Lesson: Solenoids

Contributed by: VU Bioengineering RET Program, School of Engineering, Vanderbilt University

A photograph shows a blue coil representing a solenoid.
This current-carrying coil produces a magnetic field that can be calculated.
copyright
Copyright © 2004 Microsoft Corporation, One Microsoft Way, Redmond, WA 98052-6399 USA. All rights reserved.

Summary

In this lesson about solenoids, students learn how to calculate the magnetic field along the axis of a solenoid and then complete an activity exploring the magnetic field of a metal slinky. Solenoids form the basis for the magnets of MRIs. Exploring the properties of this solenoid helps students understand the MRI machine.

Engineering Connection

Engineers use giant solenoids to create the magnetic fields of MRIs. Just as engineers must understand the properties of solenoids in order to design and build MRI machines, students must understand these properties in order to determine the safety hazards of MRI machines.

Learning Objectives

After the lesson, students should be able to:

  • Calculate the magnetic field of a solenoid from its current, radius, and number of coils.
  • Explain how a solenoid creates a magnetic field.

More Curriculum Like This

Changing Fields

Students induce EMF in a coil of wire using magnetic fields. Students review the cross product with respect to magnetic force and introduce magnetic flux, Faraday's law of Induction, Lenz's law, eddy currents, motional EMF and Induced EMF.

High School Lesson
Ampere's Law

A class demo introduces students to the force between two current carrying loops, comparing the attraction and repulsion between the loops to that between two magnets. After a lecture on Ampere's law (including some sample cases and problems), students begin to use the concepts to calculate the magn...

High School Lesson
Magnetic Fields Matter

Students are introduced to the effects of magnetic fields in matter addressing permanent magnets, diamagnetism, paramagnetism, ferromagnetism and magnetization.

High School Lesson
Biot-Savart Law

Beginning with a class demo, students are prompted to consider how current generates a magnetic field, and the direction of the field that is generated. Via a lecture, students learn Biot-Savart's law (and work some sample problems) in order to calculate, most simply, the magnetic field produced in ...

High School Lesson

Educational Standards

Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN), a project of D2L (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; e.g., by state; within source by type; e.g., science or mathematics; within type by subtype, then by grade, etc.

  • Energy can be grouped into major forms: thermal, radiant, electrical, mechanical, chemical, nuclear, and others. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?
  • Technological innovation often results when ideas, knowledge, or skills are shared within a technology, among technologies, or across other fields. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?
  • Results of scientific inquiry--new knowledge and methods--emerge from different types of investigations and public communication among scientists. In communicating and defending the results of scientific inquiry, arguments must be logical and demonstrate connections between natural phenomena, investigations, and the historical body of scientific knowledge. In addition, the methods and procedures that scientists used to obtain evidence must be clearly reported to enhance opportunities for further investigation. (Grades 9 - 12) Details... View more aligned curriculum... Do you agree with this alignment?
Suggest an alignment not listed above

Introduction/Motivation

In order to solve the challenge problem, we need to understand the magnetic field of an MRI. The MRI magnet uses a large solenoid to create its magnetic field, so learning the properties of solenoids will help us determine where the magnetic field is present and/or largest.

Engineers study solenoids in order to understand how the "giant solenoid" is going to create a large enough magnetic field for the MRI machine. Engineers continue to test and experiment with more loops and more powerful currents to create larger magnetic fields. Safety engineers make sure these larger magnetic fields do not result in any health or mechanical hazards.

Lesson Background and Concepts for Teachers

Legacy Cycle Information

This lesson fits into the research and revise phase of the legacy cycle during which students are provided with additional information enabling them to revise their initial ideas for solving the challenge. The research aspect consists of a lecture that explains how to calculate the magnetic field along the axis of a solenoid and an associated activity exploring the magnetic field of a metal slinky.

The Solenoid

Solenoids in general are also useful to engineers because their magnetic fields can be easily changed by adjusting the current or number of loops. Electromechanical solenoids can be used as switches in pneumatic or hydraulic valves in different engineering projects. After this lesson, during the associated activity, student groups use slinkies (the toy) to discover more about solenoids.

The magnetic field produced by a loop of current is relatively small, due to the size of the constant

μ0
To produce a magnetic field of a greater strength, one may use a relatively large amount of current, or several loops of current. In fact, one can use a single current and take the same wire coiled around several times to produce the same magnetic field that would be produced using several separate loops of wire. If these loops are close enough to be approximated as lying in the same plane, then the magnitude of the magnetic field generated in the center of the loops would be:

Equation B =
where N is the number of loops in the coil.

If the wire is coiled several hundred times, many loops may be stacked against each other so that the multiple loops of wire begin to take on the shape of a cylinder. This arrangement is called a solenoid.

A graph with a looped line that looks like a solenoid that is horizontal to the page, centered on the coordinate plane such that the x axis runs through the center of the solenoid.
Figure A
copyright
Copyright © 2006 Eric Appelt, Vanderbilt University

Consider the solenoid shown in Figure A. By the right hand rule, the magnetic field along the x-axis points directly to the right, assuming that the solenoid is tightly wound. To determine the magnitude of the magnetic field at a given point x along the axis, we need to separate the solenoid into clusters of loops with an infinitesimally small length dx. Using the result of the previous homework, if the small segment of loops resided at a point a along the x-axis, the field generated by this group of coils would be:

equation dB =

Where

N segment
is the number of coils in the small segment of length dx. This amount is usually given in terms of n, the number of turns per unit length of the solenoid. That is
N sesment = n dx
.

By integration, the total field is:

equation B =

Example:

Graph the field along the axis of a tightly would solenoid centered at the origin with a current of 5 A, 100 turns per unit length, a length of 1 meter, and a radius of 20 cm. Use Gauss instead of Tesla for units. (Note: Tightly wound may be taken to mean that n is much greater than I, and we can ignore the current flowing from left to right.)

Solution:

Using these values in the above equation:

equation B (x) =

This produces the graph shown in Figure B. The magnetic field produced is close to constant when the position along the x-axis is within the middle half of the length of the solenoid.

A graph of the magnetic field along the axis of a solenoid. The magnetic field versus position is a normal curve with the maximum at 0,5.8.
Figure B
copyright
Copyright © 2006 Eric Appelt, Vanderbilt University

If a solenoid is much longer than its radius, and we look at a point near the center of the solenoid, the field can be approximated as:

equation B =

This approximation can be used to a high degree of accuracy when a solenoid has a length at least 10 times its radius and when the point along the axis of the solenoid is closer to the center than it is to the edge.

An Equivalent for Gauss Law?

Unlike electricity, simple examples do not exist to enable us to easily compute a field. For an electric field, a single point of charge enables one to quickly calculate the field using Coulomb's law, but in magnetism, a steady current necessarily involves a continuous curve of current segments, so even the simplest examples require integration. For electrostatic problems involving a high degree of symmetry, the Gauss law enables one to compute the field on a carefully selected surface without integrating. Gauss law states:

Gauss law equation

that is the total electric flux, or number of field lines coming out of a closed surface, is equal to the enclosed charge divided by the permittivity of free space. If you recall, electric field lines emerge from positive charges and point into negative charges. However, magnetic field lines never begin or end, but rather form closed loops around segments of steady current. In other words, there is no magnetic charge, and since magnetic field lines never emerge from a point or halt at a point, the total magnetic flux through any closed surface must be zero, that is:

equation

However, this law is not really useful in calculating the magnitude of the magnetic field. It is sometimes called Gauss' law for magnetism or the "no-name law."

Vocabulary/Definitions

solenoid: A current-carrying coil of wire that produces a magnetic field.

Associated Activities

  • Slinkies as Solenoids - Students use metal slinkies as solenoids, and test the magnetic field created by sending currents through the slinkies.

Assessment

Embedded Assessment

Assign students the assessment provided in the associated activity, Slinkies as Solenoids. In this activity, students create a lab report in which they design experimental procedures to answer a number of questions that require students to apply what they have learned about solenoids to creating a safe MRI machine.

Contributors

Eric Appelt

Copyright

© 2013 by Regents of the University of Colorado; original © 2006 Vanderbilt University

Supporting Program

VU Bioengineering RET Program, School of Engineering, Vanderbilt University

Acknowledgements

The contents of this digital library curriculum were developed under National Science Foundation RET grant nos. 0338092 and 0742871. However, these contents do not necessarily represent the policies of the NSF, and you should not assume endorsement by the federal government.

Last modified: September 7, 2017

Comments