/*! 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 62 A solenoid contains 750 coils of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 20: Problem 62

A solenoid contains 750 coils of very thin wire evenly wrapped over a length of \(15.0 \mathrm{~cm} .\) Each coil is \(0.800 \mathrm{~cm}\) in diameter. If this solenoid carries a current of \(7.00 \mathrm{~A},\) what is the magnetic field at its center?

Short Answer

Expert verified
The magnetic field at the center of the solenoid is approximately 0.04396 T.

Step by step solution

01

Identify the Formula for Magnetic Field of a Solenoid

The magnetic field at the center of a solenoid can be calculated using the formula: \[ B = \mu_0 \cdot n \cdot I \] Where: - \( B \) is the magnetic field, - \( \mu_0 = 4\pi \times 10^{-7} \text{ T m/A} \) is the permeability of free space, - \( n \) is the number of turns per unit length, - \( I \) is the current through the solenoid.
02

Calculate the Turns per Unit Length

The solenoid has 750 turns evenly distributed over a length of 0.15 meters (since 15.0 cm = 0.15 m). The turns per unit length \( n \) can be calculated as: \[ n = \frac{750 \text{ turns}}{0.15 \text{ m}} = 5000 \text{ turns/m} \]
03

Substitute Values into the Formula

Now substitute the given values into the magnetic field equation: \[ B = \mu_0 \cdot n \cdot I = (4\pi \times 10^{-7} \text{ T m/A}) \cdot (5000 \text{ turns/m}) \cdot (7.00 \text{ A}) \]
04

Calculate the Magnetic Field

Calculate the expression: \[ B = (4\pi \times 10^{-7} \cdot 5000 \cdot 7.00) \text{ T} \] Simplifying gives: \[ B = 4\pi \times 10^{-7} \cdot 35000 \text{ T} \]\[ B = 4\pi \times 3.5 \times 10^{-3} \text{ T} \]\[ B \approx 4.396 \times 10^{-2} \text{ T} \]

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.

Understanding the Solenoid
A solenoid is a type of electromagnetic coil consisting of a wire wound into a helix. When an electric current flows through it, a magnetic field is generated. This is due to the alignment of magnetic dipoles caused by the electric current. The solenoid's primary characteristic is its ability to produce a uniform magnetic field inside its core, assuming the solenoid is long relative to its diameter and has closely wound coils.
Solenoids are widely used in numerous applications such as electromagnets, inductors, and valves. Their ability to convert electrical energy into a controlled magnetic field makes them essential components in various devices.
Magnetic Field Calculation
The strength of the magnetic field inside a solenoid is determined using the formula: \[ B = \mu_0 \cdot n \cdot I \]where:
  • \(B\) is the magnetic field strength in Tesla (T).
  • \(\mu_0\) is the permeability of free space.
  • \(n\) is the number of turns per unit length.
  • \(I\) is the current through the solenoid.
This formula highlights how the magnetic field depends on three key factors: the material through which the field is formed (\(\mu_0\)), the density of the coil turns (\(n\)), and the current passing through the solenoid (\(I\)). Understanding this relationship allows us to design solenoids with specific magnetic field strengths for particular applications.
Permeability of Free Space (\(\mu_0\))
The permeability of free space is a fundamental physical constant denoted as \(\mu_0\). Its value is approximately \(4\pi \times 10^{-7}\) T m/A. This constant is crucial in calculating the magnetic effects in a vacuum or air where the solenoid is usually placed.
The concept of permeability refers to the ability of a material to support the formation of a magnetic field within itself. For a vacuum (or air), this is a baseline measurement ensuring that magnetic field calculations can be performed uniformly across different solenoids and settings. It essentially acts as a scaling factor in determining the magnetic field's intensity generated by a given current.
Turns Per Unit Length (\(n\))
The term "turns per unit length" (\(n\)) refers to the number of wire loops, or turns, in the solenoid per meter of its length. In mathematical terms, it is calculated as:\[ n = \frac{N}{L} \]where:
  • \(N\) is the total number of turns.
  • \(L\) is the length of the solenoid (in meters).
The turns per unit length concept is vital as it directly affects the strength of the magnetic field that the solenoid can generate. More turns per meter translate to a stronger magnetic field, assuming the same current flow, due to increased density of the magnetic lines of force within the solenoid.

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 proton traveling at \(3.60 \mathrm{~km} / \mathrm{s}\)suddenly enters a uniform magnetic field of \(0.750 \mathrm{~T},\) traveling at an angle of \(55.0^{\circ}\) with the field lines (Figure 20.55 ).(a) Find the magnitude and direction of the force this magnetic field exerts on the proton.(b) If you can vary the direction of the proton's velocity, find the magnitude of the maximum and minimum forces you could achieve, and show how the velocity should be oriented to achieve these forces. (c) What would the answers to part (a) be if the proton were replaced by an electron traveling in the same way as the proton?

A particle with a charge of \(-2.50 \times 10^{-8} \mathrm{C}\) is moving with an instantaneous velocity of magnitude \(40.0 \mathrm{~km} / \mathrm{s}\) in the \(x\) - \(y\) plane at an angle of \(50^{\circ}\) counterclockwise from the \(+x\) axis. What are the magnitude and direction of the force exerted on this particle by a magnetic field with magnitude \(2.00 \mathrm{~T}\) in the (a) \(-x\) direction and (b) \(+z\) direction?

A long, straight wire carries a current \(I_{0}\) and produces a magnetic field \(B_{0}\) at a distance \(r_{0}\) from the wire's center. In terms of \(B_{0}\), determine the magnetic-field strength (a) at a distance \(r_{0} / 2\) from the wire's center and (b) at a distance \(3 r_{0}\) after the current is increased to \(6 I_{0}\).

A singly charged ion of \({ }^{7} \mathrm{Li}\) (an isotope of lithium containing three protons and four neutrons) has a mass of \(1.16 \times 10^{-26} \mathrm{~kg} .\) It is accelerated through a potential difference of \(220 \mathrm{~V}\) and then enters a 0.723 T magnetic field perpendicular to the ion's path. What is the radius of the path of this ion in the magnetic field?

Home circuit breakers typically have current capacities of around 10 A. How large a magnetic field would such a current produce \(5.0 \mathrm{~cm}\) from a long wire's center? How does this field compare with the strength of the earth's magnetic field?
See all solutions

Recommended explanations on Physics Textbooks

Electricity

Read Explanation

Electricity and Magnetism

Read Explanation

Fundamentals of Physics

Read Explanation

Atoms and Radioactivity

Read Explanation

Scientific Method Physics

Read Explanation

Solid State 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.