional area, \(A = \pi r^2\) where \(r = 1.0\ \mathrm{cm}\). Then, find the inductance using the formula \(L = \frac{\mu_0 N^2 A}{l}\) with given values \(\mu_0 = 4\pi \times 10^{-7}\ \mathrm{T\cdot m/A}, N = 240, A = \pi\ \mathrm{cm^2},\) and \(l = 8.0\ \mathrm{cm}\). Finally, use the formula \(X = 2 \pi f L\) with the given frequency \(f = 15.0 \mathrm{kHz}\) to find the reactance, \(X\ \Omega\).

Step by step solution

01

Calculate the solenoid's inductance

To find the inductance of an air core solenoid, we can use the formula \(L = \frac{\mu_0 N^2 A}{l}\), where \(L\) is the inductance, \(\mu_0\) is the permeability of free space \((4\pi \times 10^{-7} \mathrm{T\cdot m/A})\), \(N\) is the number of turns, \(A\) is the cross-sectional area, and \(l\) is the solenoid's length. First, we need to calculate the cross-sectional area using the given radius: \(A = \pi r^2.\)
02

Calculate the cross-sectional area

Find the cross-sectional area by substituting the given radius \((r = 1.0\ \mathrm{cm})\) into the formula: \(A = \pi r^2 = \pi (1.0 \mathrm{cm})^2 = \pi \mathrm{cm^2}.\)
03

Calculate the inductance

Now, use the inductance formula to find the value of inductance, with \(\mu_0 = 4\pi \times 10^{-7}\ \mathrm{T\cdot m/A}\), \(N = 240\), \(A = \pi\ \mathrm{cm^2} = 0.0001\ \mathrm{m^2}\), and \(l = 8.0\ \mathrm{cm} = 0.08\ \mathrm{m}\): \( L = \frac{\mu_0 N^2 A}{l} = \frac{(4\pi \times 10^{-7}\ \mathrm{T\cdot m/A})(240)^2 (0.0001\ \mathrm{m^2})}{0.08\ \mathrm{m}}.\) Calculate the inductance, \(L.\)
04

Find the reactance

Using the inductance value \(L\) and the given frequency \((f = 15.0 \mathrm{kHz} = 15,000\ \mathrm{Hz})\), find the reactance \((X)\) of the solenoid using the formula: \(X = 2 \pi f L.\) Calculate the reactance, \(X.\)
05

Present the result

The reactance of the air core solenoid at a frequency of \(15.0\ \mathrm{kHz}\) is \(X\ \Omega.\) (Replace \(X\) with the value calculated in Step 4.)

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

A 4.00 -mH inductor is connected to an ac voltage source of \(151.0 \mathrm{V}\) rms. If the rms current in the circuit is \(0.820 \mathrm{A},\) what is the frequency of the source?
In an RLC series circuit, these three elements are connected in scries: a resistor of \(60.0 \Omega,\) a 40.0 -mH inductor, and a \(0.0500-\mathrm{F}\) capacitor. The series elements are connected across the terminals of an ac oscillator with an rms voltage of \(10.0 \mathrm{V}\). Find the resonant frequency for the circuit.
Show, from \(X_{\mathrm{C}}=1 /(\omega C),\) that the units of capacitive reactance are ohms.
A computer draws an rms current of \(2.80 \mathrm{A}\) at an \(\mathrm{rms}\) voltage of \(120 \mathrm{V}\). The average power consumption is 240 W. (a) What is the power factor? (b) What is the phase difference between the voltage and current?
A European outlet supplies \(220 \mathrm{V}\) (rms) at \(50 \mathrm{Hz}\). How many times per second is the magnitude of the voltage equal to $220 \mathrm{V} ?$
See all solutions

Recommended explanations on Physics Textbooks

Space Physics

Read Explanation

Fluids

Read Explanation

Measurements

Read Explanation

Electromagnetism

Read Explanation

Electricity and Magnetism

Read Explanation

Dynamics

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.

Sign-up for free