/*! 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 50 A circular wire coil consists of... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 19: Problem 50

A circular wire coil consists of 100 turns and is wound tightly around a very long iron cylinder with a radius of \(2.5 \mathrm{~cm}\) and a relative permeability of \(2200 .\) The loop has a current of \(7.5 \mathrm{~A}\) in it. Determine the magnetic field strength produced by the coil (a) at the center of the coil and (b) at a location on the central axis of the iron cylinder \(5.0 \mathrm{~cm}\) above the center of the circular coil.

Short Answer

Expert verified
At the coil center, the magnetic field is roughly 0.207 T; calculations outside are complex.

Step by step solution

01

Understand the Physics

We're dealing with a solenoid-like system, where the magnetic field inside can be calculated. The magnetic field inside a solenoid is uniform and directed along its axis.
02

Using Ampere's Law for a Solenoid

For a solenoid, the magnetic field strength \( B \) can be given by \( B = \mu n I \), where \( \mu \) is the permeability of the medium (\( \mu = \mu_0 \mu_r \)), \( n \) is the number of turns per unit length (\( n = \frac{N}{L} \)), and \( I \) is the current.
03

Calculate Permeability and Length

First, calculate \( \mu \). The permeability of free space \( \mu_0 = 4\pi \times 10^{-7} \, \text{T}\cdot\text{m/A} \). Relative permeability \( \mu_r = 2200 \), so \( \mu = \mu_0 \mu_r = 4\pi \times 10^{-7} \times 2200 \, \text{T}\cdot\text{m/A} \). Assume the solenoid is infinitely long, so \( L \rightarrow \infty \).
04

Number of Turns per Unit Length

Since it's tightly wound, \( n = \frac{N}{L} \). However, for the calculation, \( n \) remains simply the number of turns \( n = 100 \) in this context of infinite length.
05

Calculate Magnetic Field at the Center

The magnetic field at the center is \( B = \mu n I = (4\pi \times 10^{-7} \times 2200) \times 100 \times 7.5 = 2.07 \times 10^{-1} \, \text{T} \). This is the field strength inside a supposed infinitely long solenoid, dominated by the permeability of the iron.
06

Magnetic Field Above the Center

The field reduces when away from the center. However, over a long iron core, this diminishes far slower; calculations aren't trivial analytically at offsets without further assumptions (current simplifications ignore boundary and end corrections). But as it trends similar to the core measure, the field remains around but often drops off markedly in practical terms unless approximated for finite lengths.

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.

circular wire coil
A circular wire coil is a loop or series of loops made by winding a wire in a circular shape. It forms the core of many electromagnetism applications. The magnetic field produced by a circular wire coil is determined by the flow of electric current through it and the number of loops it holds.
  • Theoretical beauty lies in its simplicity, as it can create strong magnetic fields with fewer turns compared to other shapes.
  • Special configurations can be used to concentrate the magnetic field in a directed path, making it useful for instruments like transformers and inductors.
In our scenario, the coil has 100 turns, tightly wound to form a solenoid, enabling it to produce a significantly strong magnetic field due to its configuration and the inherent ampere-turn factor.
Ampere's Law
Ampere's Law is a fundamental principle in electromagnetism used to calculate the magnetic field generated by an electric current. According to this law, the integrated magnetic field \( B \) around a closed loop is proportional to the electric current \( I \) passing through the loop. Mathematically, it's expressed as \( \oint B \cdot dl = \mu I \).
This law is particularly useful for calculating magnetic fields in symmetrical situations like solenoids and toroids.
  • For solenoids, it simplifies to \( B = \mu n I \) where \( n \) is the turn density.
  • It's instrumental in deriving the uniform magnetic field inside a solenoid, making it easier to predict and utilize in practical setups.
Through the event of choosing the right length \( L \) and turns \( N \), one can calculate the field based on such known numbers, adhering to this electrodynamic principle.
relative permeability
Relative permeability is an essential factor that determines how much more effective a medium is in conducting magnetic fields compared to a vacuum. It's symbolized by \( \mu_r \) and is dimensionless.
In the formula \( \mu = \mu_0 \mu_r \), the relative permeability \( \mu_r = 2200 \) suggests that the iron cylinder can enhance the magnetic field substantially more than a vacuum.
  • The higher the relative permeability, the better the material is at concentrating magnetic lines of force within itself.
  • This concept allows us to understand the difference between different materials in electromagnetic applications, transforming weaker setups into powerful electromagnets.
In applications such as our solenoid-based coil, the choice of a high permeability core augments the field produced, ensuring stronger and more effective magnetic flux through the central axis.
solenoid
A solenoid is a long coil of wire, usually cylindrical, that generates a magnetic field when an electric current passes through it. Its field inside is strong and fairly uniform if it is long and closely wound, making it one of the fundamental constructs in electromagnetism.
By understanding its properties:
  • We use solenoids for generating controlled magnetic fields in a linear direction, often serving in motors, inductors, or transformers.
  • The length and coil density directly affect the field strength, governed by \( B = \mu n I \). If extremely long, it behaves as an infinite solenoid, simplifying field calculations as variation outside the axis becomes negligible.
In our example problem, the long solenoid setup creates a concentrated magnetic field tailored by the current \( I \) through the coil and enhanced by the iron's permeability. It models the scenarios where knowing precise magnetic influences is crucial for scientific and practical engineering applications.

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

Two long, straight, parallel wires carry the same current in the same direction. (a) Use both the right-hand source and force rules to determine whether the forces on the wires are (1) attractive or (2) repulsive. (b) If the wires are \(24 \mathrm{~cm}\) apart and experience a force per unit length of \(24 \mathrm{mN} / \mathrm{m}\), determine the current in each wire. (c) What is the magnetic field strength midway between the two wires?

A circular loop of wire with a radius of \(5.0 \mathrm{~cm}\) carries a current of 1.0 A. Another circular loop of wire is concentric with (that is, has a common center with) the first and has a radius of \(10 \mathrm{~cm}\). The magnetic field at the center of the loops is double what the field would be from the first one alone, but oppositely directed. What is the current in the second loop?

1 lies on the \(x\) -axis and its north end is at \(x=+1.0 \mathrm{~cm},\) while its s鈥� # Two identical bar magnets of negligible width are located in the \(x\) -y plane. Magnet #1 lies on the \(x\) -axis and its north end is at \(x=+1.0 \mathrm{~cm},\) while its south end is at \(x=+5.0 \mathrm{~cm}\). Magnet #2 lies on the \(y\) -axis and its north end is at \(y=+1.0 \mathrm{~cm},\) while its south end is at \(y=+5.0 \mathrm{~cm} .\) (a) In what direction would a compass point if it were located at the origin? (b) Repeat part (a) for the situation where magnet #1 is reversed in polarity. [Hint: Make a sketch of the two magnets and their individual fields at the origin.

A coil of four circular loops of radius \(5.0 \mathrm{~cm}\) carries a current of 2.0 A clockwise, as viewed from above the coil's plane. What is the magnetic field at the center of the coil?

An ionized deuteron (a bound proton-neutron system with a net \(+e\) charge) passes through a velocity selector whose perpendicular magnetic and electric fields have magnitudes of \(40 \mathrm{mT}\) and \(8.0 \mathrm{kV} / \mathrm{m}\), respectively. Find the speed of the ion.
See all solutions

Recommended explanations on Physics Textbooks

Radiation

Read Explanation

Linear Momentum

Read Explanation

Solid State Physics

Read Explanation

Nuclear Physics

Read Explanation

Kinematics Physics

Read Explanation

Rotational 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.